diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlmck" "b/data_all_eng_slimpj/shuffled/split2/finalzzlmck" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlmck" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and observations}\n\nThis paper is a sequel to the paper by Hutchings, Maddox, Cutri, and\nNelson (2003: paper 1), which presented optical imaging of QSOs of redshift\n0.3 and lower, selected from the Two Micron All Sky Survey (2MASS). HST\nimaging of a different sample of 2MASS QSOs is given by Marble et al (2003).\nIn this paper we present optical imaging of a sample of 2MASS QSOs at \nredshifts above 0.3. The sample of QSOs is the same as in paper 1, but \nwith the higher redshift bin.\n\nThe QSOs are identified by spectroscopy after selection by colours\nJ-K larger than 2.0, and are classified as types 1, 2 or intermediate,\nbased on the ratio of narrow to broad line emission. These do not include\nany previously known AGN. 2MASS QSO selection is described in Cutri et al.\n(2002). The present study includes objects between\n-30$^o$ and 60$^o$ declination, and thus suitable for observation with the \nCFHT. Paper I included data on 76 of 243 eligible objects, and this paper\nincludes data on 21 of 64 eligible\nobjects. All but 2 of the eligible objects have redshifts below 0.7.\n\n\n\nThe observations were taken as service-observing `snapshots' with\nthe Megaprime camera and r-band filter, with exposures of 340 secs. The pixel\nsampling is 0.187 arcsec. (The paper 1 data were taken with R band, and 0.206\narcsec sampling, and 200 sec exposures, with the CFHT 12K camera. The CFHT\nwebsite gives details of the different filters.) All\nobjects had two independent images, 5 had four images, and two had 8 images.\nThe data were obtained over the period April through June 2004, in seeing\nconditions that varied from 0.6 to 1.3 arcsec FWHM. Table 1 lists the\nobjects observed, and various of their properties, from the 2MASS database\nand also measures from the present work.\n\nAs the selection of observed objects was dictated mainly by scheduling,\nwe show in Figure 1 the distribution of observed objects within the whole\nsample of 64 high redshift objects, and also include the lower redshift\nsample from paper 1.\nThe observed subsample is slightly skewed to higher luminosity, but there\nis K-band luminosity overlap of about 15 new objects with 21 of the highest\nluminosity objects from the lower redshift sample. In terms of J-K colour,\nthe observed subset is slightly\nskewed to redder objects than the average by about 0.5 magnitudes.\nThe spectral classifications among the new sample\nare representative of the 2MASS QSOs in this redshift range.\n\n\\section{Data processing and measurements}\n\nAfter removal of the instrumental signature, subimages were generated of\neach QSO and 3 PSF stars from the same CCD and observation, as near as\npossible to the\nQSO. The two (or more) images of each field were combined, to help remove\ncosmic rays and to increase the signal. In a couple of cases, some of\nthe images had very different FWHM as they were taken on different nights,\nand such images were not combined.\n\nPSF stars were chosen to have more signal than the QSOs, and to be\nfree of nearby companions and image flaws. Any other stars near the PSF\nstars were edited out of the images before generating the profiles, to\nget the most representative PSF profiles. The images and luminosity\nprofiles were compared among the PSF stars, and generally the cleanest\none was picked as the PSF to be used. Occasionally\ntwo PSF stars were co-added, to increase the signal, after checking that\nthe profiles agreed.\n\nMeasurements were made of every individual image, as well as the\ncombined images, to give independent values of the fluxes and PSF-removal.\nThe measurements made were the flux of the QSO and the PSF stars, and the\nimage FWHM from profiles. Profiles were generated by the IRAF task ellipse,\nafter careful sky removal. Companions near the QSOs were included in\ntheir profiles,\nalthough in some cases, well separated companions were edited out to better\nstudy the extended flux centered on the QSO. Figure 2 shows profiles\nfrom one object as an example.\n\nThe host galaxy fluxes were estimated in two ways. First, the flux difference\nwas measured between the QSO and PSF profiles, after scaling the PSF\nprofile to the same peak value as the QSO. This yields a measure\nof the resolved flux which is a lower limit to the QSO host flux. Second,\ndifferently scaled PSF images were subtracted from the QSO to yield a\nresulting\nprofile that increases monotonically to the nucleus. The value for\nthe profile that just turns over near the nucleus yields an upper\nlimit for the host galaxy flux. The best value was taken to be the\nintermediate case where the difference flux follows a smooth profile for\nits inner part. While this was subjective, and depends in principle\non whether the difference follows an exponential or de Vaucouleurs\nprofile, or combination, the QSO nuclear flux was generally small\nenough that the spread in these difference fluxes was small. Error\nbars were taken for each host galaxy flux as the upper and lower\nvalues described above.\n\nThe same procedure was repeated for the co-added images for each QSO.\nTable 1 shows the final adopted values for the ratio of nuclear to\nresolved flux for each QSO. The agreement between the individual and\nthe combined images was in all cases within 20\\%, and the ranges as\ndescribed above have similar values. Thus, in Table 1 we show the\nmean values (average discarding outliers) for the nuclear to flux ratio, \nand we assign an\noverall uncertainty of 30\\% in those values. Note that the resolved flux\ngiven does not include any separate close companions within the\nradius of 10 arcsec: where these are present they were not included.\nThe resolved flux does include any features that appear to be part of\nor connected to of the host galaxy, even though they introduce irregularities \ninto the azimuthally averaged profile. Many of the profiles are\nirregular, which means that fit to either standard model is\nsomewhat arbitrary. We have noted where the profiles do have some\nregion of good fit to one or other model. There are ten with some\nspheroidal component and four with a good exponential. Four others\nhave so many companions that no fit is good, and another three are clean\nbut do not have clean regions of good fit. In several cases, the host\ngalaxy appears to have both bulge and disk components to the azimuthally\naveraged profile. \n\nThe morphology of the QSO images, both raw and PSF-subtracted, was\ninspected to make an estimate of the degree of tidal disturbance.\nAs in paper 1, this value takes into account the presence of connecting\nbridges to companions, single asymmetric arms or extensions, radial\njet-like features, warps, and other asymmetries. This is quantified as\nan interaction strength index from 0 to 3, as in paper 1. Two objects have\narchival HST images from Marble et al 2003 (the last and third last ones \nin Table 1). In the first of these, no interaction is seen in either\ntelescope image. In the second, our CFHT image has some asymmetry (interaction\nindex 1), but the HST image shows considerable structure that clearly\nindicates a significant disturbance on small scales (see Figure 3). \nThus, our interaction indices may well be conservative. Figure 3 shows \nexamples of the three levels of interaction, plus an HST image for comparison.\n\n\\section{Results}\n\n All the QSOs in the sample are resolved, most of them easily, with \nratio of nuclear to\nresolved flux ranging from 0.17 to 25. We discuss the properties\nof the resolved flux in the subsections below, but begin with a look\nat possible biases and selection effects within the dataset.\n\n\\subsection{Biases and systematics}\n\n Since there is a range of image quality in the sample (from 0.6 to\n1.3 arcsec), we looked for measured quantities that may depend on it.\nThe image quality shows no correlation with object redshift.\nThe interaction index shows no envelope or correlation. The ratio of\nnuclear to resolved flux also does not show any trend, and certainly not\nin the expected sense of lower ratios for better seeing. The dynamic range\nof `useful' QSO image does show an upper envelope where the maximum range is\nsmaller for larger FWHM images. This is largely the effect of reducing the\npeak central signal with poor image resolution, and the limiting signal is \nnot affected. Overall, for the sample and measurements we discuss, the image\nquality is not very important and is not introducing any significant bias\nto the results.\n\n Another observational variable is the total exposure time. While most\nobjects have 680 secs integration, three have twice that, and two have\nfour times that. The interaction index, and useable dynamic range are\nnot correlated with the exposure, and the two best-resolved objects\nhave the minimum exposure. Thus, exposure time is not biassing the\noverall results of the work.\n\n The mix of spectral types is not correlated with object redshift. The\nmagnitude of the QSO is not correlated with the dynamic range of the\nimages. The least resolved objects are not the highest redshift ones, so\nthere is no observational bias obvious. The median ratio of nuclear to\nresolved flux is 4, and values near this are spread evenly across the\nredshift range. There is no systematic colour change with redshift, or\nchange of scale length with redshift, so the mix of objects appears to\nbe unrelated to redshift.\n\n\\subsection{Nucleus to host ratio}\n\n Figure 4 shows the ratio of nuclear to resolved flux with spectral\nclass. The ratio increases with the dominance of the broad emission\nline components, which is as expected for the general model of\ncentral obscuration by a torus for type 2 objects.\n\n The lowest dynamic range observations have the lowest nuclear light\ndomination. This means that the resolved flux is in the inner parts\nof the host galaxies, and that low nuclear domination is caused by\nobscuration within\nthe central host galaxy. However, overall QSO colour is loosely\ncorrelated with nuclear domination, so the obscuration is connected with\nreddening of the nuclear light - a separate phenomenon from the\nobscuration of line emission by the torus.\n\nThe result that obscuration is connected with the diminishing\nof the nuclear light in the optical is supported by the findings\nof Francis, Nelson and Cutri (2004) who found that a number of\nnear-ir flux-selected AGN were missed in SDSS because their optical colors\nwere indistinguishable from normal galaxies. They speculated this\nwas because the nuclei were preferentially reddened and thus better\nvisible in the near-ir.\n\n Scale length of the host galaxy is not correlated with nuclear domination,\nso the nuclear region is obscured without affecting the outer parts of the\nhost galaxy. Scale length is also not related to the spectral class or the\ncolour of the QSO. Further mention is made of the scale length below.\n\n\\subsection{Interaction status}\n\n The level of interaction seen is not strongly related with the QSO colour,\nbut there is a trend towards redder colour for more strongly interacting\nsystems. There is a clear dependence and upper envelope with the nuclear\nfraction, in the sense that the more interacting objects have more \nobscured nuclei (see Figure 5).\n\n It is also clear that interactions are less obvious in higher redshift\nobjects, as expected as the signal to noise and the angular scales decrease.\nWe note that the object with HST imaging, given interaction class 1 from\nthe CFHT data, is clearly interacting in the HST images, and would have\nindex 2. Thus, the CFHT interaction indices should be regarded as lower limits\noverall, and particularly for the higher redshift objects.\n\n The host galaxy scale length shows a trend where they are larger\nfor weaker interacting systems, although there is considerable spread.\nThis will reflect the presence and decay of disks, tidal arms, and\neventual increase in spheroid structure during a major interaction.\n\n Comparison with the z$<$0.3 sample in paper 1 is of interest. Figure 6\nshows the fractions with interaction index with redshift in different\nQSO samples,\nwhere the 2MASS lower redshift sample is large enough to split at\nz=0.2. There is a\nsystematic shift towards lower interaction index with increasing redshift.\nEven allowing for lowering the index by one grade point for 1\/2 of them\n(which would allow for missed signatures seen only with higher resolution,\nas is Figure 3),\nwe find the interactions are stronger at lower redshifts. Since the\nhigher redshift objects are more luminous, it may be that the more interacting\nand hence more obscured objects are not detected in the higher redshift\nbins because of flux limits. The ratio of counts of objects above and below\nz=0.3 (64 to 243, plus lack of LINERS at z$>$0.3), shows the 2MASS flux\nlimitation clearly.\n\n\n\\subsection{Radio flux}\n\n Six of the sample of 21 appear in the FIRST radio catalogue (although\n2 of them\nare too far south to appear in the FIRST catalogue: see Becker et al 1995).\nThey are\nall unresolved, and faint - see Table 1 for the fluxes. The mean redshift\nof the radio sources is the same as that for the others ($\\sim$0.40).\nThere is no correlation between radio flux and colour. The mean ratio\nof nucleus to host for the radio sources is 2.5, compared with 7.2 for\nthe others (medians 1.3 and 3.2). The spectroscopic types have the same\n(full) spread for radio and radio-quiet objects. The radio sources\nhave an average interaction index of 2 while the others have value 1,\nand this difference is significant at the 95\\% level, if the\ndistributions\nare gaussian. This suggests that\nnuclear radio sources occur in recently activated nuclei where the\nsigns of interaction, and dust obscuration are higher. This is similar\nto the result on IRAS galaxies investigated by Neff and Hutchings\n(1992).\n\n\\subsection{Asymmetries and and profiles}\n\n The ellipticities of the contours in this sample are lower than\nthose in paper 1, and follow the same upper limit which decreases with\nincreasing redshift. This is largely the effect of the diminishing scale and\nsurface brightness with redshift, but the values are less determinate as the \npresence of line of sight companions increases too.\n\n Profiles were classified as good fits to R$^{1\/4}$ law or\nexponentials\nif they had significant linear sections in the relevant plots. Many\nobjects have nearby companions or asymmetries which made these simple\nclassifications impossible. However, there were 3 objects with good\nexponential\nprofile sections and 9 with good bulge components. Their mean redshifts\nare 0.38 and 0.43, respectively, so that the outer exponential tails may\nsimply be less detectable in the fainter higher redshift objects.\n\n It is of interest to compare the profiles of an object with HST imaging\n(1715+281).\nThe HST image shows strong asymmetry of the resolved flux and is traceable to\na radius of 2.4\", while the CFHT image shows less of the structure but\nthe same asymmetry out to 7\" (see Figure 2). \nThe profiles agree and both indicate\nthat r$^{1\/4}$ fits quite well, while an exponential does not. Thus, there\nis good agreement in this one case of overlap with HST data.\n\n\\subsection{Scale lengths}\n\n The scale lengths given in Table 1 are derived from the slopes of the\nradius-magnitude plots for the images, outside the unresolved cores. They\ndo not include any signficant companions, and are converted to Kpc for\na Hubble constant of 75. Cases where the slope cannot be measured\nwith any reliability are left blank. Scale length is not correlated\nwith the dynamic range of the images, although the latter range only\nby about 25\\%. It is also not correlated with the image quality - as\nexpected\nsince the profiles are azimuthally averaged over values far greater than\nthe image quality.\n\n The scale length is larger for the more interacting systems. This\nis a quantitative measure of the extended arms and asymmetries that\nlead to the higher interaction indices. In general the scale length values\nare comparable with large nearby galaxies.\n\n\n\\section{Discussion}\n\n In paper 1 we noted a number of differences between the 2MASS low\nredshift QSOs and standard blue QSOs. Generally speaking, the 2MASS obects\nare redder and have more obscured nuclei, and the host galaxies show a far\ngreater proportion of tidal interactions.\n\n We note that the 2MASS sample requires detection in all three of the\nJHK passbands, so that there is a bias against highly reddened objects.\nThe J-K colours average at 2.2 mag in both the lower and higher\nredshift subsamples, but the B-R colours are different, as seen in Table 2.\nThe selection will also lose the less luminous objects as the redshift\nincreases. While this is clearly true from Figure 1, it is interesting to \nnote the large fraction of more luminous objects that appear at higher\nredshifts. This increase is just what is expected from the 6-fold increase \nin volume of space sampled, so there may not be significant selection effects\nbetween the present sample and the lower redshift objects in paper 1.\n\n\n In most of the aspects considered, the higher redshift 2MASS objects are\nmore similar to `normal' blue-selected QSOs, but they are intermediate between\nthem and the low redshift 2MASS QSOs. Table 2 shows some key comparison\nnumbers, based on paper 1 and the optically selected sample of Hutchings\nand Neff (1991).\n\n The higher redshift sample of this paper correspond to the highest\nluminosity objects in paper 1. The subset of the paper 1 objects that\nmatch this K-band luminosity are also shown in Table 2, and they have higher\nnucleus to host ratio and bluer colour than the full paper 1 sample,\nbut not as high or blue as the sample in this paper. Thus, luminosity\nis a relevant parameter as well as redshift. The single high redshift object\n(at z=2.37) has very high luminosity (although optical emisssion lines \nshifted into the 2MASS bandpasses probably contribute), and is hardly\nresolved in our data. Thus, there is probably nothing very remarkable\nin this object from the data presented here.\n\n The higher redshift (and luminosity) sample of this paper is\ngenerally similar to normal blue QSOs, but still have a higher fraction of\ninteraction evidence, although the fraction of highly interacting objects\nis similar to that for blue QSOs. This is seen in both morphology and\nluminosity profiles. It seems\nlikely that the higher luminosity QSOs blow away the circumnuclear dust\nfaster, but are still relatively young AGN. The unresolved nature of the \nradio sources is consistent with this idea.\n\n The sample does not reach the faint luminosities of the low redshift\nobjects in paper 1, so the evolution of these sources cannot be\ntraced until we have a deeper NIR QSO survey. We note that the Spitzer\ntelescope has reported finding a large population of lower luminosity\nAGN in their deeper survey. It will be important to follow those up\nwith high resolution imaging. \n\nThis publication makes use of data products from the Two Micron All Sky\nSurvey, which is a joint project of the University of Massachusetts and the\nInfrared Processing and Analysis Center\/California Institute of Technology,\nfunded by the National Aeronautics and Space Administration and the\nNational Science Foundation.\n\n\\newpage\n\\scriptsize\n\\begin{deluxetable}{lrllrrccrlrc}\n\\tablenum{1}\n\\tablecaption{2MASS QSO sample}\n\\tablehead{\\colhead{RA} &\\colhead{Dec}\n&\\colhead{z} &\\colhead{Typ} &\\colhead{B, R, J, H, K} &\\colhead{N\/H}\n&\\colhead{Int} &\\colhead{Sc L} &\\colhead{Range} &\\colhead{20cm}\n&\\colhead{IQ} &\\colhead{M$_K$}\\\\\n&&&\\colhead{\\tablenotemark{a}} &&\\colhead{\\tablenotemark{b}}\n&\\colhead{\\tablenotemark{c}} &\\colhead{Kpc\\tablenotemark{d}}\n&\\colhead{mag\\tablenotemark{e}} &\\colhead{mJy} &\\colhead{\"\\tablenotemark{f}}}\n\\startdata\n11 03 12.93 & 41 41 54.9 &0.403 &1.2\n&16.6,16.1,15.2,14.2,13.0 &16.6&1. &-- &9.5&-- &1.1 &-29.1\\\\\n13 32 31.17 & 03 59 28.0 &0.346& 1.\n&17.4,17.3,16.0,14.8,13.4 & 4.3& 1.&--& 11.3&1.5\n& 0.9 &-28.3\\\\\n13 45 17.89 & -08 29 57.3 &0.473& 1.\n& --, ~~~~--, 15.6,14.2,12.8\n& 3.9& 2.&7.7 &9.6&-- & 0.8 &-29.6\\\\\n14 32 04.62 & 39 44 38.9 &0.349&1.5\n&16.4,16.3,16.0,15.4,14.4\n& 5.4& 1. &4.4 &8.5&-- & 0.8 &-27.3\\\\\n14 35 15.66 & 02 32 21.7 &0.305&1.2\n&16.6,16.7,15.6,14.4,12.9\n& 12.& 1.&--& 11.5&-- & 0.9 &-28.5\\\\\n14 38 27.94 & -11 22 49.5 &0.401&1.5\n&19.6,18.7,16.3,15.1,13.6\n&1.2& 1.&8.6 & 9.0 &-- & 1.1 &-28.4\\\\\n14 41 18.87 & -11 31 47.5 & 0.330 &1.2\n&16.6,16.8,16.2,15.2,14.0\n& 2.5& 1.&4.8 & 10.5&-- &1.0 &-27.6\\\\\n14 42 02.95 & 14 55 39.3 &0.307&1.2\n&17.6,17.0,16.2,15.2,14.2\n& 1.8& 3. &5.9 & 10.4& 104. & 0.9 &-27.2\\\\\n14 50 00.90 & 14 29 48.7 &0.358&1.2\n&17.7,17.1,16.4,15.6,14.4\n& 12.& 0.&--&12.0 &-- & 0.7 &-27.4\\\\\n15 00 13.40 & 12 36 45.2 &0.407& 1.9\n&19.5,18.9,16.8,15.9,14.6\n& 0.3& 3. &4.7 &8.1&-- & 0.8 &-27.5\\\\\n15 01 50.51 & 49 33 38.2 &0.337&1.9\n&18.1,17.1,16.4,14.9,13.5\n&0.26& 3.&4.8 & 8.0 &3.1 & 0.8 &-28.2\\\\\n15 31 07.19 & 12 08 14.4 &0.542&1.5\n&19.8,18.7,17.1,15.7,14.6\n& 3.2& 2.&6.7& 10.2&-- &1.0 &-28.1\\\\\n15 36 44.92 & 14 12 29.4 &0.399&1.2\n&16.7,17.3,16.1,15.4,14.0\n& 4.8& 2. &6.6 & 10.5&-- & 0.9 &-28.0\\\\\n15 40 19.57 & -02 05 05.3 &0.319& 1.\n& 16.0,15.8,15.3,14.3,13.2\n&6.& 2.&4.7 &9.3&4.7 &1.0 &-28.3\\\\\n15 49 38.73 & 12 45 09.2 & 2.37& 1.9\n&18.7,17.3,15.8,14.5,13.5\n& 25.& 0. &-- &12.5 &-- & 0.7 &-33.0\\\\\n15 50 59.30 & 21 28 08.8 &0.373& 1.\n&17.3,16.8,16.3,15.7,14.3\n& 8.8& 0.&--& 12.8&-- & 0.6 &-27.6\\\\\n16 18 09.74 & 35 02 08.9 &0.446& 1.9\n&18.8,18.2,16.8,15.4,14.1\n& 2.6& 2.&6.9&10.0 &14. & 0.9 &-28.2\\\\\n16 44 20.14 & 56 36 44.6 &0.329&1.5\n&19.8,18.2,17.2,15.9,14.6\n&0.17& 1.&4.7 &9.5&-- & 1.3 &-27.0\\\\\n17 00 02.99 & 21 18 23.3 &0.596 &1.5\n& --, ~~~~--, 17.4,15.9,14.9\n&1.4 & 0.&3.8 &8.3&-- & 1.1 &-28.1\\\\\n17 00 56.01 & 24 39 28.2 &0.509&1.5\n&16.8,16.8,16.0,15.3,14.3\n& 10.& 1. &8.7 & 10.3&-- & 0.7 &-28.3\\\\\n17 15 59.77 & 28 07 16.9 &0.524 &1.8\n& --, ~~~~--, 17.2,15.8,14.6\n&0.32& 1. &4.4 &7.8&1.6 & 0.8 &-28.0\\\\\n\\enddata\n\\tablenotetext{a}{ The range from 1 to 2 reflects the ratio of narrow to\nbroad emission lines.}\n\\tablenotetext{b}{ The ratio of nuclear to host flux in r-band, from\nthis work. Uncertainties are estimated to be 30\\%.}\n\\tablenotetext{c}{ Strength of host galaxy interaction based on observed\nmorphological features, as in paper 1.}\n\\tablenotetext{d}{ 1\/e profile flux drop}\n\\tablenotetext{e}{ Total dynamic range of profile in magnitudes}\n\\tablenotetext{f}{FWHM of stellar images in arcsec}\n\n\\end{deluxetable}\n\n\\newpage\n\\normalsize\n\\begin{deluxetable}{lcccc}\n\\tablenum{2}\n\\tablecaption{2MASS QSO sample comparisons}\n\\tablehead{\\colhead{Property} &\\multicolumn{3}{c}{2MASS QSOs}\n&\\colhead{Blue QSOs}\\\\\n&\\colhead{z$<$0.3} &\\colhead{z$<$0.3,luminous} &\\colhead{z$>$0.3} }\n\\startdata\nAverage z &0.2 &0.2 &0.4 &0.2\\\\\nSample size &76 &12 &21 &28\\\\\n\\\\\nB-R &1.1 &0.8 &0.5 &0.3\\\\\nNuc\/Host &0.5 &1.2 &6.0 &5.0\\\\\nInt strong &33\\% &42\\%&14\\% &11\\%\\\\\nInt (any) &75\\% &58\\% &81\\% &39\\%\\\\\nExp profile &16\\% &-- &14\\% &18\\%\\\\\nBulge profile &20\\% &-- &48\\% &54\\%\\\\\nMessy profile &64\\% &-- &38\\% &28\\%\\\\\nRadio detected &40\\% &50\\% &29\\% &43\\%\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\newpage\n\\centerline{References}\n\nBecker, R., White R.L., and Helfand D.J., 1995, ApJ, 450, 559\n\nCutri,R.M. et al, 2002, ASP Conf Ser vol 284, 127\n\nFrancis P.J., Nelson B.O., and Cutri, R.M., 2004, AJ, 127, 646\n\nHutchings J.B., Maddox N., Cutri R.M., Nelson B.O., 2003, AJ, 126, 63\n(paper 1)\n\nHutchings J.B. and Neff S.G., 1992, AJ, 101, 434\n\nMarble A.R. et al., 2003, ApJ, 590, 707\n\nNeff S.G. and Hutchings J.B., 1992, AJ, 103, 1992\n\n\n\\newpage\n\\normalsize\n\\centerline{Captions to figures}\n\n1. Sample selection in this paper (z$>$0.3) and paper 1, from the full\n2MASS sample observable from CFHT. Filled circles are the objects observed. \nTop: dashed lines are contours of constant luminosity. Bottom: spectral\ntypes range from broad to narrow emission lines, with intermediate\nvalues. Dashes are the full sample and dots are those observed.\nThe paper 1 observed sample is grouped as types 1 and 2 only, as given\nin that paper.\n\n2. Azumuthally averaged profiles of QSO 1715+281, with the CFHT PSF \nand also an HST image (with slightly different filter).\n\n3. Images of representative objects from the sample, in decreasing levels\nof interaction - top left (1500+12) level 3, top right (1442+14) level 2, \nbottom (1715+28) level 1. The lower right is the HST image of the object \nat lower left. The images are 11 arcsec on a side except for the HST which\nis 6 arcsec.\n\n4. Correlations with nuclear light fraction. Top: the type 1 objects\nhave higher nuclear flux, consistent with the orientation expectation\nfor broad-line objects. Bottom: With the high value exception, there\nis correlation with overall QSO colour, consistent with the nuclear\nfraction being reduced by dust reddening. In the lower panel, open dots \nare B-K and filled dots are 3(B-R). The line is the linear fit \nthrough all except the top right object.\n\n5. Correlations with interaction level of host galaxies. The interaction\nscale was estmated on a 5-point scale but is reduced to 3 here to match\nthe values from paper 1. Top: high\nlevels of interaction are not seen in the highest redshift objects,\npresumably because of signal level and angular scale. Bottom: Highly\ninteracting hosts have low flux nuclei, presumably because they are\nobscured by dust.\n\n6. Fraction of hosts in different interaction levels in three redshift\nbins with about equal sample numbers. There is a systematic change with\nredshift whereby lower redshift objects are more interacting. This may\nnot all be due to detectability changes with redshift.\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\n\n\\subsection{Background and Motivation}\n\nAs one studies \\emph{differential structures} on a manifold such as Milnor \\cite{milnor}, \\emph{affine structures} on a scheme, taken as counterparts, will be introduced and discussed in this paper. Here, we will obtain several properties on affine structures in a rigour and systematic manner. These results in fact will have been applied to the\ndiscussions on the complex analytical space of an algebraic scheme over a number field.\n\n\\subsubsection{An affine covering of a scheme}\n\nAs well-known, a scheme (or a projective scheme, respectively) is defined to be a\nringed space that can be covered by a family of affine (or\nprojective, respectively) schemes, called \\emph{an affine covering} of the scheme $X$. An affine scheme is the spectrum of a\ncommutative ring equipped with the sheaf in an evident manner (\\cite{ega,hartshorne}). In the paper it will be seen that such a family of affine schemes determines a unique \\emph{affine\nstructure} on the scheme.\n\n\\subsubsection{Each affine covering produces a complex analytical space}\n\nFixed an algebraic scheme $X$ over a number field $K$. Let $\\{A_{\\alpha}\\}_{\\alpha \\in\\Gamma}$\nbe a family of finitely generated algebras over $K$ such that their spectra\n$Spec{A_{\\alpha}}$ cover $X$, i.e., $$\\bigcup_{\\alpha} Spec{A_{\\alpha}}\\supseteq X$$ holds. Here, each $A_{\\alpha}$ is isomorphic to the\nquotient of some polynomial ring $K[t_{1},t_{2},\\cdots,t_{n_{\\alpha}}]$ by a\nfinite number of polynomials $$f_{1},f_{2},\\cdots,f_{r_{\\alpha}}$$ over $K$ in the variables.\n\nBy Serre's\n\\emph{GAGA} (\\cite{sga1,serre-gaga}), each open subscheme $Spec{A_{\\alpha}}$ has an analytical space\n$X^{an}_{\\alpha}$ that is defined by the common zeros of the polynomials\n$$f_{1},f_{2},\\cdots,f_{r_{\\alpha}}$$ mentioned above. Gluing these analytical spaces\n$X^{an}_{\\alpha}$, we will obtain an analytical space $X^{an}$, called the\n\\emph{complex analytical space} of $X$. It has been seen that such a process has several functorial\nproperties with respect to $X$ (\\cite{sga1,serre-gaga}).\n\nHence, every affine covering of $X$ produces a complex analytical space $X^{an}$ of $X$.\n\nHowever, in general, an algebraic scheme $X$ can have many affine coverings. Then what about the complex analytical spaces of $X$ produced by different affine coverings of $X$?\n\n\\subsubsection{Different affine coverings can produce different complex analytical spaces}\n\nLet's take an example raised by Serre (\\cite{serre-example}):\n\\begin{quotation}\n\\emph{Let $V$ be the\nnonsingular projective variety over a number field $K$ as defined in \\emph{\\cite{serre-example}}. Suppose that\n$V_{\\phi}$ is a conjugate variety of $V$ defined by an isomorphism $\\phi$.\nThen there is such an isomorphism $\\phi$ that the complex analytic spaces $V^{an}$ and $V^{an}_{\\phi}$ are not of\nthe same homotopy type.}\n\\end{quotation}\n\nDenote by $A$ and $A_{\\phi}$\nthe homogeneous coordinate rings of $V$ and $V_{\\phi}$, respectively. Then\nwe have projective schemes $$X=Spec{A}; \\, X_{\\phi}=Spec{A_{\\phi}}$$\nwith complex analytical spaces $$X^{an}=V^{an}; \\, X_{\\phi}^{an}=V^{an}_{\\phi}$$ respectively.\n\nThis shows that different affine coverings of the same scheme can produce different complex analytical spaces.\n\nFor this phenomenon, there are also some more examples arising from abelian varieties and Shimura varieties (\\cite{deligne-milne,milne-suh}).\n\nNow we come to a conclusion that there do exist evidences, such as related examples in \\cite{deligne-milne,sga1,milne-suh,serre-example}, that different affine\ncoverings can produce different complex analytical spaces for a fixed algebraic scheme.\n\nWhy there does exist such a phenomena? It is an interesting problem. Such related topics will be\ndiscussed in our subsequent papers.\n\n\\subsubsection{Several problems}\n\nIn the above it has been seen that different affine coverings of an algebraic scheme can produce different complex analytical space. Then it is natural for one to have several questions (similarly for projective schemes)\nsuch as the following:\n\\begin{itemize}\n\\item \\emph{How can we give a restrictive definition for such a family of affine schemes to\n patch a scheme?}\n\n\\item \\emph{Does there exist another family of affine schemes covering\n the fixed scheme and making it into a scheme?}\n\n\\item \\emph{Given another family of affine schemes which cover the fixed scheme.\n Will we obtain the same scheme?}\n\n\\item \\emph{Given a ringed space. How many families of affine\n schemes do patch it? How many schemes do there exist on the same underlying\n topological space?}\n\n\\item \\emph{In particular, given an algebraic scheme $X$ over a number field. There can be\n many families of affine schemes covering $X$. Each such a family produces an analytical\n space $X^{an}$ of $X$. When are these analytical spaces $X^{an}$ either diffeomorphic to each other or of\n the same homotopy types?}\n\\end{itemize}\n\nSeveral questions related to the above will be in part discussed in the paper.\n\nAt the same time, affine structures have also been encountered by us during the discussions on a type of Galois covers of algebraic and arithmetic schemes, where such a scheme is said to be \\emph{Galois closed} if it has only one affine\nstructures.\n\nThe Galois closed schemes have several nice properties with applications to\nclass fields, for example, their Galois groups of rational fields are\nisomorphic to their groups of automorphisms (for instance, see \\cite{an}).\n\n\n\\subsection{Techniques in the Paper}\n\nAs a counterpart, an affine structure on a scheme behaves exactly like a differential\nstructure on a manifold.\n\n\\subsubsection{A smooth manifold can have many differential structures}\n\nIn a classical way, a differential manifold is a\ntopological space covered by a family of open subsets in some Euclidean space,\nwhich is obtained by glueing such a family of open sets as patches. Under some\ntechnical conditions, such a family of open sets is called a \\emph{differential\nstructure} on the manifold.\n\nNowadays there have been many well-known facts about\nmanifolds and their differential structures:\n\\begin{itemize}\n\n\\item \\emph{There exist differential manifolds which have many\ndifferential structures on them that are not diffeomorphic to each\nother} (\\cite{milnor}).\n\n\\item \\emph{There exist topological spaces that have no differential structures on\nthem}.\n\n\\item \\emph{There exist differential\nmanifolds will be of different properties if we establish different differential\nstructures on the (same) underlying spaces}.\n\\end{itemize}\n\n\\subsubsection{Affine structure v.s. differential structure}\n\nMany approaches and skills in topology can be applied here to schemes. The techniques in the\npresent paper, which we borrowed from differential topology, is thus not new to\nsome certain degree.\n\n\\subsubsection{Known related results on affine coverings}\n\nFor affine coverings, there have been several informal\ndiscussions, for example, see \\cite{hartshorne,shafarevich}, on how to patch a scheme, that is, how to glue a given\nfamily of affine schemes into a scheme; for a more abstract case of categories,\nthere have been fibered categories and groupoids (\\cite{sga3-1,sga1}) which can be\napplied to discuss such coverings.\n\nHowever, all those discussions involved in \\cite{sga3-1,sga1,hartshorne,shafarevich} deal only with \\emph{coverings},\nthat is, a family of abstract objects over a fixed object.\n\n\\subsubsection{Further results obtained in the paper}\n\nIn the paper here we will have further discussions on\nsuch affine coverings and several results will be obtained. We will introduce and discuss affine\nstructures in a rigor and systematic manner, where an affine structure is an affine covering that is taken as maximal families of objects covering a given object and satisfying the certain properties.\n\nIn fact, the affine structures will afford a platform to\nus to discuss the problems mentioned above \\S 1.1. In particular, an\nalgebraic scheme over $\\mathbb{C}$ can have a unique associative analytical\nspace if there exists only one affine structure on its underlying space.\n\nFurthermore, in a rigor manner, a scheme is a locally ringed space with a\nspecified affine structure on it; it follows that in such a case an algebraic\nscheme over a number field can be associated exactly with a unique complex\nanalytical space.\n\nThe main results obtained in the paper are that by the whole of affine structures on a space,\nit will be seen whether\ntwo spaces are homeomorphic and whether two schemes are isomorphic. In other words, the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively.\n\nSuch results can be applied to complex analytical spaces of algebraic schemes and arithmetic schemes.\n\n\\subsection{Outline of the Paper}\n\nAt last we give an outline of the paper.\nIn \\S 2 we will use \\emph{pseudogroups $\\Gamma$ of affine transformations} to\ndefine an \\emph{affine $\\Gamma-$atlas} on a topological space, which consists of a\nfamily of \\emph{affine charts}. An \\emph{affine $\\Gamma-$structure} on a space is an affine\n$\\Gamma-$atlas which is maximal. Our discussion can be regarded as an\nalgebraic version of differential structures (\\cite{str,kobayashi}).\n\nIn \\S 3 an affine $\\Gamma-$structure on a space is said to be\n\\emph{admissible} if there is a sheaf on the space such that they are coincide with\neach other on each affine chart. Here, such a sheaf is called an \\emph{extension} of\nthe given affine structure.\n\nAn affine structure which is not admissible will be\nof no practical use.\n\nGiven a scheme $(X,\\mathcal{O}_{X})$ in the usual manner (\\cite{ega,hartshorne}). In \\S 4\nwe will discuss the special types of affine structures on the space $X$, called\nthe \\emph{canonical} and the \\emph{relative canonical} affine structures in the scheme\n$(X,\\mathcal{O}_{X})$ respectively. Their extensions are called the \\emph{associate\nschemes} of $(X,\\mathcal{O}_{X})$.\n\nEvery scheme has an associate scheme. In\nparticular, a scheme itself is an associate scheme of it. As schemes, a fixed\nscheme and their associate schemes are isomorphic with each other.\n\nNow put\n\\begin{itemize}\n\\item \\emph{$\\mathbb{A}\\left( X\\right)\\triangleq$ the set of all admissible affine structures\non a topological space $X$;}\n\n\\item \\emph{$ \\mathbb{A}_{0}\\left(\nX,\\mathcal{O}_{X}\\right) \\triangleq$ the set of all the relative\ncanonical affine structures in a scheme} $\\left( X,\\mathcal{O}_{X}\\right)$.\n\\end{itemize}\n\nUsing the set of affine structures on a space, in \\S 5 we will give the statements of the two main\ntheorems in the paper:\n\n\\textbf{Theorem 5.1.} \\emph{Let $X$ and $Y$ be two\ntopological spaces such that either $\\mathbb{A}\\left(\nX\\right)\\not=\\emptyset$ or $\\mathbb{A}\\left( Y\\right)\\not=\\emptyset$\nholds. Then $X$ and $Y$ are homeomorphic if and only if there is $$ \\mathbb{A}\\left(\nX\\right) =\\mathbb{A}\\left( Y\\right).$$}\n\n\\textbf{Theorem 5.2.} \\emph{Any two schemes $\\left(\nX,\\mathcal{O}_{X}\\right) $ and $ \\left( Y, \\mathcal{O}_{Y}\\right) $ are\nisomorphic if and only if we have $$ \\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right)\n\\cong \\mathbb{A}_{0}\\left( Y, \\mathcal{O}_{Y}\\right) .$$}\n\n\nFrom the two main theorems it will be seen that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively. Such results will be applied to complex analytical spaces of algebraic schemes and arithmetic schemes. The two theorems will be proved in \\S 7 and \\S 8, respectively.\n\n\nAs a conclusion, in \\S 6 we will give several concluding remarks. Particularly, we will come to a conclusion that to be\nprecisely defined, a scheme should be a locally ringed space together with a\ngiven admissible affine structure on it if the affine structures are in action in a\nparticular case.\n\n\\bigskip\n\n\\textbf{Acknowledgment}\nThe author would like to express his sincere\ngratitude to Professor Li Banghe for his advice and instructions\non algebraic geometry and topology.\n\n\n\n\n\n\\section{Definitions for Affine Structures}\n\nIn this section we will introduce affine structures on a space in a evident manner as one does for differential structures on a space (for instance, see \\cite{str,kobayashi}).\n\n\\subsection{Pseudogroup of Affine Transformations}\n\nLet $\\mathfrak{Comm}$ be the category of commutative rings with identities,\nand $\\mathfrak{Comm}\/k$ the category of finitely generated algebras over a\nfield $k.$ Here, a pseudogroup (or groupoid) is a small category in which every\nmorphism is invertible (\\cite{maclane}).\n\n\\begin{definition}\nA \\textbf{pseudogroup $\\Gamma$ of affine transformations}, as a subcategory of $\\mathfrak{Comm}$, is a pseudogroup of isomorphisms between commutative rings satisfying the\nconditions 1-5:\n\\begin{enumerate}\n\\item Each $\\sigma\\in\\Gamma$ is an isomorphism from\na ring $dom\\left( \\sigma\\right) $ onto a ring $rang\\left( \\sigma\\right)$\ncontained in $\\Gamma$, called the \\textbf{domain} and\n\\textbf{range} of $\\sigma$, respectively.\n\n\\item If $\\sigma\\in\\Gamma$, the inverse\n$\\sigma^{-1}$ is contained in $\\Gamma.$\n\n\\item The identity map $id_{A}$ on $A$ is contained in \n\\Gamma$ if there is some \n\\delta\\in\\Gamma$ with $dom\\left( \\delta\\right) =A.$\n\n\\item If $\\sigma\\in\\Gamma$, the isomorphism\ninduced by $\\sigma$ defined on the localization $dom\\left(\n\\sigma\\right)_{f}$ of the ring $dom\\left( \\sigma\\right)$ at any\n $0\\not= f\\in dom\\left( \\sigma\\right) $ is contained in $\\Gamma.$\n\n\\item Given any $\\sigma,\\delta\\in\\Gamma$. Then\nthe isomorphism factorized by $dom\\left( \\tau\\right) $ from\n$dom\\left( \\sigma\\right) _{f}$ onto $rang\\left( \\delta\\right) _{g}$\nis contained in $\\Gamma$ if for some\n$\\tau\\in\\Gamma$ there are isomorphisms $dom\\left( \\tau\\right) \\cong\ndom\\left( \\sigma\\right) _{f}$ and $dom\\left( \\tau\\right) \\cong\nrang\\left( \\delta\\right) _{g}$ with $0\\not= f\\in dom\\left(\n\\sigma\\right) $ and $0\\not= g\\in rang\\left( \\delta\\right) .$\n\\end{enumerate}\n\nSuch a pseudogroup $\\Gamma$ is said to be a \\textbf{pseudogroup of } $k-$\\textbf{affine transformation} if $\\Gamma$\nis contained in the category $\\mathfrak{Comm}{\/k}$, or equivalently,\nif each isomorphism in $\\Gamma$ is an isomorphism of finitely\ngenerated algebras over a field $k.$\n\\end{definition}\n\n\n\\subsection{Affine Charts and Affine Atlas}\n\nFor a topological space, we give the notions of affine charts and affine atlas.\n\n\\begin{definition}\nLet $X$ be a topological space and $\\Gamma$ a\npseudogroup of affine transformations. Then an \\textbf{affine }$\\Gamma-$\\textbf{atlas}\n $\\mathcal{A}\\left( X,\\Gamma\\right) $ on $X$ is a collection of pairs $\\left( U_{j},\\varphi_{j}\\right)\n$ with $j\\in\\Delta$, called \\textbf{affine charts}, satisfying the conditions\n1-3:\n\\begin{enumerate}\n\\item {For every pair} $\\left( U_{j},\\varphi\n_{j}\\right) \\in\\mathcal{A}\\left( X,\\Gamma\\right) ,${\n}$U_{j}${ is an open subset of }$X${ and\n}$\\varphi_{j}${ is an homeomorphism of }$U_{j}${ onto\n}$Spec\\left( A_{j}\\right) ,${ where }$A_{j}${ is a\ncommutative ring contained in $\\Gamma$.}\n\n\\item $\\bigcup_{j\\in\\Delta} U_{j}\\supseteq X${ is an open covering\nof }$X.$\n\n\\item {Given any} $\\left( U_{i},\\varphi\n_{i}\\right) ,\\left( U_{j},\\varphi_{j}\\right) \\in\\mathcal{A}\\left(\nX,\\Gamma\\right) $ {with} $U_{i}\\cap U_{j}\\not\n=\\varnothing${. There exists a pair }$\\left(\nW_{ij},\\varphi_{ij}\\right) \\in\\mathcal{A}\\left( X,\\Gamma\\right) $\n{such that }$W_{ij}\\subseteq U_{i}\\cap U_{j}${ and that the\nisomorphism from the localization }$\\left( A_{j}\\right)\n_{f_{j}}${ onto the localization }$\\left( A_{i}\\right)\n_{f_{i}}${ that is induced by the restriction\n}$$\\varphi_{j}\\circ\\varphi_{i}^{-1}\\mid_{W_{ij}}:\\varphi_{i}(W_{ij})\n\\rightarrow\\varphi_{j}(W_{ij})$$\n{is also contained in }$\\Gamma$. {Here} $A_{i}$ and $A_{j}$\n{are commutative rings contained in $\\Gamma$ such that}\n$$\\varphi _{i}\\left( U_{i}\\right) =SpecA_{i}\\, and \\,\n\\varphi_{j}\\left( U_{j}\\right) =SpecA_{j}$$ {hold and that there\nare homeomorphisms}\n$$\\varphi_{i}\\left( W_{ij}\\right) \\cong Spec\\left( A_{i}\\right)\n_{f_{i}}{ and }\\varphi_{j}\\left( W_{ij}\\right) \\cong Spec\\left(\nA_{j}\\right) _{f_{j}}$${for some} $f_{i}\\in A_{i}$ \\, and \\, \nf_{j}\\in A_{j}.$\n\\end{enumerate}\n\nMoreover, $\\mathcal{A\n\\left( X,\\Gamma\\right) ${ is said to be a $k-$\\textbf{affine }$\\Gamma- $\\textbf{atlas}}{ on }$X$ {if\n}$\\Gamma$ is a subcategory of $\\mathfrak{Comm}\/k.$\n\n{An affine }$\\Gamma-${atlas }$\\mathcal{A}\\left( X,\\Gamma\\right) \n{ on }$X${ is said to be \\textbf{complete} (or\n\\textbf{maximal}) if it can not be contained properly in any other\naffine }$\\Gamma-${atlas of }$X.$\n\\end{definition}\n\n\\begin{remark}\n{The above construction in Definition 2.2 is\nwell-defined since the open covering }$\\{U_{j}\\}$ {such that}\n$\\left( U_{j},\\varphi _{j}\\right) \\in\\mathcal{A}\\left(\nX,\\Gamma\\right) ${ is a base for the topology on }$X.$\n\n{Let }$\\mathcal{A}\\left( X,\\Gamma\\right) $ \nand }$\\mathcal{A}\\left( X,\\Gamma^{\\prime}\\right) $ {be atlases\non a\nspace }$X.$ {Then }$\\Gamma\\supseteq\\Gamma^{\\prime}${ holds if }\n\\mathcal{A}\\left( X,\\Gamma\\right) \\supseteq\\mathcal{A}\\left(\nX,\\Gamma^{\\prime}\\right) .$\n\\end{remark}\n\n\\subsection{Affine Structures}\n\nAs one has differential structures on a manifold, here we have affine structures on a space such as the following.\n\n\\begin{definition}\n{Let }$X${ be a topological space and }$\\Gamma${ a\npseudogroup of affine transformations. Then two affine }$\\Gamma-$\natlases }$\\mathcal{A}$ {and} $\\mathcal{A}^{\\prime}${ on }$X$\n\\ are said to be $\\Gamma-$\\textbf{compatible}} {if the\n condition below is satisfied:}\n\\begin{quotation}\nFor any $\\left( U,\\varphi\\right) \\in\\mathcal{A}$ {and }\n\\left( U^{\\prime},\\varphi^{\\prime}\\right) \\in\\mathcal{A}^{\\prime}$\n{with} $U\\cap U^{\\prime}\\not =\\varnothing$ {there\nexists an affine chart }$\\left( W,\\varphi^{\\prime\\prime}\\right) \\in\\mathcal{\n}\\bigcap\\mathcal{A}^{\\prime}$ {such that }$W\\subseteq U\\cap U^{\\prime}\n{ and that the isomorphism from the localization }$\\left( A\\right)\n_{f}${ onto the localization }$\\left( A^{\\prime}\\right)\n_{f^{\\prime}}${\ninduced by the restriction }$\\varphi^{\\prime}\\circ\\varphi^{-1}\\mid _{W}\n{ is also contained in }$\\Gamma$. {Here }$A$ and $A^{\\prime}$\n{are commutative rings contained in $\\Gamma$ such that}\n$\\varphi\\left( U\\right) =SpecA {\\, and \\, } \\varphi^{\\prime}\\left(\nU^{\\prime}\\right) =SpecA^{\\prime}$ {hold and that there are homeomorphisms}\n$\\varphi\\left( W\\right) \\cong Spec\\left( A\\right)_{f}{\\, and \\,\n}\\varphi^{\\prime}\\left( W\\right) \\cong Spec\\left( A^{\\prime}\\right)\n_{f^{\\prime}}$ {for some} $f\\in A$ {and} $f^{\\prime}\\in\nA^{\\prime}.$\n\\end{quotation}\n\\end{definition}\n\n\\begin{proposition}\n{Let }$X${ be a topological space and let }$\\Gamma${ be a\npseudogroup of affine transformations. Then for any given affine }$\\Gamma-\n{atlas $\\mathcal{A}$ on }$X,${ there is a unique complete affine \n$\\Gamma-${atlas }$\\mathcal{A}_{m}${ on }$X${ such\nthat}\n\\begin{itemize}\n\\item $\\mathcal{A\\subseteq A}_{m};$\n\n\\item $\\mathcal{A}$ {and }$\\mathcal{A}_{m}$ \nare }$\\Gamma-${compatible.}\n\\end{itemize}\n\\end{proposition}\n\n{In such a case, we will say that $\\mathcal{A}$ is a \\textbf{base} for\n}$\\mathcal{A}_{m}$ {and} $\\mathcal{A}_{m}${ is the\n\\textbf{complete affine }$\\Gamma-$\\textbf{atlas}}{ determined\nby $\\mathcal{A}.$}\n\n\n\\begin{proof}\nProve the existence. Let $\\Sigma$ be the collection of affine $\\Gamma-\natlases $\\mathcal{A}_{\\alpha}$ on $X$ such that\n$\\mathcal{A}\\subseteq\\mathcal{A}_{\\alpha}$ and that\n$\\mathcal{A}$ and $\\mathcal{A}_{\\alpha}$ are $\\Gamma-$compatible.\n\nThen $\\Sigma$ is a partially ordered set together with the inclusions of sets\n$\\mathcal\nA}_{\\alpha}\\subseteq\\mathcal{A}_{\\beta}$ for any $\\mathcal{A}_{\\alpha}\n\\mathcal{A}_{\\beta}\\in\\Sigma.$ It is clear that every totally\nordered subset of $\\Sigma$ has a upper bound in $\\Sigma$. By Zorn's\nLemma, $\\Sigma$ has maximal elements.\n\nProve the uniqueness. Let $\\mathcal{A}_{m}$ and $\\mathcal{A}_{m}^{\\prime}$\nbe two maximal elements of $\\Sigma.$ Then we must have $\\mathcal{A}_{m}\n\\mathcal{A}_{m}^{\\prime}.$ Otherwise, hypothesize $\\mathcal{A}_{m}\\not \n\\mathcal{A}_{m}^{\\prime}.$ It is seen that $\\mathcal{A}_{m}$ and $\\mathcal{A\n_{m}^{\\prime}$ are $\\Gamma-$compatible since they are $\\Gamma\n-$compatible respectively\nwith $\\mathcal{A}$; then the union $\\mathcal{A}_{m}\\cup\\mathcal\nA}_{m}^{\\prime}$ is contained in $\\Sigma,$ where we will obtain a\ncontradiction.\n\\end{proof}\n\n\\begin{definition}\n{Let }$X${ be a topological space. An \\textbf{affine }}\n\\Gamma -${\\textbf{structure} on }$X$ {is a complete affine }\n\\Gamma -${atlas }$\\mathcal{A}\\left( \\Gamma \\right) $ {on\n}$X,$ {where $\\Gamma $ is a given pseudogroup of affine\ntransformations.}\n\\end{definition}\n\nLikewise, we define a {$k-$\\textbf{affine\n}$\\Gamma-$\\textbf{structure}} if $\\Gamma\\subseteq\\mathfrak{Comm}\/k.$\n\n\n\n\n\n\\section{Admissible Affine Structures}\n\n\n\nBy Proposition 2.1 it is seen that an affine atlas on a topological space\n$X$ determines a unique affine structure on it. From this view of point, we\nsometimes identify an affine atlas on $X$ with its determined complete\naffine structure on $X.$ In this section will discuss admissible affine structures on a space. On a given space, only admissible affine structures are interesting and are of the practical uses.\n\n\\begin{definition}\n{Let $\\mathcal{A}$}$\\left( \\Gamma \\right) ${ be an affine }\n\\Gamma -${structure on a topological space }$X${. Suppose\nthat there exists a locally ringed space }$\\left(\nX,\\mathcal{F}\\right) $ such that\n$\\varphi _{\\alpha \\ast }\\mathcal{F}\\mid\n_{U_{\\alpha }}\\left( SpecA_{\\alpha }\\right) =A_{\\alpha }$ holds for each $\\left( U_{\\alpha },\\varphi\n_{\\alpha }\\right) \\in ${$\\mathcal{A}$}$\\left( \\Gamma \\right) $,\n{where} $A_{\\alpha }$ {is a commutative ring contained in\n$\\Gamma$ with} $\\varphi _{\\alpha }\\left( U_{\\alpha }\\right)\n=SpecA_{\\alpha }.$\n\n{Then }$\\mathcal{A}\\left( \\Gamma\n\\right) ${ is said to be an \\textbf{admissible affine structure} on }\nX$ {and }$\\left( X,\\mathcal{F}\\right) $ {is said to be an\n\\textbf{extension}} {of the affine }$\\Gamma -${structure }$\\mathcal\nA}\\left( \\Gamma \\right) .$\n\\end{definition}\n\n\\begin{proposition}\n{All extensions of an admissible affine\nstructure on a topological space are schemes which are isomorphic with each other.}\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{A}$ be an admissible affine structure on a topological space $X$.\nIt is evident that each extension of $\\mathcal{A}$ on $X$ is a scheme.\n\nNow fixed any extensions $\\left( X,\\mathcal{F}\\right) $ and $\n\\left( X,\\mathcal{G}\\right) $ of $\\mathcal{A}$ on $X$.\nWe prove $\\mathcal{F}\\cong\\mathcal{G}.$\n\nIn deed, let $U_{\\alpha }$ be an open subset of $X$ contained in\n$\\mathcal{A}$. From the assumption we have $$\\Gamma \\left(\n\\mathcal{F},U_{\\alpha }\\right) =\\Gamma \\left( \\mathcal{G},\nU_{\\alpha }\\right) .$$\n\nTake any open subset $U$ of $X$. We have $$U=\\bigcup_{\\alpha\n}U_{\\alpha }$$ with $U_{\\alpha }\\in \\mathcal{A}$.\nDefine a map $$\\phi:\\Gamma \\left(\n\\mathcal{F},U\\right) =\\Gamma \\left( \\mathcal{G},\nU\\right),\\\\ t\\mapsto \\phi (t)$$\nwhere $\\phi (t)\\in \\Gamma \\left( \\mathcal{G},\nU\\right)$ is the section on $U$ determined by $$t\\mid _{U_{\\alpha}}=\n\\phi (t)\\mid _{U_{\\alpha}}.$$\nThen $\\phi$ is an isomorphism for every open subset $U$ of $X$.\n\nBy $\\phi$\nwe obtain an\nisomorphism $\\mathcal{F}_{x}\\cong \\mathcal{G}_{x}$ at every $x\\in X$ and\nhence $\\mathcal{F}\\cong\\mathcal{G}$ holds.\n\\end{proof}\n\n\\begin{corollary}\nFor affine structures, there are the following statements:\n\\begin{enumerate}\n\\item {Let }$\\left( X,\\mathcal{F}\\right) $ {be an\nextension of the affine }$\\Gamma -${structure }$\\mathcal{A}\\left(\n\\Gamma \\right) $ {on a space }$X.$ {Then we have}\n$$\n\\left( U,\\mathcal{F}\\mid _{U}\\right) \\cong \\left( SpecA,\\widetilde{A\n\\right) \\text{ {and }}\\mathcal{F}\\mid _{U}=\\varphi _{\\ast }^{-1\n\\widetilde{A}\n$$\n{for every affine chart }$\\left( U,\\varphi \\right) \\in ${$\\mathcal\nA}$}$\\left( \\Gamma \\right) $ {with }$\\varphi \\left( U\\right)\n=Spec\\left( A\\right) ,${ where }$A$\n{is a commutative ring contained in }\n\\Gamma .$\n\n\\item {An affine structure }$\\mathcal{A}${\non a space }$X${ is admissible if and only if\n}$\\mathcal{A}${ can be\nextended to be a sheaf }$\\mathcal{F}${ on }$X$ {such that }\n\\left( X,\\mathcal{F}\\right) $ {is a locally ringed space.}\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nIt is immediate from Definition 3.1 and Proposition 3.1.\n\\end{proof}\n\n\n\\section{Canonical Affine Structures}\n\nIn this section it will be seen that for a given scheme there\ncan be many different admissible affine structures on the underlying\ntopological space of the scheme. That is, a given scheme can have many associate schemes. All associate schemes of a given scheme are isomorphic as schemes but have different affine structures.\n\n\\subsection{Canonical Pseudogroups of Affine Transformations}\n\nTo start with, let's consider an example.\n\n\\begin{example}\n\\textbf{(Different Affine Structures)} {Let} $k$ {be a field.}\n\\begin{enumerate}\n\\item {Put} $$\\Gamma _{1}=\\{\\text{{the\nidentity }}1_{k}:k\\rightarrow k\\};$$\n\\begin{equation*}\n\\begin{array}{c}\n\\Gamma _{2}=\\{\\text{{the identity }}1_{k}:k\\rightarrow k\\}\\bigcup \\{\\text{{a field isomorphism\n}}\\sigma\n:k\\rightarrow k^{\\prime }\\}\\\\\n\n \\bigcup \\{\\text{{the inverse }}\\sigma\n^{-1}:k^{\\prime }\\rightarrow\nk\\}.\\\\\n\\end{array}\n\\end{equation*}\n{Then }$\\Gamma _{1}${ and }$\\Gamma _{2}${ are both\npseudogroups of\\ affine transformations.}\n\n\\item Let\n$$\\mathcal{A}\\left( \\Gamma _{1}\\right) =\\{\\left( U,\\varphi\n\\right) \\};$$ $$ \\mathcal{A}\\left( \\Gamma _{2}\\right)\n=\\{\\left( U,\\varphi \\right) ,\\left( V,\\eta \\right) \\},$$ {where\n}$$U=V=Spec(k),\\text{ }\\varphi \\left( U\\right) =Spec(k),\\text{ and }\\eta \\left( V\\right) =Spec(k^{\\prime }).$$\n{Then }$Spec(k)${ is an extension of the affine structure }\n\\mathcal{A}\\left( \\Gamma _{1}\\right) $.\n{In general, it is not true that\n}$Speck${ is an extension of }$\\mathcal{A}\\left( \\Gamma\n_{2}\\right) .$\n{For example, let }$\\sqrt[3]{2},\\xi ,\\overline{\\xi }$ {be\nthe roots of the equation }$X^{3}-2=0$ {in }$\\mathbb{C}.$\n{Consider }$k=\\mathbb{Q}\\left( \\sqrt[3]{2}\\right); k^{\\prime } =\\mathbb{Q}\\left( \\xi \\right) .$\n\\end{enumerate}\n\\end{example}\n\n\\begin{definition}\n{Let }$\\left( X,\\mathcal{O}_{X}\\right) $ be a scheme.\n{Denote by $\\Gamma_{ 0}$ (respectively, $\\Gamma^{max}$)\nthe union of\nthe set of some (respectively, all) identities of commutative rings}\n$$id_{A_{\\alpha }}:A_{\\alpha }\\rightarrow A_{\\alpha }$$ {and\nthe set of some (respectively, all) isomorphisms of commutative rings}\n$$\\sigma _{\\alpha \\beta }:\\left( A_{\\alpha }\\right) _{f_{\\alpha\n}}\\rightarrow \\left( A_{\\beta }\\right) _{f_{\\beta }},$$\n{satisfying the conditions 1-2:}\n\\begin{enumerate}\n\\item {Each} $ A_{\\alpha },A_{\\beta },A_{\\gamma }\\in Comm$\n{are commutative rings such that there are affine open subsets}\n$$U_{\\alpha },U_{\\beta }, { and \\, }U_{\\gamma }\\subseteq U_{\\alpha\n}\\cap U_{\\beta }$${of} $X$ {satisfying the conditions}\n$$\\varphi _{\\alpha }\\left( U_{\\alpha }\\right) =SpecA_{\\alpha\n}, \\varphi _{\\beta }\\left( U_{\\beta }\\right) =SpecA_{\\beta\n},{ and \\,} \\varphi _{\\gamma }\\left( U_{\\gamma }\\right)\n=SpecA_{\\gamma }.$$\n\n\\item {Each} $\\sigma _{\\alpha \\beta }:\\left( A_{\\alpha\n}\\right) _{f_{\\alpha }}\\rightarrow \\left( A_{\\beta }\\right)\n_{f_{\\beta }}${is induced from the homeomorphism}\n$$\\varphi _{\\alpha }\\circ \\varphi _{\\beta }^{-1}\\mid _{U_{\\gamma}}:\n\\varphi _{\\beta }({U_{\\gamma}})\\rightarrow\\varphi _{\\alpha}({U_{\\gamma}})$$\n{such that}\n$$\n\\varphi _{\\alpha }\\left( U_{\\gamma }\\right) \\cong Spec\\left( A_{\\alpha\n}\\right) _{f_{\\alpha }}\\text{{ and }}\\varphi _{\\beta }\\left( U_{\\gamma\n}\\right) \\cong Spec\\left( A_{\\beta }\\right) _{f_{\\beta }}\n$$\n{hold for some} $f_{\\alpha }\\in A_{\\alpha }$ {and\n}$f_{\\beta }\\in A_{\\beta }.$\n\\end{enumerate}\n\n{Then the pseudogroup generated by\n$\\Gamma_{0}$ in $\\mathfrak{Comm}$, denoted by\n$\\Gamma_{X,\\mathcal{O}_{X}}$, which is the smallest pseudogroup\ncontaining $\\Gamma_{0}$ in $\\mathfrak{Comm}$, is called \\textbf{a\npseudogroup of affine transformations} in $\\left(\nX,\\mathcal{O}_{X}\\right) $.}\n\n{The pseudogroup generated by\n$\\Gamma^{max}$ in $\\mathfrak{Comm}$, denoted by\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$, is called \\textbf{the maximal\npseudogroup of affine transformations} in $\\left(\nX,\\mathcal{O}_{X}\\right) $.}\n\n{For any given $\\Gamma_{X,\\mathcal{O}_{X}}$, define} $$\\mathcal{A}^{\\ast }\n\\left( \\Gamma_{ X,\\mathcal{O}_{X}}\n\\right) =\\{\\left( U_{\\alpha },\\varphi _{\\alpha }\\right) :\\varphi _{\\alpha\n}\\left( U_{\\alpha }\\right) =SpecA_{\\alpha } \\text{ and }A_{\\alpha }\\in\n\\Gamma_{ X,\\mathcal{O}_{X}} \\}$${where each $U_{\\alpha}$ is an\naffine open subset in the scheme $X$.}\n\\end{definition}\n\n\\begin{definition}\n{Let }$\\left( X,\\mathcal{O}_{X}\\right) $ be a scheme.\n{Given such a pseudogroup $\\Gamma_{\nX,\\mathcal{O}_{X}}$ in $\\left(\nX,\\mathcal{O}_{X}\\right)$. Suppose that $\\mathcal{A}^{\\ast }\\left(\n\\Gamma_{ X,\\mathcal{O}_{X}}\n\\right)$ is an affine $\\Gamma_{ X,\\mathcal{O}_{X}}\n-$atlas on the space $X$.} {Then $\\Gamma_{\nX,\\mathcal{O}_{X}}$ is said to be a \\textbf{canonical pseudogroup of affine transformations}\nin\nthe scheme $\\left( X,\\mathcal{O}_{X}\\right)$ and $\\mathcal{A}^{\\ast }\\left(\n\\Gamma_{ X,\\mathcal{O}_{X}}\n\\right)$ is said to be an \\textbf{affine atlas} in the scheme $(X,\\mathcal{O}_{X})$.}\n\\end{definition}\n\nIt is immediate that $\\Gamma_{X,\\mathcal{O}_{X}}$ is a sub-pseudogroup of $\\Gamma^{max}_{X,\n\\mathcal{O}_{X}}$.\nThere can be many canonical pseudogroups of affine transformations in the scheme\n$(X,\\mathcal{O}_{X})$. By Zorn's Lemma it is seen that\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$ is maximal among these pseudogroups.\n\n\nTake an example. Let $X=Spec(\\mathbb{Z})$ and $Y$ be the disjoint union of $X$.\nThen there are three canonical pseudogroups of affine transformations in the\nscheme $Y$,\nwhich are generated respectively by $\\mathbb{Z}$ and its localisations, by\n$\\mathbb{Z}\\oplus\\mathbb{Z}$\nand its localisations, and by $\\mathbb{Z}$ and $\\mathbb{Z}\\oplus\\mathbb{Z}$\nand their localisations.\n\n\\subsection{Canonical Affine Structures}\n\nIn general, the underlying space of a scheme can have many affine structures on it.\n\n\\begin{definition}\n{Let $\\Gamma$ be a canonical\npseudogroup of affine transformations in a scheme} $\\left(\nX,\\mathcal{O}_{X}\\right) .$\n\n{An affine $\\Gamma-$atlas $\\mathcal{A}$ on the space $X$ is\nsaid to be \\textbf{a canonical affine structure} in the scheme $(X,\\mathcal{O}_{X})$\nif $\\mathcal{A}$\nis the affine $\\Gamma-$structure on $X$ determined by the\naffine $\\Gamma-$atlas $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right)$.}\n\n{An affine $\\Gamma-$atlas $\\mathcal{A}$ on the space $X$\nis said to be \\textbf{a relative canonical affine structure} in the scheme\n$(X,\\mathcal{O}_{X})$ if\n$\\mathcal{A}$ is maximal among all the\naffine $\\Gamma-$atlases in $(X,\\mathcal{O}_{X})$ which contain the\naffine $\\Gamma-$atlas $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right)$ and are\n$\\Gamma-$compatible.}\n\n{A scheme is said to have \\textbf{a unique (}respectively,\n\\textbf{relative}\\textbf{)\ncanonical affine structure} if there exists only one (respectively, relative)\ncanonical affine\nstructure in it.}\n\\end{definition}\n\n\\begin{proposition}\n{Let }$\\Gamma ${ be the maximal pseudogroup of affine\ntransformations in a scheme} $\\left( X,\\mathcal{O}_{X}\\right) .$ {Then }\n\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) ${ is a relative canonical affine $\\Gamma\n-$structure in} $\\left(\nX,\\mathcal{O}_{X}\\right) .$\n\\end{proposition}\n\n\\begin{proof}\nProve $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ is a $\\Gamma -$atlas\nof the space $X.$ In fact, it is clear that $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ is $\\Gamma -$compatible. Hence, it suffices to prove that\n$\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ affords us a base for the topology on\nthe space $X.$\n\nFixed any point $x\\in X$. Take any affine open subset $U\\ni x$ in $\\left( X\n\\mathcal{O}_{X}\\right) $ such that there is an isomorphism $$\\left( \\varphi ,\n\\widetilde{\\varphi \n\\right) :\\left( U,\\mathcal{O}_{X}\\mid _{U}\\right) \\cong \\left( SpecA\n\\widetilde{A}\\right) $$ where $A\\in \\mathfrak{Comm}.$\n\nThere is some $f\\in A$ such that $A_{f}$ is contained in $\\Gamma$.\nHypothesize that $A_{f}\\not\\in \\Gamma $ holds for any $f\\in A.$ Then\n$\\{\\left( U,\\varphi \\right) \\}$and $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ are\n$\\Gamma -$compatible; it follows that $\\{id_{A}\\}\\bigcup\\Gamma$ is a\npseudogroup of affine transformations in $(X,\\mathcal{O}_{X})$,\nwhere $id_{A}:A\\rightarrow A$ is the identity map; hence, we have\n$$\\Gamma \\subsetneqq \\{id_{A}\\}\\bigcup\\Gamma,$$ which will be in\ncontradiction with the assumption.\n\nNow take any $f \\in A$ such that $A_{f}\\in \\Gamma $. Let $SpecA$ be\nirreducible without loss of generality. We have $$Spec(A_{f})\\cong\nD(f)\\subseteq SpecA.$$ Then $\\{\\left(\nU,\\varphi \\right) \\}$ and $\\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $ are $\\Gamma -\ncompatible.\n\nAs $U$ is an affine open subset in $X$, we have $\\left(\nU,\\varphi \\right)\\in \\mathcal{A}^{\\ast }\\left( \\Gamma \\right) $; as $\\Gamma $ is\nmaximal in $\\left( X,\\mathcal{O}_{X}\\right) ,$ it is seen that $A$ is\ncontained in $\\Gamma .$\n\nThis proves that for any $x\\in X$ there is an affine chart $\\left(\nU,\\varphi \\right) \\in \\mathcal{A}^{\\ast }\\left( \\Gamma \\right)$ such that\n$x\\in U$.\n\\end{proof}\n\n\\begin{remark}\nLet $\\left( X,\\mathcal{O}_{X}\\right) $ be a scheme. Then\n$\\mathcal{A}^{\\ast }\\left( \\Gamma_{X,\\mathcal{O}_{X}} \\right)\n\\subseteqq \\mathcal{A}^{\\ast }\\left( \\Gamma^{m}_{X,\\mathcal{O}_{X}}\n\\right).$\nIn particular, each relative canonical affine\n$\\Gamma_{X,\\mathcal{O}_{X}}-$structure\nin $(X,\\mathcal{O}_{X})$ is contained in $\\mathcal{A}^{\\ast }\\left( \\Gamma^{m}_{X,\n\\mathcal{O}_{X}} \\right)$.\nIn general, there can be different relative canonical affine structures in a scheme.\n\\end{remark}\n\nFurthermore, we have the following conclusions.\n\n\\begin{proposition}\nLet $\\left( X,\\mathcal{O}_{X}\\right)$ be a scheme. There are the following statements.\n\\begin{enumerate}\n\\item {Let $\\Gamma$ be a canonical pseudogroup\nof affine transformations in $\\left( X,\\mathcal{O}_{X}\\right)$. Then there is a\nunique (respectively, relative)\ncanonical affine $\\Gamma-$structure in $\\left( X,\\mathcal{O}_{X}\\right)$.}\n\n\\quad Furthermore, given any affine open subset $U$ in $\\left( X,\\mathcal{O}_{X}\\right)$. Then\n$U$ is contained in the canonical affine $\\Gamma-$structure in $\\left( X,\\mathcal{O}_{X}\\right)$\nif and only if $U$ is contained in the relative canonical affine $\\Gamma-$structure\nin $\\left( X,\\mathcal{O}_{X}\\right)$.\n\n\\item {The scheme $\\left( X,\\mathcal{O}_{X}\\right)$ has a unique affine structure\nif and only if $\\left( X,\\mathcal{O}_{X}\\right)$\nhas a unique relative affine structure.}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n$\\emph{1}$. It is immediate from definition.\n\n$\\emph{2}$. Assume that $\\left( X,\\mathcal{O}_{X}\\right)$ has a unique affine structure.\nHypothesize that $\\mathcal{A}(\\Gamma _{1})$\nand $\\mathcal{A}(\\Gamma _{2})$ are two distinct relative canonical affine structures\nin $\\left( X,\\mathcal{O}_{X}\\right)$ together\nwith the canonical\npseudogroups $\\Gamma_{1}$ and $\\Gamma_{2}$ respectively.\n\nFrom $\\Gamma_{1}$ and $\\Gamma_{2}$ we obtain two canonical affine structures\n$\\mathcal{B}(\\Gamma _{1})$\nand $\\mathcal{B}(\\Gamma _{2})$ in $(X,\\mathcal{O}_{X})$. Then $\\mathcal{B}(\\Gamma _{1})$\nand $\\mathcal{B}(\\Gamma _{2})$ are neither $\\Gamma_{1}-$compatible nor\n$\\Gamma_{2}-$compatible. Otherwise, if they are $\\Gamma_{1}-$compatible, by $(i)$\nit will be seen that\n$\\mathcal{A}(\\Gamma _{1})$\nand $\\mathcal{A}(\\Gamma _{2})$ are $\\Gamma_{1}-$compatible.\n\nHence, there are two distinct canonical affine structures in $\\left( X,\\mathcal{O}_{X}\n\\right)$,\nwhich is in contradiction to the assumption.\n\nConversely, assume that $\\left( X,\\mathcal{O}_{X}\\right)$ has a unique relative affine\nstructure.\nIf $(X,\\mathcal{O}_{X})$ has two distinct canonical affine structures $\\mathcal{B}\n(\\Gamma _{1})$\nand $\\mathcal{B}(\\Gamma _{2})$, by (1) we will obtain two relative\ncanonical affine structures in $\\left( X,\\mathcal{O}_{X}\\right)$ which are neither\n$\\Gamma_{1}-$compatible nor\n$\\Gamma_{2}-$compatible in virtue of the property of a base for the topology on $X$,\nwhere there will be a contradiction.\n\\end{proof}\n\n\\subsection{Associate Schemes}\n\nIn the following it will be seen that a scheme can have many associate schemes.\n\n\\begin{proposition}\nAll (respectively,\nrelative) canonical affine structures in a scheme $\\left( X,\\mathcal{O}_{X}\\right)$ are admissible; moreover,\ntheir extensions are all isomorphic to $\\left( X,\\mathcal{O}_{X}\\right)$ as schemes.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $ be a (respectively, relative)\ncanonical\naffine structure on $X.$ Take any $\\left( U_{\\alpha },\\varphi _{\\alpha\n}\\right) \\in \\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) .$ There is\nthe isomorphism\n\\begin{equation*}\n\\left( \\tau _{\\alpha },\\widetilde{\\tau _{\\alpha }}\\right) :\\left( U_{\\alpha\n},\\mathcal{O}_{X}\\mid _{U_{\\alpha }}\\right) \\cong \\left( SpecA_{\\alpha }\n\\widetilde{A_{\\alpha }}\\right)\n\\end{equation*}\nwhere $$\\varphi _{\\alpha }\\left( U_{\\alpha }\\right) =\\tau _{\\alpha }\\left(\nU_{\\alpha }\\right) ;$$ $$\\widetilde{\\tau _{\\alpha }}\\left( \\widetilde\nA_{\\alpha }}\\right) =\\tau _{\\alpha \\ast }\\mathcal{O}_{X}\\mid _{U_{\\alpha }}.$$\n\nThis proves that the scheme $(X,\\mathcal{O}_{X})$ is at least an extension of\n$\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $.\nIt follows that $\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $ is admissible.\n\nTake any extension $(X,\\mathcal{F})$ of $\\mathcal{A}^{\\ast }\\left( X,\\mathcal{O}_{X}\\right) $.\nBy gluing sections, it is seen that $(X,\\mathcal{F})$ and $\\left( X,\\mathcal{O}_{X}\\right)$ are\nisomorphic schemes.\n\\end{proof}\n\n\\begin{definition}\nAn \\textbf{associate\nscheme} of a given scheme $\\left( X,\\mathcal{O}_{X}\\right) $ is an extension on the space $X$ of a\ncanonical affine structure or a relative canonical affine structure in $\\left( X,\\mathcal{O\n_{X}\\right) $.\n\\end{definition}\n\n\\begin{remark} By Proposition 4.3 we have the following statements:\n\n$1.$ Every scheme has an associate scheme.\nIn particular, a scheme is an associate scheme of itself.\n\n$2.$ All associate schemes of a given scheme are\nisomorphic as schemes.\n\\end{remark}\n\n\n\\section{Statements of the Main Theorems}\n\nIn the paper there are two main theorems that will be stated in the following.\n\n\\subsection{Definitions and Notations}\n\nLet's fix notations and terminology.\n\nFor a topological space $X$,\nput\n\\begin{itemize}\n\\item $\\mathbb{A}\\left( X\\right)\\triangleq$ \\emph{the set of all admissible affine structures\non the space} $X$.\n\\end{itemize}\nFor a scheme $\\left( X,\\mathcal{O}_{X}\\right)$, set\n\\begin{itemize}\n\\item $ \\mathbb{A}_{0}\\left(\nX,\\mathcal{O}_{X}\\right) \\triangleq$ \\emph{the set of all the relative\ncanonical affine structures in the scheme} $\\left( X,\\mathcal{O}_{X}\\right)$.\n\\end{itemize}\n\nLikewise, we can define $ \\mathbb{A}\\left( X;k\\right)$ and $\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X};k\\right) $ for $k-$affine\nstructures.\n\n\n\\begin{definition}\n{For two spaces $X$ and $Y$, we say\n$$\n\\mathbb{A}\\left( X\\right) \\subseteq\\mathbb{A}\\left( Y\\right)\n$$\nif the below condition is satisfied:}\n\\begin{quotation}\nGiven any affine chart $\\left( U_{\\alpha},\\varphi_{\\alpha}\\right) $\ncontained in an affine structure\n$\\mathcal{A}\\left( X\\right) $ belonging to $\\mathbb{A}\\left( X\\right)$. There\nis an affine chart $\\left( V_{\\alpha},\\psi_{\\alpha}\\right) $ contained in an\naffine structure $\\mathcal{A}\\left( Y\\right) $ belonging to $\\mathbb{A\n\\left( Y\\right) $ such that $B_{\\alpha}=A_{\\alpha}$. Here\n$\nA_{\\alpha},B_{\\alpha}\\in\\mathfrak{Comm}$, $\\varphi_{a}\\left( U_{\\alpha}\\right)\n=SpecA_{\\alpha}$, and $ \\psi_{\\alpha\n}\\left( V_{\\alpha}\\right) =SpecB_{\\alpha}.$\n\\end{quotation}\n\\end{definition}\n\n\\begin{definition}\nFor two spaces $X$ and $Y$, we say\n$$\n\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)\n$$\nif there are relations\n\\begin{equation*}\n\\mathbb{A}\\left( X\\right) \\subseteq\\mathbb{A}\\left( Y\\right) \\text{ and \n\\mathbb{A}\\left( X\\right) \\supseteq\\mathbb{A}\\left( Y\\right).\n\\end{equation*}\n\\end{definition}\n\nLikewise, replacing admissible by relative canonical, we define $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) =\\mathbb{A}_{0}\\left( Y,\n\\mathcal{O}_{Y}\\right) $$\nfor two schemes $\\left( X,\\mathcal{O}_{X}\\right) $ and $\\left( Y,\\mathcal{O\n_{Y}\\right) .$\n\n\\begin{definition}\nFor two spaces $X$ and $Y$, we say $$ \\mathbb{A}\\left( X\\right)\n\\trianglelefteq\\mathbb{A}\\left( Y\\right) $$ if the below conditions 1-3 are\nsatisfied:\n\\begin{enumerate}\n\\item \\textbf{(Local Isomorphism)} Given any affine chart $\\left( U_{\\alpha\n},\\varphi_{\\alpha}\\right) $ contained in an affine structure $\\mathcal{A}\n\\left( X\\right) $ belonging to $\\mathbb{A}\\left( X\\right)$.\n\n\\quad Then there is an affine chart\n$\\left( V_{\\alpha},\\psi_{\\alpha}\\right) $ contained in an\naffine structure $\\mathcal{A}\\left( Y\\right) $ belonging to $\\mathbb{A}\n\\left( Y\\right) $ such that $A_{\\alpha}$ and $B_{\\alpha}$ are isomorphic\nrings. Here $ A_{\\alpha},B_{\\alpha}\\in\\mathfrak{Comm}$,\n$\\varphi_{a}\\left( U_{\\alpha}\\right) =SpecA_{\\alpha}$, and $\\psi_{\\alpha\n}\\left( V_{\\alpha}\\right) =SpecB_{\\alpha}. $\n\n\\item \\textbf{(Covering)} Let $\\{\\left( U_{\\alpha },\\varphi_{\\alpha}\\right)\n\\}_{\\alpha\\in\\Gamma}$ be a family of affine charts $\\left( U_{\\alpha\n},\\varphi_{\\alpha}\\right) $ contained in some affine structures\n$\\mathcal{A}(\\Gamma_{\\alpha})$ belonging to $\\mathbb{A}\\left( X\\right)$\nsuch that $\\varphi_{a}\\left( U_{\\alpha}\\right)$ $ =SpecA_{\\alpha}$ and\n$\\bigcup_{\\alpha\\in\\Gamma}U_{\\alpha }\\supseteq X$.\n\n\\quad Then\n$\\bigcup_{i,\\alpha}V_{i,\\alpha }\\supseteq Y$ holds, where $V_{i,\\alpha }$\nruns through all the affine charts $(V_{i,\\alpha },\\psi_{i,\\alpha})$ contained in\nany affine structures $\\mathcal{A}(\\Gamma_{i,\\alpha})$ belonging to $\\mathbb{A\n\\left( Y\\right) $ such that $B_{i,\\alpha }\\cong A_{\\alpha}$ and\n$\\psi_{i,\\alpha}(B_{i,\\alpha })=SpecB_{i,\\alpha }$.\n\n\\item \\textbf{(Filtering)} Let $\\left( U_{\\alpha },\\varphi_{\\alpha}\\right)$ and\n$(U_{\\beta},\\varphi_{\\beta})$ be two affine charts contained in some affine\nstructures $\\mathcal{A}(\\Gamma_{\\alpha})$ and\n$\\mathcal{A}(\\Gamma_{\\beta})$ belonging to $\\mathbb{A}\\left( X\\right)$\nrespectively. Given any $x_{\\alpha}\\in SpecA_{\\alpha}$ and $x_{\\beta}\\in\nSpecA_{\\beta}$ with\n$\\varphi^{-1}_{\\alpha}(x_{\\alpha})=\\varphi^{-1}_{\\beta}(x_{\\beta})$, where\n$\\varphi_{\\alpha}\\left( U_{\\alpha}\\right) =SpecA_{\\alpha}$ and\n$\\varphi_{\\beta}\\left( U_{\\beta}\\right) =SpecA_{\\beta}$.\n\n\\quad Then there exist\naffine charts $\\left( V_{\\alpha },\\psi_{\\alpha}\\right)$ and\n$(V_{\\beta},\\psi_{\\beta})$ respectively contained in some affine structures\n$\\mathcal{A}^{\\prime}(\\Gamma^{\\prime}_{\\alpha})$ and\n$\\mathcal{A}^{\\prime}(\\Gamma^{\\prime}_{\\beta})$ belonging to\n$\\mathbb{A}\\left( Y\\right)$ such that\n$\\psi^{-1}_{\\alpha}\\circ\\sigma_{\\alpha}(x_{\\alpha})=\\psi^{-1}_{\\beta}\\circ\n\\sigma_{\\beta}(x_{\\beta})$\nholds and that there are ring isomorphisms $\\delta\n_{\\alpha}:B_{\\alpha}\\cong A_{\\alpha}\\text{ and }\\delta_{\\beta\n}:B_{\\beta}\\cong A_{\\beta},$\n where $\\psi_{\\alpha}(V_{\\alpha})=SpecB_{\\alpha}$, $\\psi_{\\beta}(V_{\\beta})=SpecB_{\\beta}$,\n and $\\sigma\n_{\\alpha}:SpecA_{\\alpha}\\rightarrow SpecB_{\\alpha}\\text{ and }\\sigma_{\\beta\n}:SpecA_{\\beta}\\rightarrow SpecB_{\\beta}$ are the isomorphisms induced\nfrom $\\delta _{\\alpha}$ and $\\delta_{\\beta }$, respectively.\n\\end{enumerate}\n\\end{definition}\n\n\n\nLikewise, replacing admissible by relative canonical, we define $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\unlhd\\mathbb{A}_{0}\\left( Y\n\\mathcal{O}_{Y}\\right) $$\nfor two schemes $\\left( X,\\mathcal{O}_{X}\\right) $ and $\\left( Y,\\mathcal{O\n_{Y}\\right) .$\n\n\nSuch an isomorphism $\\delta_{\\alpha}:A_{\\alpha}\\cong B_{\\alpha}$ is\ncalled a \\emph{deck transformation} from $X$ into $Y.$\n\n\\begin{definition}\nFor two spaces $X$ and $Y$, we say\n$$\n\\mathbb{A}\\left( X\\right) \\cong\\mathbb{A}\\left( Y\\right)\n$$\nif there are relations $$\\mathbb{A}\\left( X\\right) \\trianglelefteq\\mathbb{A}\\left(\nY\\right) \\text{ and }\\mathbb{A}\\left( X\\right) \\trianglerighteq\\mathbb{A}\\left(\nY\\right) .$$\n\\end{definition}\n\nLikewise, replacing admissible by relative canonical, we define $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y\n\\mathcal{O}_{Y}\\right) $$\nfor two schemes $\\left( X,\\mathcal{O}_{X}\\right) $ and $\\left( Y,\\mathcal{O\n_{Y}\\right) .$\n\n\n\\subsection{Statements of the Main Theorems}\n\nNow we give the statements of the two main theorems of the present paper.\n\n\\begin{theorem}\nLet $X$ and $Y$ be two topological spaces such\nthat either $\\mathbb{A}\\left( X\\right)\\not=\\emptyset$ or $\\mathbb{A}\\left(\nY\\right)\\not=\\emptyset$ holds. Then $X$ and $Y$ are homeomorphic if and\nonly if there is $$ \\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right).$$\n\\end{theorem}\n\n\\begin{theorem}\nAny two schemes $\\left( X,\\mathcal{O}_{X}\\right)\n$ and $ \\left( Y, \\mathcal{O}_{Y}\\right) $ are isomorphic if and only if we have $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y,\n\\mathcal{O}_{Y}\\right) .$$\n\\end{theorem}\n\nWe will prove Theorems 5.1 and 5.2 in \\S 7 and \\S 8, respectively.\n\n\\begin{remark}\nFrom the two main theorems above it is seen that the whole of affine structures on a space and the underlying space of a scheme, as local data of the space, encode the global data of the space and the scheme, in particular, the global topology of the space and the scheme, respectively.\n\\end{remark}\n\n\\begin{remark}\nIn Theorem 5.2 the condition $$ \\mathbb{A}_{0}\\left(\nX,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y,\n\\mathcal{O}_{Y}\\right)$$ can not be replaced by $$ \\mathbb{A}\\left(\nX,\\mathcal{O}_{X}\\right) = \\mathbb{A}\\left( Y, \\mathcal{O}_{Y}\\right).$$\nFor example, consider $X=Spec({\\mathbb{Q}})$ and\n$Y=Spec({\\mathbb{Q}}(\\sqrt{2}))$.\n\\end{remark}\n\n\n\n\\section{Concluding Remarks}\n\nFrom Remark 4.2 and Theorem 5.2 we have the following proposition, a comparison between two schemes of the same underlying space.\n\n\\begin{proposition}\n{Let $\\left( X,\\mathcal{O}_{X}\\right) $ and $ \\left(\nX,\\mathcal{O}^{\\prime}_{X}\\right)$ be two schemes. The\nfollowing statements are equivalent.}\n\\begin{itemize}\n\\item $\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong\n\\mathbb{A}_{0}\\left(X, \\mathcal{O}^{\\prime}_{X}\\right) $ holds.\n\n\\item $\\left( X,\\mathcal{O}_{X}\\right) $ and $ \\left(\nX,\\mathcal{O}^{\\prime}_{X}\\right)$ are isomorphic schemes.\n\n\\item There is an isomorphism $ \\left(\nX,\\mathcal{O}_{X}^{\\symbol{94}}\\right) \\cong \\left(\nX,\\mathcal{O}_{X}^{\\prime \\symbol{94}}\\right)$ for any associate\nschemes $\\left( X,\\mathcal{O}_{X}^{\\symbol{94}}\\right)$ of $\\left(\nX,\\mathcal{O}_{X}\\right) $ and $\\left( X,\\mathcal{O}_{X}^{\\prime\n\\symbol{94}}\\right) $ of and $\\left( X,\\mathcal{O}_{X}^{\\prime\n}\\right) $.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{example}\n{Let $K\/k$ be a Galois extension. Then $SpecK$ has a unique\nassociate scheme and there exists a unique admissible $k-$affine\nstructure in the scheme $SpecK$.}\n\\end{example}\n\n\\begin{definition}\n{A scheme $\\left( X,\\mathcal{O}_{X}\\right)$ is said to\n\\textbf{have a property} $P$ \\textbf{for an admissible affine\nstructures} $\\mathcal{A}$ on $X$ if as a scheme any extension\n$\\left( X,\\mathcal{O}_{ \\mathcal{A}\\left( \\Gamma \\right) }\\right)$\nof $\\mathcal{A}$ has that property $P$.}\n\\end{definition}\n\n\\begin{remark}\nLet $\\left( X,\\mathcal{O}_{\\mathcal{X}}\\right) $ be a scheme. There are the following conclusions.\n\\begin{itemize}\n\\item Fixed an associate scheme\n$(X,\\mathcal{O}_{\\mathcal{A}})$ of\n$(X,\\mathcal{O}_{X})$. In general, it is not true that $\\left( X,\\mathcal{O}_{\\mathcal{A\n}\\right)=(X,\\mathcal{O}_{X}) $ although they are isomorphic.\n\n\\item {There can be a scheme }$\\left( X,\\mathcal{O}_{X}\\right) $\n and an admissible affine structure $\\mathcal{A}$ on\nthe space $X$ such that there is some property $P$\nthat $\\left(\nX,\\mathcal{O}_{X}\\right) $ holds but an extension $\\left( X,\\mathcal{O}_{\\mathcal{A}}\\right) $ of $\\mathcal{A}$ does not hold.\n\n\\item One says that a scheme $\\left( X,\\mathcal{O}\n_{X}\\right) $ has a property $P$. But it is not specified that the\nproperty $P$ holds for some certain or all the admissible affine\nstructures on the space $X$.\n\n\\quad This situation is very similar to that in differential\ntopology. As usual, a differential manifold is said to have some\nproperty if the property holds for all the differential structures\nuntil such a structure is especially specified.\n\n\\quad It has been known that\nthere is some property $P$ on some manifold $X$ which does not hold for any\nother differential structures on $X$.\n\\end{itemize}\n\\end{remark}\n\n\\begin{remark}\nIn a precise and rigour manner, a scheme is defined to be a ringed space\ntogether with a specified admissible affine structure on it if the affine structures\nare in action in a particular case.\n\\end{remark}\n\n\\begin{remark}\n{Let $\\left( X,\\mathcal{O}_{X}\\right) $ be a\nscheme. For any $\\mathcal{A},\\mathcal{B}\\in \\mathbb{A\n\\left( X\\right) $, we say $$\\mathcal{A}\\sim \\mathcal{B}$$ if and only\nif there is an isomorphism $\\left(\nX,\\mathcal{O}_{\\mathcal{A}}\\right) \\cong \\left(\nX,\\mathcal{O}_{\\mathcal{B}}\\right) .$}\n\n{Then the quotient set $$\\mathbb{A\n\\left( X\\right) \/\\sim $$ is the whole of the schemes on the space $X$\nupon isomorphisms.}\n\\end{remark}\n\n\n\n\n\n\\section{Proof of the First Main Theorem}\n\n\nIn this section we give the proof of Theorems 5.1.\n\n\\begin{proof}\n\\textbf{(Proof of Theorem 5.1)}\nAs $\\mathbb{A}\\left( X\\right)\\not=\\emptyset$ and $\\mathbb{A}\\left(\nY\\right)\\not=\\emptyset$, we can choose two admissible affine structures\non $X$ and $Y$ respectively. Fixed their extensions $\\left(\nX,\\mathcal{O}_{X}\\right) $ and $\\left( Y, \\mathcal{O}_{Y}\\right)$, which are\nschemes.\n\n\n\n$\\implies$. Prove that $\\mathbb{A}\\left( X\\right) \\subseteq\n\\mathbb{A}\\left( Y\\right)$ holds.\n\nIn deed, take any affine chart $\\left( U_{\\alpha },\\varphi _{\\alpha }\\right)\n$ contained in an affine structure $\\mathcal{A}\\left( X\\right) $\nbelonging to $\\mathbb{A}\\left( X\\right) ,$ where $\\varphi _{a}\\left(\nU_{\\alpha }\\right) =SpecA_{\\alpha }$ and $A_{\\alpha }\\in\n\\mathfrak{Comm}.$\n\nThen $\\left( \\tau \\left( U\\right) ,\\varphi _{\\alpha }\\circ\n\\tau ^{-1}\\right) $ is an affine chart contained in an affine structure $$\n\\mathcal{A}\\left( \\tau \\left( X\\right) \\right) =\\mathcal{A}\\left( Y\\right) $$\nbelonging to $\\mathbb{A}\\left( Y\\right) .$ Hence, we have\n$$\\mathbb{A}\\left( X\\right) \\subseteq \\mathbb{A}\\left( Y\\right) .$$\n\nSimilarly, we have $$\\mathbb{A}\\left( X\\right) \\supseteq \\mathbb{A}\\left(\nY\\right).$$ So $$\\mathbb{ A}\\left( X\\right) = \\mathbb{A}\\left( Y\\right).$$\n\n\n$\\impliedby$. Let $\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)\n$. We will prove that there exists a homeomorphism\n$$\\tau:X\\longrightarrow Y.$$ We will proceed in several steps.\n\n\n\n$(i)$ Take an affine chart $\\left( U_{\\alpha},\\varphi_{\\alpha}\\right) $ contained\nin an affine structure\n$\\mathcal{A}\\left( X\\right) $ belonging to $\\mathbb{A}\\left( X\\right)$, where\n$\nA_{\\alpha}\\in\\mathfrak{Comm}$ is a commutative ring and $$\\varphi_{a}\\left( U_{\\alpha}\n\\right) =SpecA_{\\alpha}\n.$$ Then there\nis an affine chart $\\left( V_{\\alpha},\\psi_{\\alpha}\\right) $ contained in an\naffine structure $\\mathcal{A}\\left( Y\\right) $ belonging to $\\mathbb{A\n\\left( Y\\right) $ such that $$ \\psi_{\\alpha\n}\\left( V_{\\alpha}\\right) =SpecA_{\\alpha}.$$\n\nThe converse is true since we have $$\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)\n.$$\n\nLet $\\Sigma$ be the disjoint union of all such open sets $SpecA_{\\alpha}$. Take any\npoints $x,y\\in\\Sigma$.\n\nWe say $$x\\thicksim_{X} y$$ if there exist admissible affine\nstructures $\\mathcal{A}(\\Gamma_{\\alpha})$ and $\\mathcal{A}(\\Gamma_{\\beta})$\ncontained in $\\mathbb{A}\\left( X\\right)$ satisfying the condition:\n\\begin{quotation}\n\\emph{There are affine charts $(U_{\\alpha},\\varphi _{\\alpha })\\in\n\\mathcal{A}(\\Gamma_{\\alpha}),(U_{\\beta},\\varphi _{\\beta })\\in\n\\mathcal{A}(\\Gamma_{\\beta})$ such that $\\varphi _{\\alpha}^{-1}(x)=\n\\varphi _{\\beta }^{-1}(y)$, where $x\\in SpecA_{\\alpha}=\n\\varphi_{\\alpha}(U_{\\alpha})$ and $y\\in\nSpecA_{\\beta}=\\varphi_{\\beta}(U_{\\beta}).$}\n\\end{quotation}\n\nLikewise, we say $$x\\thicksim_{Y} y$$ if there exist admissible affine\nstructures $\\mathcal{A}(\\Gamma_{\\alpha})$ and $\\mathcal{A}(\\Gamma_{\\beta})$\ncontained in $\\mathbb{A}\\left( Y\\right)$ satisfying the condition:\n\\begin{quotation}\n\\emph{There are affine charts $(V_{\\alpha},\\psi _{\\alpha })\\in\n\\mathcal{A}(\\Gamma_{\\alpha}),(V_{\\beta},\\psi_{\\beta })\\in\n\\mathcal{A}(\\Gamma_{\\beta})$ such that $\\psi _{\\alpha}^{-1}(x)= \\psi\n_{\\beta }^{-1}(y)$, where $x\\in SpecA_{\\alpha}=\n\\psi_{\\alpha}(V_{\\alpha})$ and $y\\in\nSpecA_{\\beta}=\\psi_{\\beta}(V_{\\beta}).$}\n\\end{quotation}\n\n\n\n$(ii)$ Let $\\Sigma_{X}$ be the quotient of the set $\\Sigma$ by relation $\\thicksim_{X}$,\nand let\n$$\\pi_{X}:\\Sigma\\longrightarrow \\Sigma_{X}$$ be the canonical map.\n\nProve that there is a bijection $\\rho_{X}$ from the set $\\Sigma_{X}$ onto the set $X$.\n\nIn deed, we have a mapping\n$$\\rho:\\Sigma \\longrightarrow X$$ given by $$z\\longmapsto \\varphi_{\\alpha}^{-1}(z)$$ where\n$(U_{\\alpha},\\varphi\n_{\\alpha })$ is the affine chart contained in an affine structure belonging to\n$\\mathbb{A}(X)$ such that\n$$z\\in SpecA_{\\alpha}=\\varphi_{\\alpha }(U_{\\alpha}).$$ Then we have a map$$\\rho_{X}:\\Sigma_{X}\n\\longrightarrow X,\n\\pi_{X}(z)\\longmapsto \\rho (z).$$\n\nEvidently, $\\rho_{X}$ is a surjection. From the definition\nfor $\\thicksim_{X}$, it is easily seen\nthat $\\rho_{X}$ is an injection. This proves $\\rho_{X}$ is a bijection.\n\nHence, $\\Sigma_{X}$ is a topological space together with the topology on the space $X$.\n\nSimilarly, let $\\Sigma_{Y}$ be the quotient of the set $\\Sigma$ by relation $\\thicksim_{Y}$,\nand let\n$$\\pi_{Y}:\\Sigma\\longrightarrow \\Sigma_{Y}$$ be the canonical map.\nAs $$\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right),$$\nit is clear that there is a bijection $\\rho_{Y}$ from the set $\\Sigma_{Y}$ onto the set $Y$.\nThen\n$\\Sigma_{Y}$ is a topological space together with the topology on the space $Y$.\n\n\n\n$(iii)$ Take any $x,y\\in\\Sigma$. Prove that $$x\\thicksim_{X} y$$ holds if and only if\n$$x\\thicksim_{Y} y$$ holds.\n\nIn deed, let $x\\thicksim_{X} y$, that is, we have $$\\varphi\n_{\\alpha}^{-1}(x)= \\varphi _{\\beta }^{-1}(y)$$ for some affine charts\n$$(U_{\\alpha},\\varphi _{\\alpha })\\in\n\\mathcal{A}(\\Gamma_{\\alpha})\\text{ and }(U_{\\beta},\\varphi _{\\beta })\\in\n\\mathcal{A}(\\Gamma_{\\beta})$$ such that $$x\\in\nSpecA_{\\alpha}=\\varphi_{\\alpha}(U_{\\alpha})\\text{ and }y\\in\nSpecA_{\\beta}=\\varphi_{\\beta}(U_{\\beta}).$$\n\nLet $\\Gamma^{max}_{X,\\mathcal{O}_{X}}$be the maximal\npseudogroup of affine transformations in the scheme $\\left(X,\\mathcal{O}_{X}\\right) $.\n\nWe choose the above open sets\n$U_{\\alpha}$ and $U_{\\beta}$ to be affine open subsets of the scheme $\\left(X,\n\\mathcal{O}_{X}\\right) $. That is, $U_{\\alpha}$ and $U_{\\beta}$\nare contained in the pseudogroup $\\Gamma^{max}_{X,\\mathcal{O}_{X}}$.\n\nThen there is an affine open subset $U_{\\alpha \\beta}$ of $\\left(X,\n\\mathcal{O}_{X}\\right) $ contained in\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$\nsuch that $$\\varphi _{\\alpha}^{-1}(x)\\in U_{\\alpha \\beta} \\subseteq U_{\\alpha}\n\\bigcap U_{\\beta};$$\n $$\\varphi_{\\alpha}(U_{\\alpha \\beta})=Spec(A_{\\alpha})_{f_{\\alpha}};$$\n $$\\varphi_{\\beta}\n(U_{\\alpha \\beta})=Spec(A_{\\beta})_{f_{\\beta}}$$\nfor some $f_{\\alpha}\\in A_{\\alpha}$ and $f_{\\beta} \\in A_{\\beta}$, where the isomorphism\n$\\sigma_{\\alpha \\beta}$ from\n$(A_{\\alpha})_{f_{\\alpha}}$ onto $(A_{\\beta})_{f_{\\beta}}$ is contained in\n$\\Gamma^{max}_{X,\\mathcal{O}_{X}}$.\n\nAs $$\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right),$$ we have affine charts\n$(V_{\\alpha},\\psi_{\\alpha})$\nand $(V_{\\beta},\\psi_{\\beta})$ respectively contained in some affine structures\nbelonging to $\\mathbb{A}(Y)$, where\n$$V_{\\alpha}=\\psi_{\\alpha}^{-1}(SpecA_{\\alpha})\\text{ and\n}V_{\\beta}=\\psi_{\\beta}^{-1}(SpecA_{\\beta}).$$\n\nSet $$V_{\\alpha \\beta}=\\psi_{\\alpha}^{-1}(Spec(A_{\\alpha})_{f_{\\alpha}});$$\n$$V_{\\beta \\alpha}=\\psi_{\\beta}^{-1}(Spec(A_{\\beta})_{f_{\\beta}}).$$\nDenote by $\\psi_{\\beta \\alpha}$ the homeomorphism of $V_{\\beta \\alpha}$ onto\n$V_{\\alpha \\beta}$ which is induced\nfrom $\\sigma_{\\alpha \\beta}$.\n\nIt is easily seen that $(V_{\\alpha \\beta},\\psi_{\\beta}\\circ\\psi_{\\beta \\alpha}^{-1})$\nis an affine chart contained in\nsome admissible affine structure belonging to $\\mathbb{A}(Y)$. In fact, fixed any\nadmissible affine $\\Gamma_{0}-$structure $\\mathcal{A}(\\Gamma_{0})$ on the space $Y$ which\ncontains the affine chart $(V_{\\alpha},\\psi_{\\alpha})$. Let $\\Gamma_{1}$ be the pseudogroup\nof affine transformations\nin $\\frak{comm}$ generated by the union of $\\Gamma_{0}$ and the set of the identity on\n$(A_{\\beta})_{f_{\\beta}}$ and all\nthe possible isomorphisms between the localisations of the rings. Then\n$\\{(V_{\\alpha \\beta},\\psi_{\\beta}\\circ\\psi_{\\beta \\alpha}^{-1})\\}$\nand $\\mathcal{A}(\\Gamma_{1})$ are $\\Gamma_{1}-$compatible. Hence,\n$$(V_{\\alpha \\beta},\\psi_{\\beta}\\circ\\psi_{\\beta \\alpha}^{-1})\\in \\mathcal{A}(\\Gamma_{1}).$$\n\nConsider $$y\\in V_{\\beta\n\\alpha};$$ $$x\\in Spec(A_{\\alpha})_{f_{\\alpha}}\\subseteq SpecA_{\\alpha}.$$ It is evident that $$\\psi_{\\alpha}^{-1}(x)=\n\\psi_{\\beta \\alpha}\\circ \\psi_{\\beta}^{-1}(y)$$ holds since $$\\psi_{\\alpha}^{-1}\n(Spec(A_{\\alpha})_{f_{\\alpha}})\n=\\psi_{\\beta \\alpha}\\circ \\psi_{\\beta}^{-1}(Spec(A_{\\beta})_{f_{\\beta}})=V_{\\alpha \\beta}.$$\nThis proves $$x\\thicksim_{Y}y$$ holds.\n\nIn a similar manner, it is seen that the converse is true.\n\n\n\n$(iv)$ The map from $\\Sigma_{X}$ into $\\Sigma_{Y}$ defined by $$\\pi_{X}(z)\\longmapsto\n\\pi_{Y}(z)$$ for $z\\in \\Sigma$\ngives us a bijection $$\\tau:X\\longrightarrow Y,$$ which is well-defined from $(iii)$.\n\nAll the open sets $SpecA_{\\alpha}$ determine a topology on the set\n$\\Sigma$ in such a manner:\n\\begin{quotation}\n\\emph{A subset $W$ of $\\Sigma$ is open if and only if $\\pi_{X}(W)$ is open\nin $\\Sigma_{X}$.}\n\\end{quotation}\n\nIt follows that $\\Sigma_{X}$ is the quotient space of $\\Sigma$ by $\\pi_{X}$.\nAs $\\mathbb{A}\\left( X\\right) =\\mathbb{A}\\left( Y\\right)$, $\\Sigma_{Y}$ is\nthe quotient space of $\\Sigma$ by $\\pi_{Y}$. Hence, $$\\tau:X\\longrightarrow\nY$$ is a homeomorphism.\n\nThis completes the proof.\n\\end{proof}\n\n\n\n\\section{Proof of the Second Main Theorem}\n\n\n\nIn this section we give the proof of Theorem 5.2.\n\n\\begin{proof}\n\\textbf{(Proof of Theorem 5.2)}\n$\\implies$. Let $$\\tau:\\left( X,\\mathcal{O}_{X}\\right) \\cong \\left( Y,\n\\mathcal{O}_{Y}\\right)$$ be an isomorphism.\n\nAs $$\\tau _{\\ast }\\mathcal{O}_{X}\\cong \\mathcal{O}_{Y},$$ we have\n\\begin{equation*}\n\\begin{array}{l}\n\\left( \\varphi _{\\alpha }^{-1}\\left( SpecA_{\\alpha }\\right) ,\\left(\n\\varphi _{\\alpha }^{-1}\\right) _{\\ast }\\widetilde{A_{\\alpha }}\\right)\\\\\n\n\\cong\\left( U_{\\alpha },\\mathcal{O}_{X}\\mid _{U_{\\alpha }}\\right) \\\\\n\n\\cong \\left( \\tau \\left( U_{\\alpha }\\right) ,\\tau _{\\ast }\\mathcal{O}\n_{X}\\mid _{U_{\\alpha }}\\right) \\\\\n\n\\cong \\left( \\tau \\left( U_{\\alpha }\\right) ,\\mathcal{O}_{Y}\\mid\n_{\\tau \\left( U_{\\alpha }\\right) }\\right) \\\\\n\n\\cong\\left( \\psi _{\\alpha }^{-1}\\left( SpecB_{\\alpha }\\right) ,\\left( \\psi\n_{\\alpha }^{-1}\\right) _{\\ast }\\widetilde{B_{\\alpha }}\\right)\n\\end{array}\n\\end{equation*}\nfor any affine open set $U_{\\alpha}$ of $X$ such that $$\\varphi _{\\alpha }\n\\left( U_{\\alpha }\\right) =SpecA_{\\alpha }$$ and $\\left( U_{\\alpha },\\varphi\n_{\\alpha }\\right) $ is contained in some canonical affine structure\nbelonging\nto $\\mathbb\nA}_{0}\\left( X,\\mathcal{O}_{X}\\right)$.\n\nThen $\\left( \\tau \\left( U_{\\alpha }\\right) ,\\varphi _{\\alpha }\\right) $ is an affine\nchart with $$\\psi _{\\alpha }\\left( \\tau \\left( U_{\\alpha }\\right) \\right)\n=SpecB_{\\alpha },$$ which is contained in some canonical affine structure\nbelonging to $\\mathbb{A}_{0}\\left( Y,\\mathcal{O}_{Y}\\right) $.\n\nBy the\nisomorphism $\\tau$ it is easily seen that the conditions $\\emph{1}$-$\\emph{2}$ in\nDefinition 5.3 are satisfied.\n\nNow let $\\Gamma_{X}$ and $\\Gamma_{Y}$ be the maximal pseudogroups\nof affine transformations in the schemes $\\left( X,\\mathcal{O}_{X}\\right)$\nand $\\left( Y,\\mathcal{O}_{Y}\\right)$, respectively. Via the isomorphism\n$\\tau$, every affine chart in $\\mathcal{A}^{*}(\\Gamma_{X})$ is an affine chart\nin the $\\mathcal{A}^{*}(\\Gamma_{Y})$; the converse is true.\n\nUsing the same\nprocedure in proving Theorem 5.1, we can prove such a claim that\n\\textquotedblleft $x\\thicksim _{X}y$\\textquotedblright\\ for $X$ holds if and\nonly if \\textquotedblleft $x\\thicksim _{Y}y$\\textquotedblright\\ for $Y$\nholds.\n\nIt follows that the condition $\\emph{3}$ in Definition 5.3 is satisfied. Hence, we\nhave\n$$\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\trianglelefteq \\mathbb{A\n_{0}\\left( Y,\\mathcal{O}_{Y}\\right).$$\n\nSimilarly, we prove $$\n\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\trianglerighteq \\mathbb{A\n_{0}\\left( Y,\\mathcal{O}_{Y}\\right).$$ This proves $$\\mathbb{A}_{0}\n\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A}_{0}\\left( Y\n\\mathcal{O}_{Y}\\right).$$\n\n\n\n$\\impliedby$. Put $$\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\cong \\mathbb{A\n_{0}\\left( Y,\\mathcal{O}_{Y}\\right) .$$ We prove that there exists an\nisomorphism from $\\left( X,\\mathcal{O}_{X}\\right) $ onto $\\left(\nY,\\mathcal{O}_{Y}\\right) $.\n\nIn the following we will proceed in several steps in a manner similar to the procedure in proving\nTheorem 5.1.\n\n\n\n$(i)$ Let $\\Gamma_{Y}$ be the maximal pseudogroup of affine\ntransformations in the scheme $\\left(Y,\\mathcal{O}_{Y}\\right) $. We obtain a\nrelative canonical affine $\\Gamma_{Y}-$structure $$\\mathcal{A}\\left(\n\\Gamma_{Y}\\right)\\triangleq\\mathcal{A}^{*}\\left( \\Gamma_{Y}\\right) $$ on\n$Y$. Then $V_{\\alpha}$ is an affine open set in the scheme\n$(Y,\\mathcal{O}_{Y})$ for every $(V_{\\alpha},\\psi_{\\alpha})$ contained in $\\mathcal{A}\\left(\n\\Gamma_{Y}\\right)$.\n\nFor each $(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) $\nwe put\n$$\n\\psi_{\\alpha}(V_{\\alpha})=SpecB_{\\alpha}\n$$\nwhere $B_{\\alpha}$ is a commutative ring contained in the pseudogroup\n$\\Gamma_{Y}$.\n\nFrom Definition 4.1 we have $$\\mathcal{A}\\left( \\Gamma_{Y}\\right) \\supseteq\n\\mathcal{A}^{*}\\left( \\Gamma_{Y}\\right);$$ then\n$$\\bigcup_{(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) }\nV_{\\alpha} \\supseteq Y.$$\n\nLet $\\Sigma$ be the disjoint union of all the open sets $SpecB_{\\alpha}$\nsuch that $$\\psi_{\\alpha}(V_{\\alpha})=SpecB_{\\alpha}\\text{ and\n}(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) .$$ Take\nany points $x,y\\in\\Sigma$.\n\nWe say $$x\\thicksim_{Y} y$$ if there are affine charts $(V_{\\alpha},\\psi _{\\alpha\n}),(V_{\\beta},\\psi_{\\beta })\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) $ such that\n$$\\psi _{\\alpha}^{-1}(x)= \\psi _{\\beta }^{-1}(y),$$ where $$x\\in SpecB_{\\alpha}=\n\\psi_{\\alpha}(V_{\\alpha});$$ $$y\\in SpecB_{\\beta}=\\psi_{\\beta}(V_{\\beta}).$$\n\n\n\n$(ii)$ For each $(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left(\n\\Gamma_{Y}\\right) $, define\n$$\\{(U_{i,\\alpha},\\varphi_{i,\\alpha})\\}_{i\\in I_{\\alpha}}$$ to be the\nset of all the affine charts contained in each relative canonical\naffine structures in the scheme $(X,\\mathcal{O}_{X})$ such that\n$$\\varphi_{i,\\alpha}(U_{i,\\alpha})=SpecA_{i,\\alpha}$$ and that there\nis an isomorphism $$\\delta_{i,\\alpha}: A_{i,\\alpha}\\cong B_{\\alpha}.$$\n\nDenote by $\\Delta_{X}$ the set of all such affine charts\n$(U_{i,\\alpha},\\varphi_{i,\\alpha})$, where $i\\in I_{\\alpha}$ and\n$(V_{\\alpha},\\psi_{\\alpha})\\in \\mathcal{A}\\left( \\Gamma_{Y}\\right) $.\n\nAs $$\\mathbb{A}_{0}\\left( Y,\\mathcal{O}_{Y}\\right) \\unlhd\\mathbb{A}_{0}\\left(\nX, \\mathcal{O}_{X}\\right) ,$$ we have\n$$\\bigcup_{(U_{i,\\alpha},\\varphi_{i,\\alpha})\\in \\Delta_{X}} U_{i,\\alpha} \\supseteq X.$$\n\nLet $\\Sigma^{*}$ be the disjoint union of all the open sets\n$SpecA_{i,\\alpha}$ such that $$A_{i,\\alpha}\\cong B_{\\alpha}\\text{\nand }(U_{i,\\alpha},\\varphi_{i,\\alpha})\\in \\Delta_{X}.$$ Take any\n$x,y\\in \\Sigma^{*}$.\n\nWe say $$x\\thicksim_{X} y$$ if there are affine charts\n$(U_{i,\\alpha},\\varphi _{i,\\alpha }),(U_{j,\\beta},\\varphi _{j,\\beta\n})\\in \\Delta_{X}$ such that $$\\varphi _{i,\\alpha}^{-1}(x)= \\varphi\n_{j,\\beta }^{-1}(y)$$ holds, where $$x\\in SpecA_{i,\\alpha}=\n\\varphi_{i,\\alpha}(U_{i,\\alpha});$$ $$y\\in\nSpecA_{j,\\beta}=\\varphi_{j,\\beta}(U_{j,\\beta}).$$\n\nWe say $$x\\thicksim_{\\Sigma} y$$ if there are affine charts\n$(U_{i,\\alpha},\\varphi _{i,\\alpha }),(U_{j,\\beta},\\varphi _{j,\\beta })\\in\n\\Delta_{X}$ such that $$\\sigma _{i,\\alpha}^{-1}(x)=\\sigma_{j,\\beta\n}^{-1}(y)$$ holds, where $$x\\in SpecA_{i,\\alpha}= \\varphi_{i,\\alpha}(U_{i,\\alpha}), \\\\\ny\\in SpecA_{j,\\beta}=\\varphi_{j,\\beta}(U_{j,\\beta}),$$ and\n$$\\sigma\n_{i,\\alpha}:SpecB_{\\alpha}\\rightarrow SpecA_{i,\\alpha}\\text{ and\n}\\sigma_{j,\\beta }:SpecB_{\\beta}\\rightarrow SpecA_{j,\\beta}$$ are the scheme\nisomorphisms induced from the ring isomorphisms $$\\delta\n_{i,\\alpha}:A_{i,\\alpha}\\cong B_{\\alpha}\\text{ and }\\delta_{j,\\beta\n}:A_{j,\\beta}\\cong B_{\\beta}$$ respectively.\n\n\n\n$(iii)$ Let $\\Sigma^{*}_{X}$ be the quotient of the set $\\Sigma^{*}$ by\n$\\thicksim_{X}$, and let\n$$\\pi_{X}:\\Sigma^{*}\\longrightarrow \\Sigma^{*}_{X}$$ be the canonical map.\nWe have got schemes $\\Sigma^{*}$ and $\\Sigma^{*}_{X}$ in an evident\nmanner. It is clear that $\\pi_{X}$ is a morphism of the schemes.\n\nProve that there is an isomorphism $\\rho_{X}$ from the scheme\n$\\Sigma^{*}_{X}$ onto the scheme $X$.\n\nIn deed, we have a mapping $$\\rho:\\Sigma^{*} \\longrightarrow X$$ given by\n$$z\\longmapsto \\varphi_{i,\\alpha}^{-1}(z),$$ where $(U_{i,\\alpha},\\varphi\n_{i,\\alpha })\\in\\Delta_{X}$ such that $$z\\in SpecA_{i,\\alpha}=\\varphi_{i,\\alpha\n}(U_{i,\\alpha}).$$ Then we have a mapping $$\\rho_{X}:\\Sigma^{*}_{X} \\longrightarrow X,\n\\pi_{X}(z)\\longmapsto \\rho (z).$$\n\nEvidently, $\\rho_{X}$ is a surjection. From the definition for $\\thicksim_{X}$,\nit is seen that $\\rho_{X}$ is an injection. Hence, $\\rho_{X}$ is a\nhomeomorphism from the space $\\Sigma^{*}_{X}$ onto the space $X$. By\nthe construction, it is seen that $\\rho_{X}$ is an isomorphism of the\nschemes.\n\nSimilarly, let $\\Sigma_{Y}$ be the quotient of the set $\\Sigma$ by\n$\\thicksim_{Y}$, and let\n$$\\pi_{Y}:\\Sigma\\longrightarrow \\Sigma_{Y}$$ be the canonical map.\n\nThen $\\Sigma$ and $\\Sigma_{Y}$ are schemes, and $\\pi_{Y}$ is a scheme\nmorphism. There is an isomorphism $\\rho_{Y}$ from the scheme\n$\\Sigma_{Y}$ onto the scheme $Y$.\n\nLet $\\Sigma^{*}_{\\Sigma}$ be the quotient of the set $\\Sigma^{*}$ by\n$\\thicksim_{\\Sigma}$, and let\n$$\\pi_{\\Sigma}:\\Sigma^{*}\\longrightarrow \\Sigma^{*}_{\\Sigma}$$ be the canonical map.\n\nThen $\\Sigma^{*}_{\\Sigma}$ is a scheme and $\\pi_{\\Sigma}$ is a morphism.\nThere is an isomorphism $\\rho_{\\Sigma}$ from the scheme\n$\\Sigma^{*}_{\\Sigma}$ onto the scheme $\\Sigma$.\n\n\n\n$(iv)$ Take any $x,y\\in \\Sigma^{*}$. We prove\n$$\\rho_{X}\\circ\\pi_{X}(x)=\\rho_{X}\\circ\\pi_{X}(y)$$ if and only if\n$$\\rho_{Y}\\circ\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(x)=\\rho_{Y}\\circ\n\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(y).$$\n\nIn deed, let $$\\rho_{X}\\circ\\pi_{X}(x)=\\rho_{X}\\circ\\pi_{X}(y).$$ We have\n$$\\varphi _{i,\\alpha}^{-1}(x)= \\varphi _{j,\\beta }^{-1}(y)$$ for some affine charts\n$$(U_{i,\\alpha},\\varphi _{i,\\alpha }), (U_{j,\\beta},\\varphi _{j,\\beta })\\in\n\\Delta_{X}$$ such that $$x\\in\nSpecA_{i,\\alpha}=\\varphi_{i,\\alpha}(U_{i,\\alpha});$$ $$y\\in\nSpecA_{j,\\beta}=\\varphi_{j,\\beta}(U_{j,\\beta}).$$\n\nAs $$\\mathbb{A}_{0}\\left( X,\\mathcal{O}_{X}\\right) \\unlhd\\mathbb{A}_{0}\\left(\nY, \\mathcal{O}_{Y}\\right) ,$$ we have affine charts $(V_{i,\\alpha},\\psi _{i,\\alpha\n})$ and $ (V_{j,\\beta},\\psi_{j,\\beta })$ contained in\n$\\mathcal{A}(\\Gamma_{Y})$ such that\n$$\\psi^{-1}_{i,\\alpha}\\circ\\sigma\n_{i,\\alpha}^{-1}(x)=\\psi^{-1}_{j,\\beta }\\circ\\sigma_{j,\\beta }^{-1}(y)$$\nwhere\n$$\\psi _{i,\\alpha }(V_{i,\\alpha})=SpecB_{i,\\alpha},\\, \\psi_{j,\\beta\n}(V_{j,\\beta})=SpecB_{j,\\beta}$$ and\n$$\\sigma\n_{i,\\alpha}:SpecB_{i,\\alpha}\\rightarrow SpecA_{i,\\alpha}\\text{ and\n}\\sigma_{j,\\beta }:SpecB_{j,\\beta}\\rightarrow SpecA_{j,\\beta}$$\nare the isomorphisms induced from\n the ring isomorphisms\n $$\\delta\n_{i,\\alpha}:A_{i,\\alpha}\\cong B_{i,\\alpha}\\text{ and }\\delta_{j,\\beta\n}:A_{j,\\beta}\\cong B_{j,\\beta}$$ respectively.\n\nHence, we have\n$$\\rho_{Y}\\circ\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(x)=\\rho_{Y}\\circ\\pi_{Y}\n\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(y).$$\n\nIn a similar manner, it is seen that the converse is true.\n\n\n\n$(v)$ Define a map $\\tau:X\\longrightarrow Y$ by\n$$\\tau(\\rho_{X}\\circ\\pi_{X}(z))=\\rho_{Y}\\circ\\pi_{Y}\\circ\\rho_{\\Sigma}\\circ\\pi_{\\Sigma}(z)$$\nfor every $z\\in \\Sigma^{*}$.\n\nBy $(iv)$ it is seen that $\\tau$ is well-defined. It is immediate that $\\tau$ is the\ndesired scheme isomorphism from $\\left( X,\\mathcal{O}_{X}\\right) $ onto $\n\\left( Y,\\mathcal{O}_{Y}\\right)$.\n\nThis completes the proof.\n\\end{proof}\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Abstract}\nInference of the evolutionary histories of species, commonly represented by a species tree, is complicated by the divergent evolutionary history of different parts of the genome. Different loci on the genome can have different histories from the underlying species tree (and each other) due to processes such as incomplete lineage sorting (ILS), gene duplication and loss, and horizontal gene transfer. The multispecies coalescent is a commonly used model for performing inference on species and gene trees in the presence of ILS. This paper introduces Lily-T and Lily-Q, two new methods for species tree inference under the multispecies coalescent. We then compare them to two frequently used methods, SVDQuartets and ASTRAL, using simulated and empirical data. Both methods generally showed improvement over SVDQuartets, and Lily-Q was superior to Lily-T for most simulation settings. The comparison to ASTRAL was more mixed -- Lily-Q tended to be better than ASTRAL when the length of recombination-free loci was short, when the coalescent population parameter $\\theta$ was small, or when the internal branch lengths were longer.\n\n\\section{Introduction}\nThe phylogenetic inference problem is concerned with using data, including but not limited to DNA sequences, to understand the evolutionary history of a collection of species. Consider the collection of mammals shown in figure \\ref{exampleTree}. We are concerned with three aspects shown in the figure. First, is the unlabeled topology correct? In other words, for each node, how many descendants are there on the left and right branches? Second, is the labeled topology correct, or does it need to be permuted? Third, when do the speciation events occur?\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\linewidth]{\"BigTree\".png}\n \\caption{Example phylogenetic tree featuring six mammalian species.}\n \\label{exampleTree}\n\\end{figure}\n\nWe usually begin with the assumption that when a speciation event occurs each ancestral species divides into exactly two daughter species. Thus, the evolutionary history can be represented as a binary tree known as a species tree.\n\\begin{definition}\nA \\textit{species tree} is an acyclic graph $S=(V(S), E(S), \\boldsymbol{\\tau}_S)$ where $V(S)$ is the vertex set of $S$, $E(S)$ is the edge set of $S$, and $\\boldsymbol{\\tau}_S$ is a set of branch lengths. \n\\end{definition}\n\\label{Species tree}\nBiologically, internal nodes represent speciation events while the leaves represent extant species. We call this leaf set $L_S$. The leaves will be represented by lower case letters $a, b, c,$ etc., and internal nodes by letters later in the alphabet. Internal branches represent ancestral species. If we know the common ancestor of all the species under consideration, then the tree is \\textit{rooted}, and the tree becomes a directed graph from the root outward. If the root is unknown, then we only know the direction of the branches that connect to external nodes. The \\textit{degree} of a node is the number of other nodes a node is connected to. For a species tree, the leaves are of degree one, and due to the binary tree assumption, all internal nodes are of degree three (except the root, if known, which is of degree two).\n\nA common and useful assumption, known as the \\textit{molecular clock}, is that the mutation rate is constant over time (or more precisely, if $\\lambda$ represents the mutation rate there is a common $\\lambda(t)$ for all branches at time $t$). In that case, the rooted tree is \\textit{ultrametric}, meaning that each leaf will be equidistant from the root. For ultrametric trees, an equivalent parameterization to the set of branch lengths $\\boldsymbol{\\tau}_S$ is the set of node times $\\boldsymbol{\\tau}_{S_{nodes}}$. Because we assume the molecular clock, we will simplify the notation so that $\\boldsymbol{\\tau}$ represents the node times for the remainder of the paper.\n\nLet $\\mathcal{S}^{(n)}$ refer to the set of all possible trees with $n$ taxa. The superscript is in parentheses to highlight that it refers to the number of taxa rather than as an exponent. Then, for example, $\\mathcal{S}^{(4)}_1$ and $\\mathcal{S}^{(4)}_2$ can indicate the first and second 4-taxon species trees from figure \\ref{quartets}. If the labeling is unimportant, we can also use $\\mathcal{S}^{(4)}_{u1}$ and $\\mathcal{S}^{(4)}_{u2}$ to indicate the two \\textit{unlabeled} 4-taxon topologies. \n\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{\"rootedQ\".png}\n \\caption{The fifteen rooted quartets. We can label these $\\mathcal{S}^{(4)}_1, \\mathcal{S}^{(4)}_2 \\ldots \\mathcal{S}^{(4)}_{15}$. If we are unconcerned with the labels, the first column is $\\mathcal{S}^{(4)}_{u1}$ and the other columns $\\mathcal{S}^{(4)}_{u2}$. Note that each row corresponds to one of the three unrooted quartets.}\n \\label{quartets}\n\\end{figure} \n\nSpecies tree inference is complicated by the possibility for divergence between the evolutionary history of species and individual elements of their genomes. Causes for this divergence include incomplete lineage sorting (ILS), gene duplication and loss (GDL) and horizontal gene transfer (HGT). ILS is commonly modeled by the coalescent process \\cite{kingman1982}. The history of individual loci on the genome is represented by a gene tree.\n\n\\begin{definition}\nA \\textit{gene tree} is a network $G=(V(G), E(G), \\boldsymbol{t}_G)$ where $V(G)$ is the vertex set of $G$, $E(G)$ is the edge set of $G$, and $\\boldsymbol{t}_G$ is a set of branch lengths. When the molecular clock is assumed, $\\boldsymbol{t}_G$ can equivalently represent node times as with the species tree above and the notation simplified to $\\boldsymbol{t}$. These node times are subject to the constraint that the coalescent events in question must occur prior to the divergence time of the species in question.\n\\end{definition}\n\nThe gene tree is embedded within the species tree (see figure \\ref{coalescent}) and usually both trees have the same leaf set. Exceptions can occur if the gene has not been sampled for all species under consideration or if there are multiple sampled individuals per species. Thus, unless otherwise noted, we will drop the subscript and just refer to the set of species under consideration as $L$. When we need to distinguish the leaves of $G$ from the leaves of $S$ we use capital letters $A, B, C, \\ldots$. Care should be taken, however, to distinguish when $A$ or $C$ are used to denote members of the leaf set and when they are used as abbreviations for nucleotides. Internal nodes represent coalescent events, which identify the most recent common ancestor of two gene lineages. \n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[h]{.48\\linewidth}\n \\centering\n\\includegraphics[width=.9\\linewidth]{\"sym2\".png}\n\\caption{Symmetric species tree}\n\\end{subfigure}\n\\begin{subfigure}[h]{.44\\linewidth}\n\\centering\n\\includegraphics[width=.9\\linewidth]{\"asym2\".png}\n\\caption{Asymmetric species tree}\n\\end{subfigure}\n\\caption{Species trees with four taxa under the coalescent process. The green lines show example gene trees evolving within the underlying species tree.}\n\\label{coalescent}\n\\end{figure}\n\nIt is worth noting here that our definition of a gene, as is common in the phylogenetic literature, refers to a recombination-free region of the genome. Thus, the common assumption is that no recombination occurs within a gene, and all sites in a gene share a common evolutionary history, while sites on separate genes are independent conditional only on the underlying species tree. This differs from the biological definition of a gene as a segment of DNA coding for a polypeptide. There is, of course, no underlying reason why a biological gene can or should share a common history in its entirety. To avoid this confusion, \\textit{locus} is also sometimes used to describe a recombination-free region, however, it is still more common to refer to ``gene trees\" rather than ``locus trees\".\n\nA number of different approaches have been taken with regard to species tree inference in the presence of ILS. The first is essentially to ignore the problem: perform gene tree inference on concatenated data using methods such as RAxML \\cite{raml} or FastTree \\cite{FastTree}, treating all sites as if they share a single, common evolutionary history. This can be fast and accurate for estimating $S$. But, there are some concerns: concatenation has been shown to be statistically inconsistent for some values of $(S,\\boldsymbol{\\tau})$ \\cite{degnan05,roch15}, and speciation time estimates are biased since the coalescent event must naturally occur before the speciation time. Another approach is the use of summary statistics that first estimate the gene trees independently for each gene, and then use the gene tree estimates as inputs for species tree estimates. Examples of this approach include STEM \\cite{kubatko09}, ASTRAL \\cite{mirarab14,mirarab15,zhang18}, and MP-EST \\cite{MPEST}. These methods can be computationally efficient, however they depend on the accuracy of the gene tree estimates that are used as inputs as well as proper delineation of recombination-free segments of the genome \\cite{gatesy14,springer16}. A third approach uses the full data to coestimate the species tree and each of the gene trees, generally using Markov chain Monte Carlo (MCMC) methods. Examples of this approach include BEST \\cite{BEST}, *BEAST \\cite{BEAST,SB2}, and BPP \\cite{BPP}. These methods can be quite accurate but are very computationally intensive when the number of loci and\/or leaf set is large. Assessment of convergence is also a challenge, especially due to the multi-modal nature of the likelihood in the tree space \\cite{salter01}.\n\nA fourth approach, and the one we take in this paper, is to treat the gene trees as a nuisance parameter that can be integrated over. A previous example of this approach is SVDQuartets \\cite{chifman14}, which uses a rank-based methodology to infer the proper unrooted species tree for each quartet of species under consideration, and then uses an assembly algorithm to infer the final n-taxon species tree estimate taking the set of unrooted quartet trees as input. The theory behind the method assumes unlinked coalescent-independent sites (CIS) data such that the gene tree underlying each site can be treated as a random draw from the distribution of all possible gene trees given the species tree, however Wascher and Kubatko (2020)\\nocite{wascher20} recently proved that SVDQuartets is statistically consistent for multilocus data as well.\n\nOur method, \\textbf{(Li)}kelihood-based assemb\\textbf{(ly)} (\\textbf{Lily}), also assumes unlinked CIS data. From \\cite{chifman15} we have the site pattern probabilities given a species tree topology and branching times. Then, a prior on these topologies and branching times is assumed, and posterior probabilities for each set of rooted triplets (Lily-T) or unrooted quartets (Lily-Q) are calculated. These posterior probabilities are then used as weights in an assembly algorithm to infer the final n-taxon estimated topology $\\hat{S}$. The details of the procedures are described in the next section.\n\n\\section{Method}\nThe outline for the Lily-T and Lily-Q procedures are laid out in algorithms \\ref{tProd} and \\ref{Qprod}. For each triplet (Lily-T) or quartet (Lily-Q) of species, first the site pattern frequencies are found. Then, the likelihood for each rooted topology is calculated. Using Bayes's Theorem, the posterior probability of each rooted triplet or unrooted quartet is then determined. Finally, given these posteriors as inputs, the n-taxon topology is estimated using supertree assembly methods. Each step is discussed in detail in the following sections.\n\\begin{algorithm}\n\\SetAlgoLined\n \\For{$l\\gets1$ \\KwTo ${n \\choose 3}$}{\n Find site pattern frequencies $\\boldsymbol{D}_{JC}$ for the $l$\\textsuperscript{th} triplet of species (section \\ref{dataSec})\\;\n Find the site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau)}$ for each of the three rooted triplet topologies from \\cite{chifman15} (section \\ref{method1} and appendix \\ref{tripletApp})\\;\n Integrate over $\\boldsymbol{\\tau}$ using $\\theta=0.003$ and $\\beta$ as estimated from equation \\ref{distEst} to find $\\boldsymbol{\\delta}|S$ and then $L(S|\\boldsymbol{D}_{JC})$ for each rooted triplet (sections \\ref{derivation} and \\ref{robustness})\\;\n From Bayes's Theorem find the posterior probability of each of the three rooted triplets (section \\ref{derivation})\n }\n Using the $3{n \\choose 3}$ posterior probabilities as input, estimate the n-taxon rooted topology using the Triplet MaxCut algorithm \\cite{sevillya16} (section \\ref{assembly})\\;\n \\caption{Lily-T procedure}\n \\label{tProd}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\SetAlgoLined\n \\For{$l\\gets1$ \\KwTo ${n \\choose 4}$}{\n Find site pattern frequencies $\\boldsymbol{D}_{JC}$ for the $l$\\textsuperscript{th} quartet of species (section \\ref{dataSec})\\;\n Find the site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau)}$ for each of the fifteen rooted quartet topologies from \\cite{chifman15} (section \\ref{method1})\\;\n Integrate over $\\boldsymbol{\\tau}$ using $\\theta=0.003$ and $\\beta$ as estimated from equation \\ref{distEst} to find $\\boldsymbol{\\delta}|S$ and then $L(S|\\boldsymbol{D}_{JC})$ for each rooted quartet (sections \\ref{derivation} and \\ref{robustness})\\;\n From Bayes's Theorem find the posterior probability of each of the fifteen rooted quartets (section \\ref{derivation}) \\;\n Calculate the posterior probability of the three unrooted quartet topologies as the sum of the corresponding rooted quartets (see figure \\ref{rooting})\\;\n }\n Given the $3{n \\choose 4}$ posterior probabilities as input, estimate the n-taxon rooted topology using the Weighted QMC algorithm \\cite{avni14} (section \\ref{assembly})\\;\n \\caption{Lily-Q procedure}\n \\label{Qprod}\n\\end{algorithm}\n\n\\subsection{Data structure}\n\\label{dataSec}\nOur data structure and data reduction method for Lily-Q are summarized in figure \\ref{data_map}. We begin by assuming a matrix of aligned sequence data $\\textbf{D}_{raw}$ where the rows represent the species under consideration and the columns each represent an aligned site. We assume in the sequel that this alignment has been performed without error. Even with a correct alignment, there may also be sites present in one sequence and not another, either due to the data truly being missing or because of an insertion or deletion. Thus, $D_{ij} \\in \\{A,C,G,T,-\\}$. Here, $i$ is an index for the taxon, $j$ is the index for the site, the letters represent the four nucleotides, and the dash represents missing data.\n\nIf the data are iid, or if the possibility of varying rates across sites is treated as a random effect that can be integrated over, then the columns are exchangeable. We begin with one subset of four of the $n$ taxa. Let $\\delta_{TCCG}$ represent the probability that at a certain site the first species has A, the next two have C, and the fourth species has G, i.e., $\\delta_{TCCG}=P(i_A=T, i_B=C, i_C=C, i_D=G)$ where $i_x$ is the state for species $x$. As discussed before, the abbreviation C is overloaded so care should be taken to note that $i_C=C$ means the third species has cytosine at this site. Under this assumption of exchangeability, $\\delta_{TCCG}$ will be the same at every site. Then, the number of sites where the first taxon has character $i_A$, the second has character $i_B$, etc., which we label $d_{i_A, i_B, \\ldots i_n}$ will follow a binomial distribution with probability $\\delta_{i_A,i_B, \\ldots i_n}$, and the joint probability of all possible site patterns is a multinomial distribution. Since there are $4^n$ possibilities (or $5^n$ with missing data), the numbers of sites that follow each pattern $d_{i_A, i_B, \\ldots i_n}$ is a sufficient statistic. Then we can map down $\\boldsymbol{D}_{raw}$ down to a $4^n \\times 1$ vector $\\boldsymbol{D}_{ind} \\sim Multinom(J,\\boldsymbol{\\delta}_{ind})$, where each element of $\\boldsymbol{\\delta}$ represents one of the site pattern probabilities and $J$ is the total number of sites. In the sequel, we will only consider sites where all four species in the quartet have a nucleotide present -- i.e., sites where $D_{ij} \\in \\{A,C,G,T\\}$, $i=A,B,C,D$, which will not affect inference if sites are missing at random. \n\nIf we further assume the Jukes-Cantor (JC69) substitution model \\cite{jukes69}, all nucleotides have the same limiting frequency of 1\/4 and all substitution rates between nucleotides are the same. As a result, for determining probabilities, if we use the JC69 model we don't need to keep track of what nucleotides are present where; we only need to note whether or not they are the same. For example with two taxa, $P(i_A=C, i_B=C)=P(i_A=G, i_B=G)$, as is the probability of any two different nucleotides at the same site. So, we can call two identical nucleotides at the same site an XX pattern regardless of whether they represent AA, CC, GG, or TT. Similarly, XY represents the case where the two nucleotides are different. Again, the distribution of the site pattern frequencies follows a multinomial distribution, but we need to keep track of fewer cases. The number of different cases required for $n$ taxa is $\\sum_{i=1}^4 S_{n,i}$ where $S_{n,i}$ is the Sterling number of the second kind. Most relevant for our later discussion, for a four-taxon tree, we can map $\\boldsymbol{D}_{ind}$ down to a $15 \\times 1$ vector $\\boldsymbol{D}_{JC}$. We repeat this mapping for each quartet of species, creating an ${n \\choose 4}$ set of site pattern frequency vectors $\\boldsymbol{D}_{JC}$.\n\nThis data mapping process is similar for Lily-T, except that it is repeated of each of the ${n \\choose 3}$ set of triplets. $\\boldsymbol{D}_{ind}$ then is a $4^3 \\times 1$ vector and $\\boldsymbol{D}_{JC}$ is a $5 \\times 1$ vector.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{\"DataSetup\".png}\n \\caption{Data reduction (Lily-Q): For each set of four species, the raw aligned sequence data can be reduced down to $\\boldsymbol{D}_{ind}$ and then $\\boldsymbol{D}_{JC}$ under the Jukes-Cantor assumptions.}\n \\label{data_map}\n\\end{figure}\n\n\\subsection{Derivation of $L((S,\\boldsymbol{\\tau)|\\boldsymbol{D}_{JC}})$ for 3- or 4-taxon trees}\n\\label{method1}\nChifman and Kubatko (2015)\\nocite{chifman15} derived the site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau})$ for a 4-taxon tree where $\\delta_k|(S,\\boldsymbol{\\tau})$ is the probability of the $k$\\textsuperscript{th} site pattern, $k \\in \\{XXXX, XXXY, \\ldots XYZW\\}$, occurring at a given site given the species tree topology and branching times under the JC69 model and the molecular clock. Then since $\\boldsymbol{D}_{JC} \\sim Multinom(J, \\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}))$ the likelihood is given by \n\n\\begin{equation}\n L((S,\\boldsymbol{\\tau})|\\boldsymbol{D}_{JC}) \\propto \\prod_{k=1}^{K} [\\delta_k|(S,\\boldsymbol{\\tau})]^{d_k} \n \\label{lik1}\n\\end{equation}\n\nTo find $P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(S,\\boldsymbol{\\tau}))$ we first recognize that this can be factored into two processes: the coalescent process and the substitution process:\n$$ P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(S,\\boldsymbol{\\tau}))=P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(G,\\boldsymbol{t}),(S, \\boldsymbol{\\tau}))f((G,\\boldsymbol{t})|(S, \\boldsymbol{\\tau})), $$\n$$=P(\\boldsymbol{D}_{JC}=\\boldsymbol{d_{JC}}|(G,\\boldsymbol{t}))f((G,\\boldsymbol{t})|(S, \\boldsymbol{\\tau})). $$\nwhere the second equality is true under the assumption of neutral selection, whereby the substitution process and coalescent process are independent and then the first term depends directly only on the gene tree \\cite{whidden15}.\n\nWe begin by noting that:\n$$\\delta_{XXXY}|((a,(b,(c,d))),\\boldsymbol{\\tau}))=\\delta_{XYXX}|((a,(d,(c,b))),\\boldsymbol{\\tau})).$$ A similar argument can be made for all fifteen site pattern frequencies and all fifteen rooted 4-taxon topologies. Thus, we need only derive the site pattern probabilities for the two topologies shown in figure \\ref{coalescent}, $((a,b),(c,d))$ and $(a,(b,(c,d)))$ and the site pattern probabilities for the other 13 topologies is a permutation of one of these two cases. The likelihood of any other 4-taxon tree is then given by equation \\ref{lik1} with the data permuted as necessary by the permutation function $\\sigma(\\cdot)$:\n\\begin{equation}\nL((S,\\boldsymbol{\\tau})|\\boldsymbol{D}_{JC}) \\propto \\prod_{k=1}^{K} [\\delta_k|(S,\\boldsymbol{\\tau})]^{\\sigma(d_k)} \n\\label{lik2}\n\\end{equation}\n\n$\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau})$ for the lone unlabeled 3-taxon tree topology can be found be marginalizing over the fourth taxa in figure \\ref{coalescent}. The details are shown in the Appendix.\n\n\\subsection{Derivation of $P(S|\\boldsymbol{D}_{JC})$ for 3- or 4-taxon trees}\n\\label{derivation}\nFrom the Law of Total Probability, it is immediate that: \n$$ \\boldsymbol{\\delta}|S = \\int_{\\boldsymbol{\\tau}} \\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}) f(\\boldsymbol{\\tau}) d\\boldsymbol{\\tau}$$\nA wide variety of forms for the density $f(\\boldsymbol{\\tau})$ can be chosen. We choose priors to be uninformative and to allow for analytic solutions to $\\boldsymbol{\\delta}|S$. Refer to figure \\ref{coalescent} for a description of $\\boldsymbol{\\tau}$ and note that the 3-taxon case can be viewed as the asymmetric case with taxon $a$ removed. For a 3-taxon tree, we choose $\\tau_2 \\sim Exp(\\beta)$ and $\\tau_1|\\tau_2 \\sim U(0,\\tau_2)$. For the 4-taxon symmetric tree, we choose $\\tau_3 \\sim Exp(\\beta)$ and $\\tau_1|\\tau_3,\\tau_2|\\tau_3$ independently $\\sim U(0,\\tau_3)$ For the 4-taxon asymmetric tree, we choose $\\tau_3 \\sim Exp(\\beta)$ and $(\\frac{\\tau_1}{\\tau_3}, \\frac{\\tau_2}{\\tau_3}) \\sim Dirichlet(1,1,1)$. Choosing a prior on the root age gives us a prior on the age of the tree as a whole; setting exponential priors on each branch length leads to the total age of the tree being dependent on the degree of tree symmetry. Given these priors on the branching times, $\\boldsymbol{\\delta}|S$ is calculated as derived in the appendix and the likelihood of any tree can be found as before by taking these site pattern probabilities and applying the standard multinomial likelihood to a permutation of the data:\n$$L(S|\\boldsymbol{D}_{JC}) \\propto \\prod_{k=1}^{K} [\\delta_k|S]^{\\sigma(d_k)} $$\n\nAn unfortunate side effect of this choice of prior is that for four-taxon trees (by a simple application of the law of conditional expectation) $E(\\tau_1)=\\frac{1}{2}\\tau_3$ for the symmetric topology and $E(\\tau_1)=\\frac{1}{3}\\tau_3$ for the asymmetric topology. In other words, if (C,D) is a cherry, inferring an asymmetric topology automatically implies the prior assumption that this divergence time between species C and D occurs more recently. This does not appear to be a problem at first glance, since we are not concerned with inferring species divergence times here, and are treating $\\boldsymbol{\\tau} $ as a nuisance parameter. But, it turns out that this makes accurate inference of the root location of a 4-taxon tree impossible. Given our choice of model, if the true tree is symmetric -- for example $((a,b),(c,d))$ -- then various pairs of rooted asymmetric trees have the same likelihood with speciation times integrated out: $ L[(a,(b,(c,d)))]=L[(b,(a,(c,d)))]$, $ L[(c,(d,(a,b)))]=L[(d,(c,(a,b)))]$, etc. Worse, for some true values of $\\tau_1, \\tau_2, \\tau_3$, $ L[(a,(b,(c,d)))]>L[((a,b),(c,d))]$. These inference errors all concern the location of the root rather than the unrooted topology. As a result, we limit our inference on 4-taxon trees to unrooted topologies. \n\nFor the prior on topologies, we assume the tree generation follows a Yule model: there is a constant rate of species divergence over time, and the rate of species divergence is equal for all branches. This is meant to be as uninformative as possible. For 3-taxon trees, this intuitively assigns a 1\/3 probability to each of the 3 rooted 3-taxon trees. For 4-taxon trees, each of the three symmetric topologies has a 1\/9 prior probability and each of the twelve asymmetric topologies has a 1\/18 prior probability. Interestingly, while any individual symmetric topology is twice as probable as any individual asymmetric topology, in the prior it is twice as probable that the unlabeled topology will be asymmetric rather than symmetric. The details of the prior calculation are given by \\cite{harding71}.\n\nTaking both the prior on the topologies and each topology's likelihood, it is a simple application of Bayes's Theorem in the 3-taxon case to show:\n$$ P(S=s|\\textbf{D})=\\frac{L(S|\\textbf{D})P(S=s)}{\\sum_{i=1}^3 L(S_i|\\textbf{D})P(S_i=s_i)} $$ \nFor the 4-taxon case, the summation is performed over 15 rather than 3 topologies, and the final probability of the unrooted topology is the sum of the five rooted topologies compatible with it (see figure \\ref{rooting}).\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{\"rooting\".png}\n \\caption{Each unrooted 4-taxon tree corresponds to five different rooted trees, arising from placing the root on one of the five branches in the unrooted tree.}\n \\label{rooting}\n\\end{figure}\n\nFor the 3-taxon case, we can show that given a sufficient number of unlinked sites under the molecular clock, we can infer the correct topology with probability 1. For the unrooted 4-taxon case, we have demonstrated this in simulation studies for a wide variety of true trees, but it remains unproven.\n\\begin{theorem}\nGiven CIS data for three taxa evolving under the multispecies coalescent with the JC69 model and the molecular clock, if $\\hat{S}$ is the maximum a posteriori tree, $P(\\hat{S}=S) \\rightarrow 1$ as the number of sites $J \\rightarrow \\infty$ for any prior for which $0<\\tau_1<\\tau_2$ holds with probability 1.\n\\end{theorem}\nThen, using a Bonferroni adjustment, we can ensure that given a sufficient number of sites, $P(\\hat{S}=S) > 1- \\epsilon$ \\textit{uniformly} for all the triplets. It then follows that given proposition four from \\cite{steel93}, we can estimate the n-taxon topology with probability $>1-\\epsilon$.\n\n\\subsection{Estimating the n-taxon species tree}\n\\label{assembly}\nThere is no theoretical impediment to prevent extending this procedure to infer full posterior probabilities for 5-, 6-, or even n-taxon trees. But, to see the practical difficulties, consider the challenges to extend this result to just five taxa. First, for 5 taxa $\\boldsymbol{\\delta}_{JC}$ is now a $51 \\times 1$ vector for each of three unlabeled topologies. Then, a $51 \\times 105$ permutation matrix is needed to find the site pattern probabilities for all 105 5-taxon labeled gene tree topologies. A general recursive formula for the number of unordered gene histories embedded in an n-taxon species tree is provided by \\cite{rosenberg07}. We calculated this total to be 379, 313, and 208 for $S_1$, $S_2$, and $S_3$ from figure \\ref{topologies}, respectively. The probabilities of these histories need to be calculated and then integrated over to find the site pattern probabilities of the three unlabeled species tree topologies, then the same $51 \\times 105$ permutation matrix must again be applied to get the full site pattern probabilities for all species tree topologies. As a result, the full set of site pattern probabilities $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau})$ remains to be calculated for 5 taxa, and an even more complex process would be needed to extend to six or more taxa.\n\nInstead, to infer the n-taxon topology, we use an assembly algorithm that takes the posterior probabilities of the rooted triplets or unrooted quartets as inputs to infer the final n-taxon species tree estimate. We repeat the process of sections \\ref{method1} and \\ref{derivation} for each of the ${n \\choose 3}$ sets of triplets (Lily-T) or ${n \\choose 4}$ quartets (Lily-Q). Then we have a set of $3{n \\choose 3}$ posterior probabilities for each possible rooted triplet or $3{n \\choose 4}$ posterior probabilities for each possible unrooted quartet from the leaf set. These probabilities are then used in an assembly algorithm to infer the estimated n-taxon species tree $\\hat{S}$. There are many options for assembly methods. The methods we choose allow us to use the posterior probabilities as weights in the subtree inputs. Both SVDQuartets and ASTRAL are unweighted methods and thus treat all inputs equally regardless of the inference uncertainty (see figure \\ref{weighting}). For Lily-T, we chose the Triplet MaxCut (TMC) method of Sevillya et al. (2016)\\nocite{sevillya16} due to its speed, accuracy, ease of implementation, and its ability to work with rooted trees as inputs, and similarily chose the Weighted Quartet Max Cut (weighted QMC) algorithm of Avni et al. (2015)\\nocite{avni14} for Lily-Q. Further, the implementation of these algorithms are very similar to the algorithm used by SVDQuartets, and so using it reduces, but does not eliminate, one source of variation between the two methods.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\linewidth]{\"weighting\".png}\n \\caption{Unweighted tree inputs: an unweighted method treats the two cases the same, even though the right side contains far more information about the true topology.}\n \\label{weighting}\n\\end{figure}\n\nTriplet MaxCut works as follows: first we obtain posterior probabilities on all ${n \\choose 3}$ subsets from the leaf set. When we have multiple individuals from a species, posterior probabilities for each combination of individuals from each triplet of species are calculated. Then two symmetric $n \\times n$ matrices are formed, $\\textbf{G}$ and $\\textbf{B}$, called the ``good\" and ``bad\" matrices with $i,j=1,2,3 \\ldots n$ corresponding to the taxa in alphabetical order: $a, b, c,$ etc. For each triplet input, we have a set of two taxa that form a cherry and a third that is not part of the cherry, as well as a weighting for the triplet. For each of the ${3 \\choose 2}$ pairs in the triplet, if taxa $i$ and $j$ form a cherry, the weight of the triplet is added to $B_{ij}$ and $B_{ji}$ and if $i$ and $j$ do not form a cherry, the weight of the triplet is added to $G_{ij}$ and $G_{ji}$. As an example, consider the triplet $(a,(c,d))$ with weight 0.5. We would add 0.5 to $B_{34}$, $B_{43}$, $G_{13}$, $G_{31}$, $G_{14}$, and $G_{41}$. This process is repeated for all sets of input triplets. Thus, large entries in $G_{ij}$ indicate taxa $i$ and $j$ should, all else equal, be separated, and large entries in $B_{ij}$ indicate taxa $i$ and $j$ should, all else equal, be grouped together in the final tree S.\n\nNext, for the set of $n$ taxa $L=\\{1,2,3 \\ldots\\ n\\}$ we obtain all subsets of the taxa of cardinality $\\leq \\floor*{\\frac{n}{2}}$. For each subset, we can arbitrarily label the taxa in the subset $L_1$ and call the remainder $L_2=L\\setminus L_1$, resulting in a bipartitioning of $L$. Then, for each bipartition, we score the bipartition by the ratio $ \\frac{\\sum_{i \\in L_1} \\sum _{j \\in L_2} G_{ij}}{\\sum_{i \\in L_1} \\sum _{j \\in L_2} B_{ij}}$. Note that this score can be undefined if there are entries in $\\boldsymbol{B}=0$. To avoid this problem, a very small number such as $10^{-200}$ can be added to each element of $\\boldsymbol{B}$. The bipartition with the highest score is accepted. If the cardinality of either $L_1$ or $L_2$ is greater than two, the process is repeated recursively on $L_1$ and\/or $L_2$ as needed. In essence, each step of the procedure creates a node and assigns taxa to the left and right branches of the node until the final tree is resolved.\n\nThe operation of the weighted QMC is largely similar, with posteriors first being calculated for all ${n \\choose 4}$ subsets of quartets from the leaf set to use as inputs in the assembly algorithm. The major difference in the assembly is that care must be taken if one of $L_1$ or $L_2$ is a set of 3 elements. Then the set is augmented with an additional artificial taxon and the procedure is performed on this augmented set to evaluate which two elements of the set constitute a cherry.\n\n\\subsection{Uncertainty quantification}\nAn immediate concern with using an assembly procedure is that one of the key advantages of our likelihood-based approach -- the ability to produce posterior probabilities -- is lost. The assembly procedures we chose are based on heuristics that make sense -- grouping species together that have a high probability of being cherries. But, unfortunately we have no distributional results to assess. The output tree is a point estimate only, and while we can generate simulation data to say that it is reasonably accurate, once we apply it to real data we no longer have a measure of uncertainty. The difficulty lies in the same factors that led us to pursue the assembly procedure in the first place -- the inability to calculate a joint $n$-taxon set of site pattern probabilities.\n\nA standard method for measuring uncertainties of estimated phylogenies is the bootstrapping method of Efron (1979) \\cite{efron79}. We implemented a nonparametric bootstrap: for CIS data we resampled with replacement from all sites, and for multi-locus data we resampled the genes with replacement and then for each gene we resampled with replacement the sites within each gene. An advantage of the bootstrap is that it can give unbiased estimates of uncertainty without any distributional assumptions. But, a number of cautions are in order. First, we only have asymptotic guarantees about the approximation of $\\frac{\\boldsymbol{d}_k}{J}$ to $\\delta_k$ and we have no finite-sample knowledge of how good the approximation is. As a result, we don't know how much uncertainty we have about our uncertainty. Second, without parallelization, the time required to estimate the bootstrap samples increases linearly with the number of bootstrap samples taken. Last, bootstrap support values are \\textit{not} probabilities, and should not be treated as such. That said, we can compare the bootstrap support to the actual proportion of times we infer the correct tree to see if the bootstrap support reasonably ``mimics\" the true probability of being correct for reasonable parameter values.\n\n\\section{Implementation}\n\\label{implement}\nWe have written source code in C++ that implements Lily-T and Lily-Q, as well as programs that summarize the distances from the true tree for our simulation runs. The programs take as inputs alignments in PHYLIP format and output either the final output tree (Lily-T) or a properly formatted input file for the weighted QMC program (Lily-Q). We also wrote programs for calculating the number of gene tree histories in section \\ref{assembly} and a Perl wrapper for executing the simulation runs. These programs, as well as a file summarizing the results are available at \\url{https:\/\/github.com\/richards-1227\/Lily}. \n\n\\section{Results}\nData were simulated using the \\textit{ms} \\cite{MS} and Seq-Gen \\cite{seq-gen} programs in C++ using a Unix HPC platform. The \\textit{ms} program takes the node times (in coalescent units) and population parameter $\\theta$ as input and simulates a set of gene trees under the multispecies coalescent model. Seq-Gen then takes these gene trees and the mutation model parameters as input and generates the aligned sequence data $\\boldsymbol{D}_{raw}$. Caution should be taken in working with these programs for diploid organisms -- to get the correct simulated probabilities, you enter one-half the node times (in coalescent units) and $2\\theta$ rather than their actual values. 1,000,000 CIS were simulated at various settings of $\\boldsymbol{\\tau}$ and $\\theta$ and we compared the observed values of $\\boldsymbol{d}_{JC}$ to $J\\boldsymbol{\\delta}_{JC}|(S,\\boldsymbol{\\tau})$. Chi-squared goodness of fit tests were performed and there was no evidence that the simulated frequencies differed from that expected (only one individual test p-value was below 0.05 (0.006), and it was no longer significant after making a Bonferroni adjustment).\n\n\\subsection{Testing robustness to model assumptions and prior specification}\n\\label{robustness}\nWe next relaxed various assumptions of the substitution model to verify that the true tree was still estimated with high probability. One thousand runs were performed with 25,000 CIS generated with the Jukes-Cantor assumptions being progressively relaxed, first allowing for different stationary probabilities, then different relative substitution rates, then between-site rate heterogeneity, and then allowing for invariant sites. The settings are summarized in table \\ref{JCsettings}.\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n & & \\multicolumn{5}{|c|}{Relative rates vs. A-C} & & & \\\\\n \\hline\n Setting & CG content & A-G & A-T & C-G & C-T & G-T & $\\alpha$ & \\% Invariable & $P(S|\\boldsymbol{d})$\\\\\n \\hline\n 1 & 25\\% & 1 & 1 & 1 & 1 & 1 & N\/A & 0 & 0.983 \\\\ \n \\hline\n 2 & 35\\% & 1 & 1 & 1 & 1 & 1 & N\/A & 0 & 0.971\\\\ \n \\hline\n 3 & 45\\% & 3 & 1 & 1 & 3 & 1 & N\/A & 0 & 0.951 \\\\ \n \\hline\n 4 & 35\\% & 5 & 1 & 1 & 3 & 1 & N\/A & 0 & 0.951\\\\ \n \\hline\n 5 & 35\\% & 5 & 0.75 & 0.5 & 3 & 0.25 & N\/A & 0 & 0.971 \\\\ \n \\hline\n 6 & 35\\% & 2 & 0.75 & 0.5 & 1.5 & 0.25 & 3 & 0 & 0.974 \\\\ \n \\hline\n 7 & 35\\% & 2 & 0.75 & 0.5 & 1.5 & 0.25 & 9 & 0 & 0.960\\\\ \n \\hline\n 8 & 35\\% & 2 & 0.75 & 0.5 & 1.5 & 0.25 & 3 & 0.2 & 0.960 \\\\ \n \\hline\n \\end{tabular}\n \\caption{Initial simulation test settings. The final column gives the average posterior probabilities assigned to the true tree as the JC69 assumptions are progressively relaxed.}\n \\label{JCsettings}\n\\end{table}\n\nThe results shown in table \\ref{JCsettings} are typical and are presented for three taxa and each of the eight substitution model settings using $\\tau_1=5.2$, $\\tau_2=6.0$, $\\theta=0.01$, and $\\beta=0.1$. There does not appear to be a large impact from relaxing the JC69 assumptions, and for the remainder of our simulations all the data were generated under simulation setting 8, so performance of our methods is measured against a deliberately misspecified model.\n\nHere we should note that the site pattern probabilities $\\boldsymbol{\\delta}$ are also conditional on $\\theta$ as well as $(S,\\boldsymbol{\\tau})$, but keeping with the notation of \\cite{chifman15} we have ignored this conditioning. For the remainder of our work, we used $\\hat{\\theta}=0.003$ as an input to both Lily-T and Lily-Q, which we justify as follows. Most empirical values of $\\theta$ fall between 0.001 and 0.01 (see \\cite{kubatko07,rannala03,jennings05,kopp05} for estimates on species ranging from primates to finches). So, we simulated data using a true $\\theta=\\{0.0003, 0.001, 0.003, 0.01, 0.03\\}$, extending beyond the empirical range, and calculated posteriors using $\\hat{\\theta}=0.01$ and $\\hat{\\theta}=0.001$. The results for Lily-T and Lily-Q are shown in figure \\ref{ThetaEffect}. Since there is no apparent difference in performance for assuming a $\\theta$ different from the actual data-generating $\\theta$, we simply use a $\\hat{\\theta}=0.003$ as it is in the middle of the empirical range on the log scale.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.43\\textwidth}\n \\includegraphics[width=\\linewidth]{\"theta3\".png}\n \\caption{$\\hat{\\theta}=0.001$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.43\\textwidth}\n \\includegraphics[width=\\linewidth]{\"theta4\".png}\n \\caption{$\\hat{\\theta}=0.01$}\n\\end{subfigure}\n \\caption{Mean posterior probability assigned by Lily-Q to the true tree for varying numbers of sites and population parameter $\\theta$ for a symmetric 4-taxon with branch lengths of 0.5 coalescent units. The left panel (a) uses $\\hat{\\theta}=0.001$ and the right panel (b) uses $\\hat{\\theta}=0.01$.}\n \\label{ThetaEffect}\n\\end{figure}\n\nWe cannot, however, simply assume a value for the root age hyperparameter $\\beta$. Figure \\ref{betaEffect} shows the posterior assigned to the correct tree, using a $\\beta$ where $1\/\\beta$ varies from 2 orders of magnitude below to 2 orders of magnitude above the actual root age. There is very little loss in accuracy for overestimating the root age (very small values of $\\beta$) except with very few sites (see the $J=1000$ column). If the root age is underestimated, however, there can be a large loss of accuracy, even with a larger number of sites. This is to be expected given the asymmetric nature of the exponential prior -- a small $\\beta$ flattens the prior and can still have adequate prior mass near the true value, while a large $\\beta$ puts most prior mass near zero. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{\"BetaEffect\".png}\n \\caption{Mean posterior probability assigned by Lily-T to the true tree for varying numbers of sites and root age hyperparameter $\\beta$ for a 3-taxon tree with branch lengths of 0.5 coalescent units.}\n \\label{betaEffect}\n\\end{figure}\n\nWe take an empirical Bayes approach to choosing $\\beta$. From the JC69 model we can infer the well-known pairwise distance estimate for each pair of species\n$$\\hat{\\lambda} \\hat{t}=\\frac{-3log(1-\\frac{4\\hat{p}}{3})}{4} $$\nwhere $t$ is the time measured in years, $\\lambda$ is the per-year mutation rate, and $\\hat{p}$ is the fraction of discordant sites between the two species. Then, assuming a value of 0.003 for $\\theta$ and generation time $g$, we have:\n $$0.003=\\theta=4N_e\\mu=4\\lambda N_e g $$\nAfter a quick algebraic manipulation we can estimate time in coalescent units ($2N_e$ generations) as:\n \\begin{equation}\n \\frac{\\hat{t}}{2N_eg}=\\frac{-6log(1-\\frac{4\\hat{p}}{3})}{0.012} \n \\label{distEst}\n \\end{equation} \nFor each set of three or four species, we will have either three or six of these pairwise divergence estimates from equation \\ref{distEst}. Lily-T and Lily-Q use the mean of these estimates as the value for the hyperparameter $\\beta$. It is worth noting that because these estimates use concatenated data, they are biased upwards with respect to the species tree root time since they estimate the time of coalescent events and any coalescent event must occur prior to species divergence. But, here, that bias is useful as it helps reduce the risk of underestimating the root age, which has a greater performance impact than overestimation. \n\n\\subsection{Application to simulated data}\nData were simulated for 5, 8, and 12-taxon trees for a total of 12 different topologies ranging from fully symmetric to fully asymmetric. The full set of topologies simulated is displayed in figure \\ref{topologies}. For each topology, we used four different values of the population parameter: $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$. These were chosen to extend slightly above and below the empirical range of 0.001 to 0.01 and the values are linear on the log scale. We used four different settings for the minimum branch length (MBL): 0.2, 0.5, 1.0, and 2.0 coalescent units. Multi-locus data were simulated with 10, 50, and 500 genes and 50, 200, and 500 sites per gene. For CIS data, we simulated either 5,000 or 50,000 sites. 100 runs were performed at each combination of settings.\n\\begin{comment}\n\\begin{table}[]\n \\centering\n Five taxa:\n \\vskip 1mm\n \\begin{tabular}{c c c }\n $(a,(b,(c,(d,e))))$ & $(a,((b,c),(d,e)))$ & $((a,b),(c,(d,e)))$ \n \\end{tabular}\n \\vskip 1mm\n Eight taxa:\n \\vskip 1mm\n \\begin{tabular}{c c }\n $(a,(b,(c,(d,(e,(f,(g,h)))))))$ & $((a,b),((c,d),(e,(f,(g,h)))))$ \\\\ $(((a,b),c),((d,e),(f,(g,h))))$ & $(((a,b),(c,d)),((e,f),(g,h)))$ \\\\ \n \\end{tabular}\n \\vskip 1mm\n Twelve taxa:\n \\vskip 1.5mm\n \\begin{tabular}{c c }\n $(a,(b,(c,(d,(e,(f,(g,(h,(i,(j,(k,l)))))))))))$ & $((a,b),((c,(d,(e,f))),((g,h),((i,j),(k,l)))))$ \\\\ $((a,(b,c)),((d,(e,f)),((g,(h,i)),(j,(k,l)))))$ & $(((a,b),(c,d)),(((e,f),(g,h)),((i,j),(k,l))))$ \n \\end{tabular}\n \\begin{tabular}{c } $(((a,(b,c)),(d,(e,f))),((g,(h,i)),(j,(k,l))))$ \n \\end{tabular}\n \\caption{Species tree topologies used in simulation study.}\n \\label{top_used}\n\\end{table}\n\\end{comment}\n\nFocusing on the minimum branch length created a number of side effects. Comparing trees $S_4$ and $S_5$ in figure \\ref{topologies} we note that for the same MBL, the root for the asymmetric tree $S_5$ will occur much further in the past than for the symmetric tree $S_4$. Second, for trees that are neither fully symmetric nor asymmetric such as $S_{12}$, some internal branches naturally have to be longer than the MBL, in which case the internal nodes were chosen to be equally spaced. \n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{.47\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{\"topology1\".png}\n\\end{subfigure}\n\\begin{subfigure}{.47\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{\"topology2\".png}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.47\\linewidth}\n\\centering\n\\includegraphics[width=\\linewidth]{\"topology3\".png}\n\\end{subfigure}\n \\caption{The twelve topologies used in the simulation study. 5 taxa: $S_1$ through $S_3$, 8 taxa: $S_4$ through $S_7$, 12 taxa: $S_8$ though $S_{12}$. The red bar indicates scale of the MBL, which was set at 0.2, 0.5, 1.0, and 2.0 coalescent units.}\n \\label{topologies}\n\\end{figure}\n\nIn order to evaluate how well each method performed, we first need a proper metric of distance between trees. The most common is the Robinson Folds (RF) distance from \\cite{RFdist}. They define the distance between two trees $T_1$ and $T_2$ as the sum of the number of bipartitions in $T_1$ not contained in $T_2$ and vice versa. The presence of a bipartition is symmetric, since the number of bipartitions is equal to the number of internal branches, which is in turn equal to $n-2$ for a rooted tree and $n-3$ for an unrooted tree. Thus, if there is a bipartition of $T_1$ not in $T_2$, there must be a bipartition in $T_2$ not in $T_1$. As a result, the RF metric can take on any even value from zero to $2(n-2)$ (or $2(n-3)$ if unrooted). Since the metric is also a function of whether or not the two trees are rooted, to compare Lily-T to the other methods (which produce unrooted trees), we must first unroot both the estimated and true tree. For multi-locus data, we compared SVDQuartets (as implemented in PAUP* \\cite{paup}), ASTRAL (using FastTree for gene tree estimation as per the example in \\cite{chou15}), Lily-T, and Lily-Q for each run and calculated the mean RF distance and standard error for the 100 runs. For CIS data, we excluded ASTRAL from comparison since gene tree cannot be estimated for a single site.\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF5pt2\".png}\n \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF5pt5\".png}\n \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF5_1\".png}\n \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF5_2\".png}\n \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for true tree $(a,(b,(c,(d,e))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL is displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8bpt2\".png}\n \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8bpt5\".png}\n \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8b1\".png}\n \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8b2\".png}\n \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for true tree $(((a,b),c),((d,e),(f,(g,h))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL is displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF6}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8bpt2s\".png}\n \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8bpt5s\".png}\n \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8b1s\".png}\n \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF8b2s\".png}\n \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-T vs. SVDQuartets for true tree $(((a,b),c),((d,e),(f,(g,h))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- dark, 50 -- medium, 500 -- light. SVDQuartets displayed in grey shading and Lily-T in green.}\n\\label{simulation_RF6s}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF12dpt2\".png}\n \\caption{$MBL=0.2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF12dpt5\".png}\n \\caption{$MBL=0.5$}\n\\end{subfigure}\n\\newline\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF12d1\".png}\n \\caption{$MBL=1.0$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RF12d2\".png}\n \\caption{$MBL=2.0$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for true tree $(((a,(b,c)),(d,(e,f))),((g,(h,i)),(j,(k,l))))$ at different minimum branch lengths (MBL). In each panel, $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL is displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF12}\n\\end{figure}\n\nA representative selection of the simulation results are shown in figures \\ref{simulation_RF1} through \\ref{simulation_RF12} (the full plots are in the supplementary material.) One complication is that for very small number of sites and a small value of $\\theta$, there may be cases where there is no divergence between any set of three species. The proper inference in this case would be to return a polytomous tree. Each method, however, treats this case differently. SVDQuartets will either return a polytomous tree or an error. Lily-Q infers an 1\/3 probability for each of the three possible unrooted quartets involving three or more zero-divergence species, but the assembly procedure will then produce an error message. Lily-T will do the same, but will generate an error only after consuming much time and memory. ASTRAL will infer an apparently random binary tree, but that is due to FastTree incorrectly inferring binary trees for the gene tree inputs. As a result, for those settings where this occurred, we display the Lily-Q vs. ASTRAL comparison for first 100 valid runs without any set of three zero-distance taxa. We also did not use Lily-T or SVDQuartets on the 10 genes and 50 sites-per-gene setting because SVDQuartets invariably produced errors and Lily-T produced errors while materially slowing down the simulation process. Thus, the plots for Lily-T and SVDQuartets do not show results for some settings. \n\nA number of results become evident from the RF plots. First, all estimation is better with a larger $\\theta$, as there is more mutation along each branch allowing us to pick up more of the phylogenetic signal. A lone exception to this trend is that estimation gets worse for SVDQuartets for $S_8$ with $\\theta=0.0216$ and a minimum branch length of 2.0 coalescent units. It is worth noting that for these settings the root is 22.0 coalescent units in the past and the sequences may be nearing saturation and so SVDQuartets may have greater difficulty resolving the phylogeny near saturation. Second, with enough data, all methods perform well. Third, by comparing the 50 genes\/500 sites per gene and the 500 genes\/50 sites per gene case, we can see that all methods do better with more genes even when the total number of sites is held constant. At smaller sample sizes, SVDQuartets is generally outperformed by all other methods, and Lily-T is in turn outperformed by Lily-Q under most simulation settings.\n\nTherefore, we focused on the comparison between Lily-Q and ASTRAL. For MBL of 1.0 or 2.0 coalescent units, Lily-Q generally performs no worse, and in many cases, better than ASTRAL. For an MBL of 0.5 coalescent units, ASTRAL does better for 200 or 500 sites per gene and $\\theta$ of 0.0072 or 0.0216, and this effect appears to be stronger with fewer taxa. Lily-Q does as well or better for almost all cases with 50 sites per gene and 10 or 50 genes. For 0.2 CI, Lily-Q does better with fewer sites per gene and with smaller values of $\\theta$ and ASTRAL performs better with more sites per gene and larger values of $\\theta$. The comparison also seems somewhat dependent on topology, but without any clear pattern. ASTRAL outperforms Lily-T for most settings, except when the branch lengths are large and $\\theta$ is smaller. \n\nThese results match our expectations, as ASTRAL depends on accurate gene tree inputs and those gene trees are easier to resolve when the genes are long (many sites per gene) and there is more mutation along each branch (higher $\\theta$). One oddity is that for Lily-Q, with 10 genes, sometimes the RF distance was higher going from 200 to 500 sites per gene. But, the standard errors almost always overlapped so this appears to be random noise arising out of the fact that, with only 10 genes, if the gene trees differ from the species tree, increasing the number of sites only allow better inference of the mismatched gene trees rather than the underlying species tree.\n\n\\begin{comment}\nTables \\ref{det_result1} through \\ref{det_result4} show the results for a single topology $((a,b),((c,d),(e,(f,(g,h)))))$, with a minimum branch length of 0.5 coalescent units. A number of results become evident. First, all estimation is better with a larger $\\theta$, as there is more mutation along each branch allowing us to pick up more of the phylogenetic signal. Second, with enough data, all methods perform well. Third, by comparing the 50 genes\/500 sites per gene and the 500 genes\/50 sites per gene case, we can see that all methods do better with more genes even when the total number of sites is held constant. At smaller sample sizes, SVDQuartets is generally outperformed by all other methods, and Lily-T is outperformed by Lily-Q and ASTRAL. Therefore, we focused on the comparison between Lily-Q and ASTRAL and calculated a (pooled variance) T-statistic for the difference between them. As expected, ASTRAL performs better under conditions where gene tree estimation is more accurate -- when both the gene length and $\\theta$ is large. Lily-Q, on the other hand, tends to perform better when the gene lengths are shorter or $\\theta$ is smaller. \n\nOur program in C++ for calculating RF distances was written assuming binary trees. In some cases for Lily-T and especially SVDQuartets, the assembly algorithm did not output a binary tree in all 100 runs and in those cases we did not calculate the mean RF distance for comparison (represented as dashes in the tables).\n\n\\begin{table}[]\n \\centering\n \\begin{scriptsize}\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Genes} & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n \\hline\n 500 & 500 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 200 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 50 & 0.56(0.09) & 0.2(0.06) & 0.02(0.02) & 0.22(0.07) & 2.80\\\\\n \\hline\n 50 & 500 & 0.68(0.10) & 0.32(0.08) & 0.16(0.05) & 0.30(0.08) & 1.49\\\\\n \\hline\n 50 & 200 & 2.14(0.24) & 1.22(0.14) & 0.74(0.11) & 1.30(0.15) & 2.95\\\\\n \\hline\n 50 & 50 & -- & 3.86(0.27) & 2.52(0.21) & 3.34(0.20) & 2.81\\\\\n \\hline\n 10 & 500 & -- & 2.64(0.23) & 2.06(0.19) & 2.8(0.21) & 2.64\\\\\n \\hline\n 10 & 200 & -- & 4.52(0.27) & 3.06(0.20) & 4.32(0.21) & 4.30\\\\\n \\hline\n 10 & 50 & -- & -- & 6.06(0.22) & 6.94(0.21) & 2.89\\\\\n \\hline\n 25000 & 1 & 0.56(0.10) & 0.12(0.05) & 0.02(0.02) & -- & -- \\\\\n \\hline\n 5000 & 1 & -- & 1.84(0.18) & 1.00(0.14) & -- & -- \\\\\n \\hline\n \\end{tabular}\n \\end{scriptsize}\n \\caption{Mean RF distance (with standard error) from true tree $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0008$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n \\label{det_result1}\n\\end{table}\n\\begin{table}[]\n \\centering\n \\begin{scriptsize}\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Genes} & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n \\hline\n 500 & 500 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 200 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 50 & 0.06(0.03) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 50 & 500 & 0.32(0.07) & 0.12(0.05) & 0.02(0.02) & 0.12(0.05) & 1.94\\\\\n \\hline\n 50 & 200 & 0.48(0.09) & 0.26(0.07) & 0.08(0.04) & 0.34(0.08) & 3.07\\\\\n \\hline\n 50 & 50 & 3.18(0.32) & 1.48(0.17) & 0.80(0.11) & 1.60(0.15) & 4.31\\\\\n \\hline\n 10 & 500 & 1.96(0.17) & 1.58(0.16) & 1.44(0.15) & 1.42(0.14) & -0.10\\\\\n \\hline\n 10 & 200 & -- & 2.62(0.20) & 1.84(0.17) & 2.24(0.19) & 1.59\\\\\n \\hline\n 10 & 50 & -- & -- & 3.58(0.26) & 5.14(0.27) & 4.18\\\\\n \\hline\n 25000 & 1 & 0.04(0.03) & 0(0) & 0(0) & -- & -- \\\\\n \\hline\n 5000 & 1 & 0.76(0.11) & 0.44(0.08) & 0.14(0.05) & -- & -- \\\\\n \\hline\n \\end{tabular}\n \\end{scriptsize}\n \\caption{Mean RF distance (with standard error) from true tree $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0024$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n\\end{table}\n\\begin{table}[]\n \\centering\n \\begin{scriptsize}\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Genes} & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n \\hline\n 500 & 500 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 200 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 50 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 50 & 500 & 0.28(0.07) & 0.04(0.03) & 0.08(0.04) & 0(0) & -2.04\\\\\n \\hline\n 50 & 200 & 0.28(0.07) & 0.04(0.03) & 0.06(0.03) & 0.08(0.04) & 0.38\\\\\n \\hline\n 50 & 50 & 1.00(0.11) & 0.56(0.10) & 0.34(0.08) & 0.50(0.10) & 1.31\\\\\n \\hline\n 10 & 500 & 1.14(0.14) & 0.68(0.11) & 0.64(0.11) & 0.54(0.09) & -0.71\\\\\n \\hline\n 10 & 200 & 1.70(0.15) & 1.48(0.15) & 1.30(0.13) & 1.48(0.14) & 0.84\\\\\n \\hline\n 10 & 50 & -- & -- & 2.40(0.18) & 2.98(0.22) & 2.04\\\\\n \\hline\n 25000 & 1 & 0(0) & 0(0) & 0(0) & -- & -- \\\\\n \\hline\n 5000 & 1 & 0.36(0.08) & 0.06(0.03) & 0(0) & -- & -- \\\\\n \\hline\n \\end{tabular}\n \\end{scriptsize}\n \\caption{Mean RF distance (with standard error) from true tree $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0072$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n\\end{table}\n\n\\begin{table}[]\n \\centering\n \\begin{scriptsize}\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Genes} & \\textbf{Sites\/gene} & \\textbf{SVDQ} & \\textbf{Lily-T} & \\textbf{Lily-Q} & \\textbf{ASTRAL} & \\textbf{T-stat}\\\\\n \\hline\n 500 & 500 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 200 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 500 & 50 & 0(0) & 0(0) & 0(0) & 0(0) & 0\\\\\n \\hline\n 50 & 500 & 0.06(0.03) & 0(0) & 0.02(0.01) & 0(0) & -1.00\\\\\n \\hline\n 50 & 200 & 0.28(0.07) & 0.04(0.03) & 0.02(0.02) & 0(0) & -1.00\\\\\n \\hline\n 50 & 50 & 0.38(0.08) & 0.22(0.06) & 0.16(0.05) & 0.2(0.06) & 0.49\\\\\n \\hline\n 10 & 500 & 1.82(0.17) & 1.18(0.15) & 1.08(0.15) & 0.94(0.12) & -0.73\\\\\n \\hline\n 10 & 200 & 1.32(0.15) & 0.84(0.12) & 0.6(0.10) & 0.70(0.11) & 0.68\\\\\n \\hline\n 10 & 50 & -- & -- & 1.54(0.17) & 1.80(0.17) & 1.10\\\\\n \\hline\n 25000 & 1 & 0(0) & 0(0) & 0(0) & -- & -- \\\\\n \\hline\n 5000 & 1 & 0(0) & 0(0) & 0(0) & -- & -- \\\\\n \\hline\n \\end{tabular}\n \\end{scriptsize}\n \\caption{Mean RF distance (with standard error) from true tree $((a,b),((c,d),(e,(f,(g,h)))))$ for population parameter $\\theta=0.0216$ and MBL of 0.5 coalescent units. Note the maximum RF distance of an unrooted 8-taxon tree is 10.}\n \\label{det_result4}\n\\end{table}\n\nFigures \\ref{simulation_RF1} through \\ref{simulation_RF6} show a graphical comparison of Lily-Q vs. ASTRAL for all cases with an MBL of 0.5 coalescent units. These show the normalized RF distance (RF distance divided by the maximum value of $2(n-3)$). The direction of results are largely similar for other branch lengths, but tend to be larger in magnitude for MBL of 0.2 coalescent units and smaller for higher MBL. The full results is available online (see section \\ref{implement}, and a complete table of cases where there is a significant difference between Lily-Q and ASTRAL (defined as a T-statistic with an absolute value greater than 2) is in the supplemental material. With 500 genes, there is almost no difference between methods, although Lily-Q appears to do slightly better with 50 sites per gene. With 50 genes, at higher $\\theta$ values, there is little error, except again a slight outperformance by Lily-Q with 50 sites per gene. At smaller $\\theta$ values, with 50 genes, Lily-Q appears to outperform especially with smaller numbers of sites per gene. With 10 genes, Lily-Q again outperforms at small values of $\\theta$ even when the number of sites per gene large. For higher values of $\\theta$ Lily-Q has lower RF distance for few sites per gene and ASTRAL does better for many sites per gene.\n\nOne oddity is that for Lily-Q, with 10 genes, sometimes the RF distance was higher going from 200 to 500 sites per gene. But, the standard errors almost always overlapped so this appears to be random noise arising out of the fact that, with only 10 genes, if the gene trees differ from the species tree, increasing the number of sites only allow better inference of the mismatched gene trees rather than the underlying species tree.\n\nFinally, it is worth noting that the RF metric is heavily left-skewed even with moderate numbers of taxa \\cite{steel93}. As a result, even with only 10 sites and 50 sites per gene, the average estimated tree for both methods is closer to the true tree than the vast majority of the tree space. \n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot1\".png}\n \\caption{$(a,(b,(c,(d,e))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot2\".png}\n \\caption{$(a,((b,c),(d,e)))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot3\".png}\n \\caption{$((a,b),(c,(d,e)))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot4\".png}\n \\caption{$(a,(b,(c,(d,(e,(f,(g,h)))))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot5\".png}\n \\caption{$((a,b),((c,d),(e,(f,(g,h)))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot6\".png}\n \\caption{$(((a,b),c),((d,e),(f,(g,h))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF3}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot7\".png}\n \\caption{$(((a,b),(c,d)),((e,f),(g,h)))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot8\".png}\n \\caption{$(a,(b,(c,(d,(e,(f,(g,(h,(i,(j,(k,l)))))))))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot9\".png}\n \\caption{$((a,b),((c,(d,(e,f))),((g,h),((i,j),(k,l)))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot10\".png}\n \\caption{$((a,(b,c)),((d,(e,f)),((g,(h,i)),(j,(k,l)))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF5}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot11\".png}\n \\caption{$(((a,b),(c,d)),(((e,f),(g,h)),((i,j),(k,l))))$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\linewidth]{\"RFplot12\".png}\n \\caption{$(((a,(b,c)),(d,(e,f))),((g,(h,i)),(j,(k,l))))$}\n\\end{subfigure}\n\\caption{Mean normalized RF distance for Lily-Q vs. ASTRAL for trees with MBL of 0.5 coalescent units and $\\theta=\\{0.0008, 0.0024, 0.0072, 0.0216\\}$ from left to right. Number of genes: 10 -- magenta\/blue, 50 -- orange\/purple, 500 -- red\/green. ASTRAL displayed in ``hot\" colors (magenta\/orange\/red) and Lily-Q in ``cold\" (blue\/purple\/green). }\n\\label{simulation_RF6}\n\\end{figure}\n\n\\end{comment}\n\n\\subsection{Bootstrapping results}\nFor twenty different combinations of topology, $\\theta$, MBL, gene length, and number of genes, we drew 100 bootstrap samples by resampling both the genes and then resampling the sites within each gene. For both Lily-T and Lily-Q, we compared the bootstrap support for the estimated tree to the RF distance from the true tree. Figure \\ref{RFboot5} shows the bootstrap support given the RF distance between the estimated tree and the true tree. We can see that the farther the estimated tree is from the true tree, the lower the bootstrap support tends to be. Tables \\ref{boot_table5} to \\ref{boot_table12Q} in turn show the positive and negative predictive value of high or low bootstrap support. What we see is that we have very good positive predictive value -- for Lily-Q only 5 of 697 trees ($<$1\\%) with bootstrap support greater than 0.7 were incorrect, and all of them had an RF distance of 2. With lower bootstrap support, however, more trees tend to be further from the true tree but the estimated tree may still be close to the correct tree -- for Lily-Q 356 of 830 estimated trees (43\\%) with bootstrap support less than 0.4 were in fact correct.\n\nThese simulation studies suggest that the bootstrap support may be useful as a guide to triaging larger phylogenetic questions -- if the support is high, the faster results that can be obtained from Lily-T or Lily-Q can be accepted with confidence. Conversely, if the support is low, it serves as an indication that a more computationally intensive MCMC-based method may be necessary for accurate results. \n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"taxa5_boxplot\".png}\n \\caption{Lily-T (5 taxa)}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"Quart5taxa_boxplot\".png}\n \\caption{Lily-Q (5 taxa)}\n \\end{subfigure}\n \\newline\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"taxa8_boxplot\".png}\n \\caption{Lily-T (8 taxa)}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"Quart8taxa_boxplot\".png}\n \\caption{Lily-Q (8 taxa)}\n \\end{subfigure}\n \\newline\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"taxa12_boxplot\".png}\n \\caption{Lily-T (12 taxa)}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"Quart12taxa_boxplot\".png}\n \\caption{Lily-Q (12 taxa)}\n \\end{subfigure}\n \\caption{Bootstrap support for estimated trees grouped by RF distance}\n \\label{RFboot5}\n\\end{figure}\n\n\n\\begin{table}[]\n \\centering\n \\begin{minipage}{0.59\\linewidth}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} \\\\\n \\hline\n 0 & 0 & 0 & 0 & 1\\\\\n \\hline\n (0, 0.4] & 121 & 135 & 109 & 34\\\\\n \\hline\n (0.4, 0.7] & 107 & 22 & 1 & 0\\\\\n \\hline\n (0.7, 0.9] & 62 & 0 & 0 & 0\\\\\n \\hline\n (0.9, 1] & 108 & 0 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\end{minipage}\n \\begin{minipage}{0.39\\linewidth}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} \\\\\n \\hline\n 0 & 0 & 0 & 0 \\\\\n \\hline\n (0, 0.4] & 46 & 88 & 62 \\\\\n \\hline\n (0.4, 0.7] & 114 & 49 & 12 \\\\\n \\hline\n (0.7, 0.9] & 153 & 5 & 0 \\\\\n \\hline\n (0.9, 1] & 171 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\end{minipage}\n \\caption{RF distances of estimated 5-taxon trees by bootstrap support level for Lily-T (left) and Lily-Q (right).}\n \\label{boot_table5}\n\\end{table}\n\n\\begin{table}[]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} & \\textbf{RF 10} \\\\\n \\hline\n 0 & 1 & 1 & 4 & 1 & 1 & 1 \\\\\n \\hline\n (0, 0.2] & 129 & 117 & 72 & 19 & 3 & 1 \\\\\n \\hline\n (0.2, 0.9] & 205 & 41 & 3 & 0 & 0 & 0 \\\\\n \\hline\n (0.9, 1] & 100 & 0 & 0 & 0 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{RF distances of 8-taxon trees estimated by Lily-T for a given bootstrap support level}\n \\label{boot_table8T}\n\\end{table}\n\n\\begin{table}[]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|}\n \\hline\n \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} \\\\\n \\hline\n (0, 0.4] & 143 & 113 & 47 & 14 & 3 \\\\\n \\hline\n (0.4, 0.7] & 149 & 13 & 0 & 0 & 0 \\\\\n \\hline\n (0.7, 0.9] & 87 & 0 & 0 & 0 & 0 \\\\\n \\hline\n (0.9, 1] & 131 & 0 & 0 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{RF distances of 8-taxon trees estimated by Lily-Q for a given bootstrap support level}\n \\label{boot_table8Q}\n\\end{table}\n\n\\begin{table}[]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} & \\textbf{RF 10} \\\\\n \\hline\n 0 & 3 & 5 & 3 & 2 & 0 & 1 \\\\\n \\hline\n (0, 0.2] & 131 & 108 & 55 & 27 & 6 & 1 \\\\\n \\hline\n (0.2, 0.4] & 68 & 25 & 3 & 0 & 0 & 0 \\\\\n \\hline\n (0.4, 0.7] & 95 & 4 & 0 & 0 & 0 & 0 \\\\\n \\hline\n (0.7, 1] & 63 & 0 & 0 & 0 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{RF distances of 12-taxon trees estimated by Lily-T for a given bootstrap support level}\n \\label{boot_table12T}\n\\end{table}\n\n\\begin{table}[]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Bootstrap support} & \\textbf{RF 0} & \\textbf{RF 2} & \\textbf{RF 4} & \\textbf{RF 6} & \\textbf{RF 8} & \\textbf{RF 10} \\\\\n \\hline\n 0 & 1 & 2 & 1 & 1 & 0 & 0 \\\\\n \\hline\n (0, 0.2] & 75 & 52 & 36 & 8 & 6 & 1 \\\\\n \\hline\n (0.2, 0.4] & 91 & 36 & 4 & 0 & 0 & 0 \\\\\n \\hline\n (0.4, 0.7] & 129 & 7 & 0 & 0 & 0 & 0 \\\\\n \\hline\n (0.7, 0.9] & 88 & 0 & 0 & 0 & 0 & 0 \\\\\n \\hline\n (0.9, 1] & 62 & 0 & 0 & 0 & 0 & 0 \\\\\n \\hline\n \\end{tabular}\n \\caption{RF distances of 12-taxon trees estimated by Lily-Q for a given bootstrap support level}\n \\label{boot_table12Q}\n\\end{table}\n\n\\subsection{Application to empirical sequence data}\nWe applied Lily-T and Lily-Q to four empirical datasets. For all the datasets we estimated the topology and then drew 100 bootstrap samples and calculated the bootstrap support at each node. The first dataset is a somewhat toy example: a segment of 888 sites over five genes coming from the mitochondria of nine primates \\cite{yang97}. Since mitochondria are passed down solely from the mother, we performed the bootstrap as if the data were from a single 888-site gene since all sites share a common lineage. The results are displayed in figure \\ref{primate_out}. Both Lily-T and Lily-Q inferred the same tree as \\cite{yang97}. This tree is different than the primate consensus from \\cite{chou15, gatesy17} in that infers that tarsiers and lemurs are a cherry rather than that lemurs are an outgroup, likely due to the small number of sites from a single lineage. Lily-Q had higher bootstrap support than Lily-T for most nodes, especially for the (human, chimp) clade. Run time for both methods was a matter of seconds, including bootstrapping.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.4]{\"priT\".png}\n \\caption{Lily-T}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.4]{\"priQ\".png}\n \\caption{Lily-Q}\n \\end{subfigure}\n \\caption{Estimates for 9-taxon primate tree with bootstrap support for each clade. Lily-Q display assumes root location is known.}\n \\label{primate_out}\n\\end{figure}\n\nThe second dataset consisted of approximately 127,000 sites over 106 genes from eight yeast species: \\textit{S.cerevisiae} (Scer), \\textit{S. paradoxus} (Spar), \\textit{S. mikatae} (Smik), \\textit{S. kudriavzevii} (Skud), \\textit{S. bayanus} (Sbay), \\textit{S. castellii} (Scas), \\textit{S. kluyveri} (Sklu), and the outgroup \\textit{C. albicans} (Calb) \\cite{rokas03, wen18}. Both methods matched the results from \\cite{wen18}. Estimation took approximately 4 min including bootstrapping on a MacBook Air. All but one node had 100\\% bootstrap support. The exception was some uncertainty over whether \\textit{Skud} and \\textit{Sbay} represent a clade or if \\textit{Sbay} is an outgroup to all of \\textit{Scer}, \\textit{Spar}, and \\textit{Smik} (see figure \\ref{yeast_out}).\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.4]{\"YeastT\".png}\n \\caption{Lily-T}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.4]{\"YeastQ\".png}\n \\caption{Lily-Q}\n \\end{subfigure}\n \\caption{Estimates for 8-taxon yeast tree with bootstrap support for each clade. Lily-Q display assumes root location is known.}\n \\label{yeast_out}\n\\end{figure}\n\nNext, we applied Lily-T and Lily-Q to a dataset of 52 individuals from seven North American snake species consisting of aligned sequences from 18 nuclear and 1 mitochondrial genes with 190-850 characters per gene. Runtime was around a half hour on a Unix HPC platform (the increase in run time was due to the many different combination of individuals within each subset of three or four species). This is much less than the runtimes reported by \\cite{chifman14}: $<$1 day for SVDQuartets and $\\approx$10 days for *BEAST, although we acknowledge runtimes for these methods have improved in the time since those results were published. Our results are shown in figure \\ref{snake_out}. Bootstrap support was over 90\\% for all clades except the (S.m.miliarius, S.m.barbouri) clade. For that clade, bootstrap support was 34\\% for Lily-T and 67\\% for Lily-Q. SVDQuartets also exhibited low bootstrap support for this clade, so the weak support may be a function of the weak phylogenetic signal present in the data.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.4]{\"snakeT\".png}\n \\caption{Lily-T}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.4]{\"snakeQ\".png}\n \\caption{Lily-Q}\n \\end{subfigure}\n \\caption{Estimates for 7-taxon snake tree with bootstrap support for each clade. (Species names abbreviated for clarity. Full names in \\cite{chifman14}). Lily-Q display assumes root location is known.}\n \\label{snake_out}\n\\end{figure}\n\nFinally, we estimated the topology for four mosquito species using aligned sequences consisting of over 25 million sites from around 80,000 different genes. For Lily-T, we ran 100 bootstrap samples and there was 100\\% bootstrap support for each node, with total computation time under an hour using a Unix HPC platform even with this large dataset. Because we did not have the delineation of the different genes, we resampled the sites for bootstrapping in a single stage. The estimated species tree is shown in figure \\ref{mos_out} and matched the results from \\cite{tha18}. Since there were only four taxa, Lily-Q could calculate a posterior probability of 100\\% using a single run which took around 15 seconds. \n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"mosT\".png}\n \\caption{Lily-T}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\linewidth} \\centering\n \\includegraphics[scale=0.3]{\"mosQ\".png}\n \\caption{Lily-Q}\n \\end{subfigure}\n \\caption{Estimates for 4-taxon yeast mosquito with bootstrap support (Lily-T). Lily-Q display assumes root location is known (posterior probability of 100\\% not shown).}\n \\label{mos_out}\n\\end{figure}\n\n\\section{Conclusions and further work}\nLily-T and Lily-Q are two new methods for fast, accurate species tree estimation under the multispecies coalescent model along with the assumption of the molecular clock. The methods are insensitive to the value of the coalescent population parameter $\\theta$, and sensitivity to the prior on branching times can be controlled through our method for estimating root age from the data. Lily-Q is more accurate than Lily-T, but Lily-T does have certain advantages in that it can estimate the root location, and there is a theoretical guarantee as the number of unlinked sites goes to infinity.\n\nThe comparison between Lily-Q and ASTRAL follows our expectations in that ASTRAL is able to perform well under conditions when estimation of the individual genes is likely to be accurate -- when $\\theta$ is large and when there are a large number of sites per gene. It is worth noting that our comparisons are of a correctly specified ASTRAL model against an incorrectly specified Lily-Q model: we assume that we have delineated the genes correctly and that sites within each gene have a common history whereas Lily-Q assumes that all sites are unlinked. Even with this disadvantage, Lily-Q outperformed ASTRAL for many parameter settings. There is also reason to believe that the number of sites per gene needed for ASTRAL to do better than Lily-Q may not be realistic -- \\cite{hobolth11} estimated 75\\% of loci in a primate dataset have a recombination-free length of between 17 and 93bp and \\cite{springer16} reported a loci length of around 12bp for a large mammalian dataset. An avenue for further research would be to test Lily-Q against ASTRAL when the genes are not properly delimited to measure how much this degrades ASTRAL performance.\n\nWe also did not compare directly to any concatenation methods. These comparisons may indicate the limits of concatenation methods, especially for the highly asymmetric trees with short MBL. One comparison in particular that may be of interest would be to compare Lily-T to SMRT-ML \\cite{SMRTML}, which estimates rooted triplets using concatenated data. We should highlight that, unlike concatentation or coestimation methods, we can only estimate $S$, and not any other parameters of interest such as $\\boldsymbol{\\tau}$, although other methods for estimating $\\boldsymbol{\\tau}$ incorporating ILS such as that of \\cite{peng20} may be able to be used in conjunction with our work.\n\nWhile both the Lily-T and Lily-Q assembly steps generate errors in the face of three or more identical sequences, there are ways to work around this problem. For example, consider the case with six taxa where taxa $a$, $b$, and $c$ have identical sequences. As a first step $a$ and $b$ could be simply removed from consideration, inference applied to the remaining taxa (perhaps inferring the split $(c,d,(e,f))$, which results in the final inferred tree in figure \\ref{polytomy}. While such a workaround would be unwieldy for the large-scale simulation study performed here, it could be reasonably implemented on empirical data.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\linewidth]{\"polytomy\".png}\n \\caption{Handling identical sequences: After first treating the identical sequences as coming from a single species, the rest of the tree is resolved as $(c,d,(e,f))$. Then $c$ is replaced by the polytomy $(a,b,c)$.}\n \\label{polytomy}\n\\end{figure}\n\nWe have presented some data showing that high bootstrap support is indicative of high tree estimation accuracy, but we wish to highlight that we need to explore this claim further in future work. First, to minimize simulation time, we only performed the bootstrap analysis on twenty different parameter settings. Second, we would like to perform further simulation studies to investigate whether bootstrap support for a particular clade has predictive value for the clade being present in the true tree.\n\nFinally, this work has all been done under the molecular clock assumption, and we have some early signs that the methods are in fact quite sensitive to this assumption. To see why, consider a true tree $((a,b),c)$ with the branch leading to $a$ having a much larger mutation rate than other branches. This would lead to a larger number of YXX patterns than under equal rates, which if the divergence in rates was large enough, would lead Lily-T to infer $(a,(b,c))$ rather than the true tree. We hope to investigate this in greater detail in future work. Correcting for sensitivity to violations of the clock would require deriving $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau},\\boldsymbol{\\gamma}) $ where $\\boldsymbol{\\gamma}$ is the vector of relative rates along different branches of the tree. Then we would need to either estimate $\\hat{\\boldsymbol{\\gamma}}$ from the data or apply a prior to $\\boldsymbol{\\gamma}$ and integrate over it to get $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}) $. Whether these steps can be performed without substantially slowing down the methodology remains an open question.\n\n\\section{Appendix}\n\n\\subsection{Calculation of $\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}))$ for 3-taxon trees}\n\\label{tripletApp}\nThe fifteen site pattern probabilities derived in \\cite{chifman15} can be mapped down to the five site patterns for three taxa by summing over the fourth taxon we are not interested in to get the following relationships (dropping the conditioning on $(S, \\boldsymbol{\\tau})$ for clarity):\n$$p_0=\\delta_{XXX}=\\delta_{XXXX}+3\\delta_{YXXX}$$\n$$p_1=\\delta_{YXX}=\\delta_{XYXX}+\\delta_{XXYY}+2\\delta_{YZXX}$$\n$$p_2=\\delta_{XXY}=\\delta_{XYX}=\\delta_{XXXY}+\\delta_{XYYX}+2\\delta_{YXXZ}=\\delta_{XXYX}+\\delta_{XYXY}+2\\delta_{YXZX}$$\n$$p_3=\\delta_{XYZ}=\\delta_{XXYZ}+\\delta_{XYXZ}+\\delta_{XYZX}+\\delta_{XYZW}$$\nThese probabilities take the form:\n\\begin{eqnarray}\n\\label{3taxaProb1}\n \\left\\{\n \\begin{aligned}\n& p_0=c_0+3c_1+6c_2+12c_3\\\\\n& p_1=c_0+3c_1-2c_2-4c_3\\\\\n& p_2=c_0-c_1+2c_2-4c_3\\\\\n& p_3=c_0-c_1-2c_2+4c_3\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nwhere (measuring $\\tau_1$ and $\\tau_2$ in coalescent units):\n\\begin{eqnarray}\n\\label{3taxaProb2}\n \\left\\{\n \\begin{aligned}\n& c_0=1\/64\\\\\n& c_1=\\frac{e^{-8\\tau_1\\theta\/3}}{64(1+\\frac{8\\theta}{3})}\\\\\n& c_2=\\frac{e^{-8\\tau_2\\theta\/3}}{64(1+\\frac{8\\theta}{3})}\\\\\n& c_3=\\frac{e^{-4\\tau_1\\theta\/3}e^{-8\\tau_2\\theta\/3}}{128(1+\\frac{8\\theta}{3})(1+\\frac{4\\theta}{3})}\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nIt is easily verified that:\n\\begin{equation}\n\\label{3taxaProb3}\n 4p_0+12p_1+24p_2+24p_3=1\n\\end{equation}\nThe coefficients in equation \\ref{3taxaProb3} arise as follows. $X$ can represent any of $A$, $C$, $G$, or $T$. $Y$ can represent any of the remaining three characters, and $Z$ any of the remaining two. Finally, the coefficient of $p_2$ is doubled to account for both $\\delta_{XYX}$ and $\\delta_{XXY}$.\n\n\\subsection{Calculation of $\\boldsymbol{\\delta}|S$ for 3-taxon trees}\nOne can see from equation \\ref{3taxaProb1} that $\\delta_k|(S,\\boldsymbol{\\tau})$ is a linear combination of terms, so $\\delta_k|S$ is also linear combination of terms where each integral is of one of two forms (since the prior on $\\boldsymbol{\\tau}$ is $f(\\boldsymbol{\\tau}) = \\frac{1}{\\tau_2}\\beta e^{-\\beta\\tau_2}$): \n\n\\textbf{Form one:}\n\\begin{equation}\n\\label{post1}\n \\int_0^{\\infty} \\int_0^{\\tau_2} \\frac{c}{\\tau_2}e^{-a\\tau_2} d\\tau_1 d\\tau_2=\\int_0^{\\infty} ce^{-a\\tau_2}d\\tau_2=\\frac{c}{a}\n\\end{equation}\n\n\\textbf{Form two:}\n$$\\int_0^{\\infty} \\int_0^{\\tau_2} \\frac{c}{\\tau_2}e^{-b\\tau_1}e^{-a\\tau_2} d\\tau_1 d\\tau_2$$\n$$=\\int_0^{\\infty}\\frac{c}{b\\tau_2} (1-e^{-b\\tau_2}) e^{-a\\tau_2}d\\tau_2$$\n\\begin{equation}\n\\label{post2}\n =\\frac{c}{b}log(\\frac{a+b}{a})\n\\end{equation}\nTaken together, $\\delta_k|S$ has the linear form of equation \\ref{3taxaProb1} with the following terms replacing those of equation \\ref{3taxaProb2}: \n\\begin{eqnarray}\n\\label{3taxaProb4}\n \\left\\{\n \\begin{aligned}\n& c_0=1\/64\\\\\n& c_1=\\frac{\\beta}{64(1+\\frac{8\\theta}{3})(\\frac{8\\theta}{3})}log(\\frac{\\beta+\\frac{8\\theta}{3}}{\\beta})\\\\\n& c_2=\\frac{\\beta}{64(1+\\frac{8\\theta}{3})(\\beta+\\frac{8\\theta}{3})}\\\\\n& c_3=\\frac{\\beta}{64(1+\\frac{8\\theta}{3})(2+\\frac{8\\theta}{3})(\\frac{4\\theta}{3})}log(\\frac{\\beta+4\\theta}{\\beta+\\frac{8\\theta}{3}})\n \\end{aligned}\n \\right.\n\\end{eqnarray}\nThese results were verified by Monte Carlo integration with 10,000 draws from the prior on $\\boldsymbol{\\tau}$ with $\\beta=0.1$.\n\n\\subsection{Calculation of $\\boldsymbol{\\delta}|S$ for 4-taxon trees}\n\nFrom Chifman and Kubatko (2015), $P(\\boldsymbol{\\delta}|(S,\\boldsymbol{\\tau}))$ is the inner product $\\textbf{c}^{T}\\textbf{b}$ where $\\textbf{c}$ is a vector of constants (with respect to the branching times) and $\\textbf{b}$ is a vector of functions of the branching times. These two vectors (where $\\alpha=4\/3$ is a constant that comes from the mutation rate when time is measured in coalescent units) are:\n$$\\boldsymbol{b}_{sym}=\\{1,e^{-2\\alpha\\theta\\tau_1},e^{-2\\alpha\\theta\\tau_2}, e^{-2\\alpha\\theta(\\tau_1+\\tau_2)}, e^{-2\\alpha\\theta\\tau_3},\n e^{-\\alpha\\theta(\\tau_1+2\\tau3)}, e^{-\\alpha\\theta(\\tau_2+2\\tau3)},$$ $$e^{-\\alpha\\theta(\\tau_1+\\tau_2+2\\tau3)},\n e^{2\\alpha(\\tau_1+\\tau_2)-4\\theta\\tau_3(\\alpha+\\frac{1}{2\\theta})}\n \\} $$\n$$\\boldsymbol{b}_{asymm}=\\{1,e^{-2\\alpha\\theta\\tau_1},e^{-2\\alpha\\theta\\tau_2}, e^{-\\alpha\\theta(\\tau_1+2\\tau_2)}, e^{-2\\alpha\\theta\\tau_3},\n e^{-\\alpha\\theta(\\tau_1+2\\tau3)}, e^{-2\\alpha\\theta(\\tau_1+\\tau3)},$$ $$ e^{-\\alpha\\theta(\\tau_2+2\\tau3)}, e^{-\\alpha\\theta(\\tau_1+\\tau_2+2\\tau3)},\n e^{2(\\tau_1-\\tau_2)-2\\theta\\alpha(\\tau_2+\\tau3)}\n \\} $$\nSince the priors are $f(\\boldsymbol{\\tau}) = \\frac{1}{\\tau_3^2}\\beta e^{-\\beta\\tau_3}$ for the symmetric topology and $f(\\boldsymbol{\\tau}) = \\frac{2}{\\tau_3^2}\\beta e^{-\\beta\\tau_3}$ for the asymmetric topology, the integrals required to integrate out the branching times take on only eight different forms (four for each topology).\n\\subsubsection{Symmetric topology}\nForm One:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} c e^{-f\\tau_3}d\\tau_3= \\frac{c}{f}$$\nForm two:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\frac{c}{a\\tau_3}(1-e^{-a\\tau_3}) e^{-f\\tau_3}d\\tau_3= \\frac{c}{a}(log(a+f)-log(f))$$\nAfter applying integration by parts.\\\\\nForm three\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\frac{c}{b\\tau_3}(1-e^{-b\\tau_3}) e^{-f\\tau_3}d\\tau_3= \\frac{c}{b}(log(b+f)-log(f))$$\nForm four:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_3} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3= \\int_0^{\\infty} \\frac{c}{ab\\tau_3^2} (1-e^{-a\\tau_3})(1-e^{-b\\tau_3})e^{-f\\tau_3} d\\tau_3$$\nUse integration by parts to obtain (the first term vanishes):\n$$=\\frac{c}{ab}\\int_0^{\\infty}\\frac{1}{\\tau_3}(-fe^{-f\\tau_3}+(a+f)e^{-(a+f)\\tau_3}+(b+f)e^{-(b+f)\\tau_3}-(a+b+f)e^{-(a+b+f)\\tau_3}) d\\tau_3$$\nAfter applying integration by parts a second time (again, the first term vanishes):\n$$=\\frac{c}{ab}(flog(f)-(a+f)log(a+f)-(b+f)log(b+f)+(a+b+f)log(a+b+f))$$\n\\subsubsection{Asymmetric topology}\nForm One:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\frac{c}{2} e^{-f\\tau_3}d\\tau_3= \\frac{c}{2f}$$\nForm Two:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3 =\\int_0^{\\infty} \\int_0^{\\tau_3} \\frac{c}{a\\tau_3^2} (1-e^{-a\\tau_2})e^{-f\\tau_3} d\\tau_2 d\\tau_3$$\n$$=\\int_0^{\\infty} \\frac{c}{a^2\\tau_3^2}(a\\tau_3e^{-f\\tau_3}-e^{-f\\tau_3}+e^{-(a+f)\\tau_3}) d\\tau_3$$\n$$= \\int_0^{\\infty} \\frac{c}{a\\tau_3}e^{-f\\tau_3} d\\tau_3 + \\int_0^{\\infty} \\frac{c}{a^2\\tau_3^2} (e^{-(a+f)\\tau_3}-e^{-f\\tau_3})$$\nPerform integration by parts once on the second integral to obtain:\n$$= \\int_0^{\\infty} \\frac{c}{a\\tau_3}e^{-f\\tau_3} d\\tau_3 + \\int_0^{\\infty} \\frac{c}{a^2\\tau_3} (fe^{-f\\tau_3}-(a+f)e^{-(a+f)\\tau_3}) - \\frac{c}{a}$$\nGrouping terms:\n$$= \\frac{c(a+f)}{a^2} \\int_0^{\\infty} \\frac{1}{\\tau_3} (e^{-f\\tau_3}-e^{-(a+f)\\tau_3}) - \\frac{c}{a}$$\nAfter performing integration by parts again:\n$$=\\frac{c(a+f)}{a^2}(log(a+f)-log(f))-\\frac{c}{a} $$\nForm three:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\int_0^{\\tau_3} \\frac{c\\tau_2}{\\tau_3^2}e^{-b\\tau_2}e^{-f\\tau_3} d\\tau_2d\\tau_3$$\n$$=\\int_0^{\\infty} \\frac{c}{b^2\\tau_3^2}(e^{-d\\tau_3}-e^{-(b+d)\\tau_3}-b\\tau_3e^{-(b+d)\\tau_3}) d\\tau_3$$\nUsing integration by parts gives:\n$$=\\frac{c}{b}+\\int_0^{\\infty} \\frac{cf}{b^2\\tau_3} (e^{-(b+f)\\tau_3}-e^{-f\\tau_3})$$\nAfter applying integration by parts a second time:\n$$=\\frac{c}{b}+\\frac{cf}{b^2}(log(f)-log(b+f)) $$\nForm four: \n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{c}{\\tau_3^2} e^{-a\\tau_1} e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3=\\int_0^{\\infty} \\int_0^{\\tau_3} \\frac{c}{a\\tau_3^2} (1-e^{-a\\tau_2}) e^{-b\\tau_2} e^{-f\\tau_3} d\\tau_2 d\\tau_3$$\n$$=\\int_0^{\\infty} \\frac{c}{ab(a+b)\\tau_3^2} (ae^{-f\\tau_3}-(a+b)e^{-(b+f)\\tau_3}+be^{-(a+b+f)\\tau_3}) d\\tau_3$$\nApplying integration by parts (the first term vanishes) gives:\n$$= \\int_0^{\\infty} \\frac{c}{ab(a+b)\\tau_3} (-afe^{-f\\tau_3}+(a+f)(b+f)e^{(b+f)\\tau_3}-b(a+b+f)e^{-a+b+f)\\tau_3}) d\\tau_3$$\nAfter integrating by parts a second time (again the first term vanishes):\n$$=\\frac{c}{ab(a+b)}(aflog(f)-(a+b)(b+f)log(b+f)+b(a+b+f)log(a+b+f))$$\n\\subsubsection{Example}\nAs an example, we will show the calculation of the second term of $\\textbf{c}^T\\textbf{d}$ for an asymmetric topology. From above, $b_2=e^{-2\\alpha\\theta\\tau_1}$ and so the integral is:\n$$\\int_0^{\\infty} \\int_0^{\\tau_3} \\int_0^{\\tau_2} \\frac{2\\beta c_2}{\\tau_3^2} e^{-2\\alpha\\theta\\tau_1} e^{-\\beta\\tau_3} d\\tau_1 d\\tau_2 d\\tau_3 $$\nThis is of form two where $c=2\\beta c_2$, $a=2\\alpha\\theta$, and $f=\\beta$.\nThen, from above,\n$$d_2=\\frac{2\\beta c_2(2\\alpha\\theta+\\beta)}{(2\\alpha\\theta)^2}(log(2\\alpha\\theta+\\beta)-log(\\beta))-\\frac{2\\beta c_2}{2\\alpha\\theta}$$\n\\subsubsection{Final results}\n$$d_{sym,1}=1 $$\n$$d_{sym,2}=d_{sym,3}=\\beta\\frac{log(2\\alpha\\theta+\\beta)-log(\\beta)}{2\\alpha\\theta}$$\n$$d_{sym,4}= \\beta\\frac{\\beta log(\\beta)-2(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)+(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta)}{4\\alpha^2\\theta^2}$$\n$$d_{sym,5}= \\beta\\frac{1}{2\\alpha\\theta+\\beta}$$\n$$d_{sym,6}= d_{sym,7}=\\beta\\frac{log(3\\alpha\\theta+\\beta)-log(2\\alpha\\theta+\\beta)}{\\alpha\\theta}$$\n$$d_{sym,8}= \\beta\\frac{(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)\n-2(3\\alpha\\theta+\\beta)log(3\\alpha\\theta+\\beta)\n+(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta)}{\\alpha^2\\theta^2}$$\n$$d_{sym,9}=\\beta(\\beta+4\\theta(\\alpha+\\frac{1}{2\\theta}))log((\\beta+4\\theta(\\alpha+\\frac{1}{2\\theta}))\n-2(\\beta-1+4\\theta(\\alpha+\\frac{1}{2\\theta}))log((\\beta-1+4\\theta(\\alpha+\\frac{1}{2\\theta}))$$\n$$+(\\beta-2+4\\theta(\\alpha+\\frac{1}{2\\theta}))log((\\beta-2+4\\theta(\\alpha+\\frac{1}{2\\theta}))$$\n$$d_{asymm,1}=1 $$\n$$d_{asymm,2}= 2\\beta\\frac{2\\alpha\\theta+\\beta}{4\\alpha^2\\theta^2}(log(2\\alpha\\theta+\\beta)-log(\\beta))-\\frac{2\\beta}{2\\alpha\\theta}$$\n$$d_{asymm,3}= 2\\beta\\frac{\\beta}{4\\alpha^2\\theta^2}(log(\\beta)-log(2\\alpha\\theta+\\beta))+\\frac{2\\beta}{2\\alpha\\theta}$$\n$$d_{asymm,4}= \\frac{2\\beta}{6\\alpha^3\\theta^3}(\\alpha\\theta\\beta log(\\beta)-(3\\alpha\\theta)(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)+(2\\alpha\\theta)(3\\alpha\\theta+\\beta)log(3\\alpha\\theta+\\beta))$$\n$$d_{asymm,5}=\\frac{2\\beta}{4\\alpha\\theta+2\\beta}$$\n$$d_{asymm,6}=2\\beta\\frac{3\\alpha\\theta+\\beta}{\\alpha^2\\theta^2}(log(3\\alpha\\theta+\\beta)-log(2\\alpha\\theta+\\beta))-\\frac{2\\beta}{\\alpha\\theta}$$\n$$d_{asymm,7}=2\\beta\\frac{4\\alpha\\theta+\\beta}{4\\alpha^2\\theta^2}(log(4\\alpha\\theta+\\beta)-log(2\\alpha\\theta+\\beta))-\\frac{2\\beta}{2\\alpha\\theta}$$\n$$d_{asymm,8}=2\\beta\\frac{2\\alpha\\theta+\\beta}{\\alpha^2\\theta^2}(log(2\\alpha\\theta+\\beta)-log(3\\alpha\\theta+\\beta))+\\frac{2\\beta}{\\alpha\\theta}$$\n$$d_{asymm,9}=\\frac{2\\beta}{2\\alpha^3\\theta^3}(\\alpha\\theta(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)-(2\\alpha\\theta)(3\\alpha\\theta+\\beta)log(3\\alpha\\theta+\\beta)+(\\alpha\\theta)(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta))$$\n$$d_{asymm,10}=\\frac{-2\\beta}{(2\\alpha\\theta)(1+2\\alpha\\theta)}(-(2\\alpha\\theta+\\beta)log(2\\alpha\\theta+\\beta)-(2\\alpha\\theta)(1+4\\alpha\\theta+\\beta)log(1+4\\alpha\\theta+\\beta)$$\n$$+(1+2\\alpha\\theta)(4\\alpha\\theta+\\beta)log(4\\alpha\\theta+\\beta)) $$\n\n\\subsection{Proof of Theorem 1}\n\\label{mainTheorem}\n\n\\begin{proof}\nWithout loss of generality, let $S_0=(a,(b,c))$ be the true tree and $\\boldsymbol{\\tau}_0$ be the true value of $\\boldsymbol{\\tau}$. We want to show that $\\forall \\epsilon>0 \\, \\, \\exists J_0 : \\forall J>J_0 \\, \\, P(\\hat{S}=S)>1-\\epsilon$. First, note that for all values of $(\\boldsymbol{\\tau}, \\theta)$, we have $\\delta_{XXY}=\\delta_{XYX}$. \n\nSecond, we note using the results of equation \\ref{3taxaProb1}\n\\begin{equation}\n \\delta_{YXX}|((a,(b,c)),\\boldsymbol{\\tau}) > \\delta_{XYX}|((a,(b,c)),\\boldsymbol{\\tau})\n \\label{expectation}\n\\end{equation}\n$$\\leftrightarrow c_0+3c_1-2c_2-4c_3 > c_0-c_1+2c_2-4c_3$$\n$$\\leftrightarrow \\frac{e^{-2m\\tau_1\\theta}}{64(1+2m\\theta)}=c1>c2=\\frac{e^{-2m\\tau_2\\theta}}{64(1+2m\\theta)}$$\n$$\\leftrightarrow \\tau_1<\\tau_2$$\nwhich holds w.p.1 by assumption.\n\nNext, note \n$$\\int_{\\boldsymbol{\\tau}}\\delta_{YXX}|((a,(b,c)),\\boldsymbol{\\tau})f(\\boldsymbol{\\tau})d(\\boldsymbol{\\tau})=\\delta_{YXX}|(a,(b,c))$$ \n$$> \\delta_{XYX}|(a,(b,c))=\\int_{\\boldsymbol{\\tau}}\\delta_{XYX}|((a,(b,c)),\\boldsymbol{\\tau})f(\\boldsymbol{\\tau})d(\\boldsymbol{\\tau})$$\nby properties of expectations from equation \\ref{expectation} and since $f(\\boldsymbol{\\tau})>0$ a.e. under the prior. \n\nSince each topology $(a,(b,c))$, $(b,(a,c))$, and $(c,(a,b))$ has a prior probability of 1\/3, the maximum posterior topology will be the maximum likelihood topology. Recall from section \\ref{derivation} that we can permute any site pattern probability given one topology to find a site pattern probability given another topology -- e.g., $\\delta_{YXX}|(a,(b,c))=\\delta_{XYX}|(b,(a,c))$.\nSo, the likelihood of trees $(a,(b,c))$ and $(b,(a,c))$ given the data are:\n$$L((a,(b,c))|\\boldsymbol{d}) =c_{\\boldsymbol{d}} (p_{xxx})^{d_{xxx}}(p_{xxy})^{d_{xxy}}(p_{xyx})^{d_{xyx}}(p_{yxx})^{d_{yxx}}(p_{xyz})^{d_{xyz}} $$\n$$L((b,(a,c))|\\boldsymbol{d}) =c_{\\boldsymbol{d}} (p_{xxx})^{d_{xxx}}(p_{xxy})^{d_{xxy}}(p_{xyx})^{d_{yxx}}(p_{yxx})^{d_{xyx}}(p_{xyz})^{d_{xyz}} $$\nBy cancelling terms in common, one can see that the likelihood ratio comes down to permuting the number of sites that follow the XYX and YXX patterns:\n$$\\frac{L((a,(b,c))|\\boldsymbol{d})}{L((b,(a,c))|\\boldsymbol{d})}= (p_{xyx})^{d_{xyx}-d_{yxx}}(p_{yxx})^{d_{yxx}-d_{xyx}}$$\nTogether $p_{yxx}>p_{xyx}$ and $d_{yxx}>d_{xyx}$ imply that $\\frac{L((a,(b,c))|\\boldsymbol{d})}{L((b,(a,c))|\\boldsymbol{d})}>1$, which implies that $\\hat{S}=S$. As we have shown that $p_{yxx}>p_{xyx}$, it is sufficient to show that $d_{yxx} \\rightarrow p_{yxx}$ and $d_{xyx} \\rightarrow p_{xyx}$ for sufficiently large number of sites.\n\nLet $p_{yxx,0}=\\delta_{yxx}|((a,(b,c)),\\boldsymbol{\\tau}_0)$ and $p_{xyx,0}=\\delta_{xyx}|((a,(b,c)),\\boldsymbol{\\tau}_0)$. From the SLLN, we have \n$$\\forall \\epsilon_1,\\delta_1>0 \\, \\, \\exists J_1 : \\forall J>J_1 \\, \\, P(\\frac{d_{yxx}}{J}>p_{yxx,0}-\\delta_1)>1-\\epsilon_1$$\n$$\\forall \\epsilon_2,\\delta_2>0 \\, \\, \\exists J_2 : \\forall J>J_2 \\, \\, P(\\frac{d_{xyx}}{J}1-\\epsilon_2$$\nChoose $\\delta_1,\\delta_2$ such that $\\delta_1+\\delta_2p_{yxx,0}-\\delta_1, p_{xyx,0}+\\delta_2>\\frac{d_{xyx}}{J})>1-\\epsilon$ and the result holds.\n\n\\end{proof}\n\n\n\n\\def\\BibTeX{{\\rm B\\kern-.05em{\\sc i\\kern-.025em b}\\kern-.08em\n T\\kern-.1667em\\lower.7ex\\hbox{E}\\kern-.125emX}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn the field-antifield formalism \\cite{BV,BV1,BV2}, the concept of\ndeformations based on a nilpotent higher-order operator $\\Delta$ was\ndeveloped in a series of articles\n\\cite{BT3,Ak,AD,BDA,BBD1,BBD3,BM1,BM2,BM3}. Such deformations\nusually modify the Jacobi identity with BRST exact terms. In\ncontrast to that, one can, with no assumptions {\\it a priori} on\nunderlying $\\Delta$ operator, consider \"local\" deformations of the\nantibracket with a Boson deformation parameter, such that the Jacobi\nidentity holds strongly.\n\nHistorically, the deformed antibracket has been studied in the\narticles \\cite{LSh,KoT,KoT1,KoT2}. In Ref. \\cite{BB2}, the deformed\n$\\Delta$-operator has been found that differentiates the deformed\nantibracket, and the first attempt has been made to understand\nactually possible role of the deformed antibracket and\n$\\Delta$-operator in the construction of the $W-X$ version\n\\cite{BT1,BT2,BT3,BMS,BT4,BBD1,BBD2,BB3} of the Lagrange deformed\nfield-antifield Batalin-Vilkovisky (BV)-formalism \\cite{BB2}. If one\nbelieves that the deformed BV - formalism still describes\ngauge-invariant field systems, then there appears a difficult\nproblem of how the deformation can coexist with the usual\ngauge-fixing mechanism. Or, in other words, if that is possible to\nprovide for a proper solution to the deformed classical\/quantum\nmaster equation. In the present article, we will try to pay\nattention enough to seek for a possible way to resolve the mentioned\nproblem.\n\nIn principle, our present consideration is based essentially on the logic\nand mathematics of the article \\cite{BB2} of Batalin and Bering.\nRegrettably, these authors did not\naccomplish their task of construction of the nontrivially deformed field-antifield\nformalism\nbased on the deformed antibracket and $\\Delta$ operator. The main idea was to extend the\noriginal antisymplectic phase space with a single extra field-antifield pair just\ncontrolling the scale of deformation. Then, one defines a trivial deformation\nwithin the extended phase space,\nand then one should reduce effectively the scale of trivial deformation in\nsuch a way that the latter becomes nontrivial in a consistent manner. That idea seems\npromising, the same as before. However, there remains a difficult unresolved problem of\nconsistent coexistence between the properness principle and non-triviality of\nthe deformation. We would like to try again to attack that problem.\n\n\n\\section{Extended $\\Delta$-Operator}\n\nWe begin with the standard odd Laplacian operator,\n\\beq \\label{i2}\n\\Delta = \\frac{1}{2}(-1)^{\\varepsilon_{A}} \\pa_{A} E^{AB} \\pa_{B},\\quad\n\\pa_A= \\frac{\\pa}{\\pa Z^A}, \\quad\n\\varepsilon(\\Delta) = 1, \\quad \\Delta^{2} = 0, \n\\eeq where $Z^{A}$ , $\\varepsilon_{A} = \\varepsilon( Z^{A})$, are\noriginal Darboux coordinates of the field-antifield formalism, and\n$E^{AB}$ is a constant invertible antisymplectic metric with the\nusual statistics and dual antisymmetry properties,\n\\beq\n\\label{2.2}\n\\varepsilon( E^{AB} ) = \\varepsilon_{A} + \\varepsilon_{B} + 1, \\quad\nE^{AB} = - E^{BA}(-1)^{ (\\varepsilon_{A} + 1) (\\varepsilon _{B} +\n1)}.\n\\eeq\nIn what follows below, typical functions, depending only\non the $Z^{A}$ variables of the original sector, will be denoted in\nsmall letters. Now, let us extend the original $Z$-sector by\nincluding two new variables, a Boson $t$ and a Fermion $\\theta$, to\nextend the original odd Laplacian, to become \\cite{BB2} (see also\nAppendix A)\n\\beq\n\\label{i1} \\Delta_{\\tau} = t^{2} \\Delta + N_{\\tau}\n\\pa_{\\theta}, \\quad\n \\pa_{\\theta}=\\frac{\\pa}{\\pa \\theta}, \\quad\n \\varepsilon(\\Delta_{\\tau}) = 1,\\quad \\Delta^2_{\\tau} = 0,\n\\eeq\nwhere\n\\beq\n\\label{i3}\nN_{\\tau} = N + t \\pa_{t}, \\quad N = N^{A} \\pa_{A}. \n\\eeq\nIn what follows below, typical functions depending on the full set of variables,\n$Z^{A}, t, \\theta$, will be denoted in capital letters.\nNilpotency of $\\Delta_{\\tau}$ requires\n\\beq\n\\label{i4}\n[ \\Delta, N ] = 2 \\Delta, \n\\eeq\nor, in more detail,\n\\beq\n\\nonumber\n&&[ \\Delta, N ] = [ \\Delta, N^{A} ] \\pa_{A} = \\left[ ( \\Delta N^{A} ) +\n{\\rm ad}( N^{A} ) (-1)^{\\varepsilon_{A}} \\right]\\pa_{A} = \\\\\n\\nonumber\n&&=( \\Delta N^{A} ) \\pa_{A} +\n\\frac{1}{2} \\left[ ( N^{A}, Z^{B} ) -\n( A \\leftrightarrow B ) (-1)^{ (\\varepsilon_{A} +1) (\\varepsilon_{B} +1) } \\right]\n(-1)^{\\varepsilon_{A}} \\pa_{B}\\pa_{A} =\\\\\n\\label{i5}\n&&=2\\Delta = E^{AB} (-1)^{\\varepsilon_{A}} \\pa_{B} \\pa_{A}, \n\\eeq\nwhich, in turn, implies\n\\beq\n\\label{i6}\n\\Delta N^{A} = 0, \n\\eeq\nand\n\\beq\n\\label{i7}\nE^{AB} =\\frac{1}{2} \\left[ ( N^{A}, Z^{B} ) - ( A \\leftrightarrow B )\n (-1)^{ ( \\varepsilon_{A} +1 ) ( \\varepsilon_{B} + 1 ) } \\right]. \n\\eeq\nHere and below the notation\n\\beq\n\\label{ad}\n{\\rm ad}(F)(...)=(F,(...))\n\\eeq\nfor the left adjoint of the antibracket is used.\n\nIt follows immediately from (\\ref{i7}) that\n\\beq\n\\label{i10}\n( N^{A}, N^{B} ) =\\frac{1}{2}\\left[ N^{A}\\overleftarrow{\\pa}_{C} ( N^{C}, N^{B} ) -\n(A\\leftrightarrow B )(-1)^{(\\varepsilon_{A} + 1 )( \\varepsilon_{B} + 1)} \\right].\n\\eeq Here, in (\\ref{i5}), (\\ref{i7}), (\\ref{ad}), (\\ref{i10}), the usual\nantibracket, generated by the operator $\\Delta$, is used\n\\beq\n\\label{i8} ( f, g ) = (-1)^{\\varepsilon(f)}[ [ \\Delta, f ], g ]\n\\cdot 1=f \\overleftarrow{\\pa}_{A} E^{AB} \\overrightarrow{\\pa}_{B} g . \n\\eeq\nThus, the coefficients $N^{A}$ of the vector field $N$ should satisfy Eqs.\n(\\ref{i6}) and (\\ref{i7}).\nOf course, the simplest solution is obvious,\n\\beq\n\\label{i9}\nN^{A} = Z^{A}, \\quad N = Z^{A}\\pa_{A}. \n\\eeq\nThat was exactly the simplest ansatz used in the article \\cite{BB2} from the very\nbeginning. Here, in the present article, we do not restrict ourselves with any {\\it a priori}\nchoice of a special solution to Eqb. (\\ref{i6}) and (\\ref{i7}). Only these equations\nthemselves will be used in our further reasoning, nothing else.\nIt can be shown that the general solution to Eq. (\\ref{i7}) is\n\\beq\n\\label{i11}\nN^{A} = Z^{A} + 2( F, Z^{A} ), \\quad \\varepsilon( F ) = 1 \n\\eeq\nwith $F$ being arbitrary Fermion.\nIn that case, Eqs. (\\ref{i6}) and (\\ref{i7}) are satisfied as follows:\n\\beq\n\\label{i12}\nE^{AB} =\\frac{1}{2}\\left[ 2 E^{AB} + 2(F, E^{AB} ) \\right] = E^{AB},\n\\eeq\n\\beq\n\\label{i13}\n\\Delta N^{A} = 2(\\Delta F, Z^{A} ) = 0, \n\\eeq\n\\beq\n\\label{i14}\n\\Delta F = {\\rm const}( Z ). \n\\eeq\nThus, we find\n\\beq\n\\label{i15}\nN = Z^{A}\\pa_{A} + 2\\;{\\rm ad}( F ). \n\\eeq\n\nOne can rewrite Eq. (\\ref{i7}) in its natural form\n\\beq\n\\label{i16}\n\\delta^{A}_{D} - \\frac{1}{2} N^{A} \\overleftarrow{\\pa}_{D} =\n\\frac{1}{2} E^{AB} (\\overrightarrow{\\pa}_{B} N^{C } ) E_{CD}, \n\\eeq\nwhere $E_{AB}$ is the inverse to $E^{AB}$,\n\\beq\n\\label{i17}\nE^{AB} E_{BC} = \\delta^{A}_{C}. \n\\eeq\nNow, the super trace of (\\ref{i16}) yields\n\\beq\n\\label{i18}\n\\sTr I = \\delta^{A}_{A} (-1)^{\\varepsilon_{A}} = 0, \n\\eeq\nthat is fulfilled identically due to equal number of Bosons and Fermions\namong the variables $Z^{A}$.\nThe supertrace imposes no restrictions for the divergence of vector field $N$,\n\\beq\n\\label{i19}\n\\div N = \\pa_{A} N^{A} (-1)^{\\varepsilon_{A}}, \n\\eeq\nNote that $E^{AB}$ and $E_{AB}$ on the right-hand side of\n(\\ref{i16}) do enter in the form of a similarity transformation (\nthat is just the meaning of the naturalness of (\\ref{i16}) ) and,\ntherefore, they cancel each other when taking the supertrace\nor superdeterminant. Notice also, that (\\ref{i16}) is a \"bridge\"\nbetween the initial equation (\\ref{i7}) and its \"dual\" form, \\beq\n\\label{i20} E_{AB} = \\frac{1}{2} \\left[ E_{AC} ( N^{C}\n\\overleftarrow{\\pa}_{B} ) -\n( A \\leftrightarrow B ) (-1)^{ \\varepsilon_{A} \\varepsilon_{B} } \\right]. \n\\eeq\nIn turn, by substituting\n\\beq\n\\label{i20.7}\nN^{A} = - 2 E^{AB} V_{B}\n = 2 V_{B} E^{BA}, \n\\eeq\n\\beq\n\\varepsilon( V_{A} ) = \\varepsilon_{A} + 1, \n\\eeq into (\\ref{i20}), the latter takes the form\n\\beq\n\\label{i20.8}\nE_{AB} = \\pa_{A} V_{B} -\n\\pa_{B} V_{A} (-1)^{\\varepsilon_{A}\\varepsilon_{B}},\n\\eeq\nwhich tells us that $V_{A}$ is just the antisymplectic potential,\ngenerating a constant invertible\nmetric in its covariant components $E_{AB}$. Thereby, one realizes\nthat the arbitrariness in $N^{A}$\nis generated by the natural geometric arbitrariness in the choice of\nthe antisymplectic potential.\nIt can be shown that the general solution to Eq. (\\ref{i20.8}) is\n\\beq\n\\label{i.V}\nV_{B} = \\frac{1}{2} Z^{A} E_{AB} + \\pa_{B} F,\n\\eeq\nwith $F$ being arbitrary Fermion, so that (\\ref{i.V})\nis consistent with (\\ref{i20.8}), (\\ref{i20.7}).\nOf course, it follows from (\\ref{i6}), (\\ref{i20.7}), that the antisymplectic potential\n$V_{A}$ should satisfy the condition\n\\beq\n\\label{i20.9}\n\\Delta V_{A} = 0, \n\\eeq\nwhich is consistent with (\\ref{i14}).\nThe relation (\\ref{i20.8}) is invariant under the shift,\n\\beq\n\\label{i2.31}\nV_{A} = V'_{A} + \\pa_{A} F' ,\\quad \\varepsilon( F' ) = 1.\n\\eeq\nOn the other hand, we have,\n\\beq\n\\label{i2.32}\n\\Delta V_{A}=\\Delta V'_{A}+\\pa_{A} \\Delta F'(-1)^{\\varepsilon_A} = 0, \n\\eeq\n\\beq\n\\label{i2.33}\nN^{A} = N^{'A} - 2 E^{AB} \\pa_{B} F', \n\\eeq\n\\beq\n\\label{i2.34}\n\\Delta N^{A} = \\Delta N^{'A} +\n2 E^{AB} \\pa_{B} \\Delta F' (-1)^{\\varepsilon_{A}} = 0, \n\\eeq\n\\beq\n\\label{i2.35}\nN = N' + 2 {\\rm ad}( F' ) , \n\\eeq\n\\beq\n\\label{i2.36}\n\\div N = \\div N' - 4 \\Delta F' . \n\\eeq\nSo, if one chooses in (\\ref{i2.31}) - (\\ref{i2.35}) for the $F'$ to satisfy the relation\n\\beq\n\\label{i2.37}\n\\div N' = \\div N + 4 \\Delta F' = 0, \n\\eeq\nthen the new value (\\ref{i2.37}) of the $\\div N'$ is zero. Thereby, the new\noperator\n\\beq\n\\label{i2.38}\nN' = N - 2 {\\rm ad}(F') = - N'^{T}, \n\\eeq is an antisymmetric one. The transposed operation is defined\nvia\n\\beq\n\\int [dZ] (A^TF)G=(-1)^{\\varepsilon(A)\\varepsilon(F)}\\int [dZ]\nF(AG). \\eeq So far, the condition (\\ref{i2.37}) seems to be the only\nrestriction on $F'$. However, let us consider the commutator \\beq\n\\label{i2.39}\n[ \\Delta, N' ]=[ \\Delta,N - 2 {\\rm ad}( F' ) ] = 2 \\Delta -2 {\\rm ad}( \\Delta F' ).\n\\eeq\nSo, if we would like for the new operator $N'$ to maintain the relation\n\\beq\n\\label{i2.40}\n[ \\Delta, N' ] = 2 \\Delta, \n\\eeq\nthen there should be\n\\beq\n\\label{i2.41}\n\\Delta F' = {\\rm const} (Z). \n\\eeq\nDue to the latter, it follows from (\\ref{i2.34})\n\\beq\n\\label{i2.42}\n\\Delta N^{A} = \\Delta N'^{A} = 0. \n\\eeq\nIn turn, due to (\\ref{i2.41}), it follows from (\\ref{i2.37}) that\n\\beq\n\\label{i2.43}\n\\div N = {\\rm const} (Z). \n\\eeq\nThus, we see that the deviation from zero allowed for $\\div N$ is not so\narbitrary. That is because the condition (\\ref{i2.42}) is rather restrictive.\nWe see that the new antisymmetric $N'$ does maintain all the basic\nconditions (\\ref{i2.40}) and (\\ref{i2.42}), provided the condition (\\ref{i2.43}) holds.\nIn what follows below, we do mean that our $N'$-operator is chosen just in its\nantisymmetrical form (\\ref{i2.38}), \"from the very beginning\". For brevity,\nin all further formulae we omit the prime of $N'$.\n\nThere exists a crucially important consequence of (\\ref{i7}),\nthat the operator $( N - 2 )$ does differentiate the antibracket,\n\\beq\n\\label{i26}\n( N - 2 ) ( f, g ) = ( ( N - 2 ) f, g ) + ( f, ( N - 2 ) g ). \n\\eeq\nThat goes as follows,\n\\beq\n\\nonumber\n&&N ( f, g ) = ( N f, g ) + ( f, N g) - f \\overleftarrow{\\pa}_{A} \\left[ (\nN^{A}\\overleftarrow{\\pa}_{C} ) E^{CB} -\n( A \\leftrightarrow B ) (-1)^{ ( \\varepsilon_{A} + 1 ) ( \\varepsilon_{B} + 1\n) } \\right]\\overrightarrow{\\pa} _{B} g =\\\\\n\\label{i27}\n&&=( N f, g ) + ( f, N g ) - 2 ( f, g ), \n\\eeq\nwhich is equivalent to (\\ref{i26}).\n\n\\section{Extended Antibracket}\n\n\nNow, let us consider the antibracket generated by the extended operator\n$\\Delta_{\\tau}$ \\cite{BB2},\n\\beq\n\\label{i28}\n( F, G )_{\\tau} = (-1)^{\\varepsilon(F)}[ [ \\Delta_{\\tau}, F], G ]\\cdot 1=t^{2} ( F, G ) +\n( N_{\\tau} F ) \\pa_{\\theta} G - F \\overleftarrow{\\pa}_{\\theta} N_{\\tau} G, \n\\eeq\nwhere $( F, G )$ is the ( usual ) antibracket (\\ref{i8}) in the original $Z^{A}$ -sector,\nalthough functions $F, G$ themselves,\nstanding for $f, g$, respectively, do depend on $t, \\theta $ as well. $N_{\\tau}$ is\ndefined in (\\ref{i3}). Due to (\\ref{i26}), the\noperator $( N_{\\tau} - 2 )$ does differentiate the usual antibracket $( F, G )$,\nas well. In fact, we will use the relation equivalent to that,\n\\beq\n\\label{i30}\nN_{\\tau} ( F, G ) = ( N_{\\tau} F, G ) + ( F, N_{\\tau} G ) - 2 ( F, G).\n\\eeq\nOne can state that the extended antibracket (\\ref{i28}) does satisfy the strong\nJacobi identity, provided the usual\nantibracket has that property. In particular, one assumes that the strong\nJacobi identity holds for any Boson $B$,\n\\beq\n\\label{i31}\n( ( B, B ) , B ) = 0, \\quad \\varepsilon( B ) = 0.\n\\eeq In its general form, the Jacobi identity can be reproduced from\n(\\ref{i31}) via the differential polarization procedure. To do this,\none has to choose a specific form for $B$\n\\cite{B},\n\\beq\n\\label{i32}\nB = \\sum_{i = 1}^{3} m_{i} n_{i}, \\quad n_{1} = F,\\; n_{2} = G, \\; n_{3} =H.\n\\eeq\nThen, the operator\n\\beq\n\\label{i33}\n\\pa_{1} \\pa_{2} \\pa_{3} (-1)^{ ( \\varepsilon_{1} + 1 ) ( \\varepsilon_{3}\n+ 1 ) + \\varepsilon_{2} }, \n\\eeq\nshould be applied to (\\ref{i31}), where $\\pa_{i}$ are partial $m_{i}$ -derivatives,\n$\\varepsilon_{i}$ are Grassmann parities,\n\\beq\n\\label{i34}\n\\varepsilon_{i} = \\varepsilon( n_{i} ) = \\varepsilon( m_{i} ). \n\\eeq\n\nIt is our task now, to prove that the extended antibracket (\\ref{i28}) satisfies\nthe compact form of the strong Jacobi identities,\n\\beq\n\\label{i35}\n( ( B, B )_{\\tau}, B )_{\\tau} = 0, \\quad \\varepsilon( B ) = 0, \n\\eeq\nprovided similar compact form (\\ref{i31}) holds. We have\n\\beq\n\\label{i36}\n( B, B )_{\\tau} = t^{2} ( B, B ) + 2 ( N_{\\tau} B ) ( \\pa_{\\theta} B).\n\\eeq\nBy substituting that in (\\ref{i35}), one gets\n\\beq\n\\nonumber\n&&(( B, B)_{\\tau}, B )_{\\tau} = t^{2}( t^{2}( B, B )+2(N_{\\tau}B)(\\pa_{\\theta}B),B)+\n( N_{\\tau} ( t^{2} ( B, B ) +\\\\\n\\label{i37}\n&&+\n2( N_{\\tau}B)(\\pa_{\\theta} B ) ))( \\pa_{\\theta} B ) -\n( \\pa_{\\theta} ( t^{2} ( B, B ) + 2 ( N_{\\tau} B ) ( \\pa_{\\theta} B )\n) ) ( N_{\\tau} B ), \n\\eeq\nwhere the relations (\\ref{i30}), (\\ref{i31}) will be used, together with\n\\beq\n\\label{i38}\n( \\pa_{\\theta} B )^{2} = 0 \n\\eeq\nThus, the right-hand side of (\\ref{i37}) takes the form\n\\beq\n\\nonumber\n&&( 2 t^{2} ( B, B ) + 2 t^{2} ( N_{\\tau} B, B ) - 2 t^{2} ( B, B ) )\n( \\pa_{\\theta} B ) +\n2 t^{2} ( N_{\\tau} B ) ( \\pa_{\\theta} B, B )-\\\\\n\\nonumber\n &&- 2 t^{2} ( N_{\\tau}\nB, B ) ( \\pa_{\\theta} B ) -\n2 t^{2} ( \\pa_{\\theta} B, B ) ( N_{\\tau} B ) + \\\\\n\\label{i39}\n&&2 ( N_{\\tau} B ) (\nN_{\\tau} \\pa_{\\theta} B ) ( \\pa_{\\theta} B ) -\n2 ( \\pa_{\\theta} N_{\\tau} B ) ( \\pa_{\\theta} B ) ( N_{\\tau} B ) = 0. \n\\eeq\nHere, the first and third, the second and fifth, the fourth and sixth, the seventh and\neight terms, compensate each other in\nevery pair mentioned. Finally, the strong Jacobi identity (\\ref{i35}) for the\nextended antibracket (\\ref{i28}) is proven.\n\n\\section{ Nontrivial Deformation in the Sector of Original Variables}\n\nIn turn, let us study the role of the operator $N$ in construction of a\nnontrivially deformed antibracket in the original\n$Z^{A}$-sector. So, let $\\kappa$ be a deformation parameter. Consider the\noperator\n\\beq\n\\label{i40}\nK = K(N) = \\kappa ( N - 2 ),\\quad K(N + 2) = \\kappa N.\n\\eeq\nWe have\n\\beq\n\\label{i41}\nK(N + 2) - K(N) = 2 \\kappa, \n\\eeq\n\\beq\n\\label{i42}\nK(N) ( f g )=( K(N + 2) f ) g + f( K(N) g)=( K(N) f ) g+f( K(N + 2)g ), \n\\eeq\n\\beq\n\\label{i43}\nK ( f, g ) = ( K f, g ) + ( f, K g ). \n\\eeq\nBy using the well-known Witten formula for the usual antibracket,\n\\beq\n\\label{i44}\n( f, g) = \\Delta( f g ) (-1)^{\\varepsilon(f)} - [ f (\\Delta g) + (\\Delta f )\ng (-1)^{\\varepsilon(f)} ], \n\\eeq\nwe define the deformed antibracket by the relation \\cite{BB2},\n\\beq\n\\label{i45}\n( f, g )_{*} = \\Delta ( f g ) (-1)^{\\varepsilon(f)} -\n( 1 - K ) [ f (\\Delta_{*} g) + ( \\Delta_{*} f ) g (-1)^{\\varepsilon(f)} ] ,\n\\eeq\nwhere\n\\beq\n\\label{i46}\n\\Delta_{*} = \\Delta (1 - K )^{-1} = ( 1 - K( N + 2 ) )^{-1} \\Delta.\n\\eeq\nBy using (\\ref{i42}) and (\\ref{i44}), the relation (\\ref{i45}) can be rewritten in the form\n\\beq\n\\label{i47}\n( f, g )_{*} = ( f, g ) + ( K f ) ( \\Delta_{*} g ) + ( \\Delta_{*} f ) ( K g\n) (-1)^{\\varepsilon(f)}. \n\\eeq\nUsually, the deformation of the antibracket is defined by that formula.\nIt follows from (\\ref{i45})\n\\beq\n\\label{i48}\n\\Delta_{*} ( f, g )_{*} = - \\Delta \\left[ f ( \\Delta_{*} g) + ( \\Delta_{*} f) g\n(-1)^{\\varepsilon(f)} \\right]. \n\\eeq\n\\beq\n\\label{i49}\n( ( \\Delta_{*} f ), g )_{*} = \\Delta ( ( \\Delta_{*} f ) g)\n(-1)^{\\varepsilon(f) + 1} -\n( 1 - K ) [ ( \\Delta_{*} f ) ( \\Delta_{*} g ) ], \n\\eeq\n\\beq\n( f, ( \\Delta_{*} g ) )_{*} = \\Delta ( f ( \\Delta_{*} g ) )\n(-1)^{\\varepsilon( f )} -\n( 1 - K ) [ ( \\Delta_{*} f ) ( \\Delta_{*} g ) ] (-1)^{\\varepsilon( f )} ,\n\\eeq\n\\beq\n\\label{i50}\n( ( \\Delta_{*} f ), g )_{*} - ( f, ( \\Delta_{*} g) )_{*}\n(-1)^{\\varepsilon(f)} =\n\\Delta \\left[ - ( \\Delta_{*} f ) g (-1)^{\\varepsilon(f)} - f ( \\Delta_{*} g )\n\\right] = \\Delta_{*} ( f, g )_{*}. \n\\eeq\nThe latter means that the deformed operator (\\ref{i46}) does differentiate the\ndeformed antibracket (\\ref{i45}), or (\\ref{i47}) \\cite{BB2}.\n\nFinally, one can state that the nontrivially deformed antibracket (\\ref{i45}), or\n(\\ref{i47}), does satisfy the strong\nJacobi identity. Indeed, we have\n\\beq\n\\nonumber\n&&( ( B, B )_{*}, B )_{*} = ( ( B, B ) + 2 ( K B ) ( \\Delta_{*} B ), B ) +\n( K ( ( B, B ) + 2 ( K B ) ( \\Delta_{*} B ) ) ) ( \\Delta_{*} B ) -\\\\\n\\nonumber\n&&-\n2 ( ( \\Delta_{*} B ), B )_{*} ( K B ) = 2 ( K B ) ( ( \\Delta_{*} B ) , B\n) - 2 ( ( K B ), B ) ( \\Delta_{*} B ) +\n2 ( ( K B ), B ) ( \\Delta_{*} B )+\\\\\n\\label{i51}\n&& + 2 ( K ( ( K B ) ( \\Delta_{*} B ) ) ) (\n\\Delta_{*} B ) -\n2 ( ( \\Delta_{*} B ), B ) ( K B ) - 2 ( K \\Delta_{*} B ) ( \\Delta_{*} B )\n( K B ) = 0. \n\\eeq\nHere, on the right-hand side of the last equality, the first and fifth, the\nsecond and third, the fourth and\nsixth terms do compensate each other in every pair mentioned. Thus, the\nstrong Jacobi identity for the\nnontrivially deformed antibracket (\\ref{i45}), or (\\ref{i47}), in the original $Z^{A}$\n-sector is proven.\n\n\n\\section*{5\\;\\; Trivial $\\tau$-Extended Deformation\n\\footnote{This section represents the main results and formulae of\nRef. \\cite{BB2}, related in general to trivial deformations\nin $\\tau$ -extended phase space.}}\n\\setcounter{section}{5}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\setcounter{equation}{0}\n\nNow, we have to consider a trivial $\\tau$ extended deformation in the extended\nphase space including\nthe variables $t$ and $\\theta$. Let us introduce the operator\n\\beq\n\\label{i52}\n K_{\\tau} = \\kappa N_{\\tau}, \\quad [ K_{\\tau}, \\Delta_{\\tau} ]= 0,\n\\eeq\nand define trivially deformed extended operator\n\\beq\n\\label{i53}\n\\Delta_{\\tau *} = \\Delta_{\\tau} ( 1 - K_{\\tau} )^{-1},\n\\quad \\Delta^2_{\\tau *} = 0. \n\\eeq\nNow, introduce the operator\n\\beq\n\\label{i54}\nT = 1 + \\kappa \\theta \\Delta_{\\tau *} , \n\\eeq\nand its inverse\n\\beq\n\\label{i55}\nT^{-1} = 1 - \\kappa \\theta \\Delta_{\\tau}. \n\\eeq\nThen, one finds that\n\\beq\n\\label{i56}\n\\Delta_{\\tau *} = T^{-1} \\Delta_{\\tau} T. \n\\eeq\nAlso, it follows that the operator T does satisfy the equations (see also App. D)\n\\beq\n\\label{5.6}\n[ \\Delta_{\\tau}, T ] = \\Delta_{\\tau} T K_{\\tau}, \\quad [ T, K_{\\tau} ] =0.\n\\eeq\nTogether with (\\ref{i1}) and (\\ref{i52}), these equations constitute what we\ncall \"T-algebra\".\n\nNext, define a trivially deformed extended antibracket,\n\\beq\n\\nonumber\n&&( F, G )_{\\tau *} = T^{-1} ( ( T F ), ( T G ) )_{\\tau} = \\\\\n\\nonumber\n&&=( F, G\n)_{\\tau} +\n( K_{\\tau} F ) ( \\Delta_{\\tau *} G ) + ( \\Delta_{\\tau *} F )\n(K_{\\tau} G ) (-1)^{\\varepsilon(F)} =\\\\\n\\label{i57}\n&&=\\Delta_{\\tau} ( F G ) (-1)^{\\varepsilon(F)} - ( 1 - K_{\\tau} )\\left [\nF ( \\Delta_{\\tau *} G ) +\n( \\Delta_{\\tau *} F ) G (-1)^{\\varepsilon(F)} \\right].\n\\eeq\nThe latter formula allows for a natural rewriting in terms of the $*$-\nmodified double-commutator formula generalizing (\\ref{i8}) and (\\ref{i28}),\n\\beq\n\\label{i57.3}\n( F, G )_{\\tau *} = (-1)^{\\varepsilon(F)} [ [ \\Delta_{\\tau *},\n( T F)_{*} ], ( T G )_{*} ]\\cdot 1 , \n\\eeq\nwhere we have used (\\ref{i28}) and (\\ref{i56}), and\n\\beq\n\\label{i57.4}\n( T F )_{*} = T^{-1} ( T F ) T, \\quad ( T G )_{*} = T^{-1} ( T G ) T. \n\\eeq\nHere in (\\ref{i57.4}), on the right-hand sides, the operator $T$ in the middle\nfactors applies only to the function\nstanding to the right within the respective round bracket.\n\nNotice that within the class of functions,\n\\beq\n\\label{i57.1}\n F = t^{-2} f, \\quad G = t^{-2} g, \n\\eeq\nthe trivially deformed $\\tau$ -extended operator (\\ref{i56}) and antibracket (\\ref{i57})\nreduces, respectively, to the\nnon-trivially deformed operator (\\ref{i46}) and antibracket (\\ref{i45})\nin the original sector,\n\\beq\n\\label{i57.2}\n\\Delta_{\\tau *}F=\\Delta_{*}f,\\quad ( F, G )_{\\tau *} = t^{-2} ( f, g )_{*} . \n\\eeq\n\nBy construction, the trivially deformed extended antibracket (\\ref{i57}) does\nsatisfy the strong Jacobi identity.\nIn turn, define a trivial associative and commutative star-product\n\\beq\n\\label{i58}\n( F * G ) = T^{-1} ( ( T F ) ( T G ) ) = F G - \\kappa \\theta ( F, G\n)_{ \\tau *} (-1)^{\\varepsilon(F)}. \n\\eeq\nIt is worthy to mention here that the operators (\\ref{i57.4}) apply to a function\nas to yield the left adjoint of the symbol multiplication (\\ref{i58}),\n\\beq\n\\label{i58.1}\n( (T F)_{*} G ) = F * G,\\quad (T F)_{*} = F -\n\\kappa\\!\\; \\theta\\!\\; {\\rm ad}_{\\tau_{*}} ( F ) (-1)^{\\varepsilon( F )}. \n\\eeq\n\n Due to (\\ref{i57.2}), within the class of functions (\\ref{i57.1}),\n the star-product (\\ref{i58}) reduces as follows\n\\beq\n\\label{5.13}\nF * G = ( t^{-2} f ) ( t^{-2} g ) - \\kappa \\theta t^{-2} ( f, g )_{*}\n(-1)^{\\varepsilon(f)}. \n\\eeq\nThen, with respect to the star-product (\\ref{i58}), we have the\ntrivially deformed extended Witten formula \\footnote{The same as in\nthe undeformed case, the deformed extended Witten formula\n(\\ref{i59}) follows directly from the double-commutator formula\n(\\ref{i57.3}) with (\\ref{i57.4}), (\\ref{i58}), (\\ref{i58.1}) taken\ninto account. } \\beq \\label{i59} ( F, G )_{\\tau *} = \\Delta_{\\tau *}\n( F * G ) (-1)^{\\varepsilon(F)} - F\n* ( \\Delta_{\\tau *} G ) -\n( \\Delta_{\\tau *} F ) * G (-1)^{\\varepsilon(F)}, \n\\eeq\nthe Leibnitz rule,\n\\beq\n\\label{i60}\n( ( F * G ), H )_{\\tau *} = F * ( G, H )_{\\tau *} +\nG * ( F, H )_{\\tau *} (-1)^{\\varepsilon(F) \\varepsilon(G)},\n\\eeq\nthe Getzler identity \\cite{G} providing for the absence of higher antibrackets in\nthe BV -algebra\n\\beq\n\\nonumber\n&&\\Delta_{\\tau *} ( F * G * H ) - \\Delta_{\\tau *} ( F * G ) * H -\nF * \\Delta_{\\tau *} ( G * H ) (-1)^{\\varepsilon(F)} -\\\\\n\\nonumber\n&&- \\Delta_{\\tau *} ( F * H ) * G (-1)^{\\varepsilon(G)\n\\varepsilon(H)}+( \\Delta_{\\tau *} F ) * G * H-\\\\\n\\label{i61}\n&&- F * ( \\Delta_{\\tau *} G )* H\n(-1)^{\\varepsilon_(F)} +\nF * G * ( \\Delta_{\\tau *} H ) (-1)^{\\varepsilon(F) +\n\\varepsilon(G)} = 0. \n\\eeq\nThe star exponential is defined as (see also App. C)\n\\beq\n\\nonumber\n&& \\exp_{*}\\{ B \\} =\n1 + B + \\frac{1}{2} B * B + \\frac{1}{3!} B * B * B + ... =\\\\\n\\label{i62}\n&&=T^{-1} \\exp\\{( T B )\\} =\n\\exp\\left\\{ B -\\frac{1}{2} \\kappa \\theta ( B, B )_{\\tau *} \\right\\}. \n\\eeq\nThe latter satisfies\n\\beq\n\\label{i63.1}\n&&\\exp_{*}\\{ - B \\} * \\exp_{*}\\{ B \\} = 1, \\\\\n\\label{i63.2}\n&&\\exp_{*}\\{ - B \\} * (\n\\Delta_{\\tau *} \\exp_{*}\\{ B \\} ) =\n( \\Delta_{\\tau *} B ) + \\frac{1}{2} ( B, B )_{\\tau *}, \\\\\n\\label{i63.3}\n&&\\exp_{*}\\{ B + B'\\} =\\exp_{*}\\{ B \\} * \\exp_{*}\\{ B'\\}, \\\\\n\\label{i63.4}\n&&\\delta \\exp_{*}\\{ B \\} = \\exp_{*}\\{ B \\} * \\delta B,\\quad\n\\varepsilon(B) = \\varepsilon(B') = 0. \n\\eeq\nThe trivially deformed extended quantum master equation has the form\n\\beq\n\\label{i64}\n\\Delta_{\\tau *} \\exp_{*}\\left\\{ \\frac{i}{\\hbar}\\; W \\right\\} = 0,\n\\eeq\nor, equivalently,\n\\beq\n\\label{i65}\n\\frac{1}{2} ( W, W )_{\\tau *} = i \\hbar \\Delta_{\\tau *} W. \n\\eeq\nOne has to seek for a solution to that equation in the form\n\\beq\n\\label{i5.21}\nW = \\sum_{k = - 2}^{\\infty} W_{ (k|0) } t^{k} + \\theta \\sum_{k = 1}^{\\infty}\nW_{ (k|1) } t^{k}, \n\\eeq\nwhere the component $W_{ (-2|0) } = S$ is identified with the classical\nnontrivially deformed proper action, (see also App. B)\n\\beq\n\\label{i5.22}\n( S, S )_{*} = 0. \n\\eeq\nThe detailed form of the equations for coefficients in (\\ref{i5.21}), together with\nthe corresponding formal techniques, can be found in Ref. \\cite{BB2}.\n\nThe trivially deformed path integral with a measure $d\\mu$ in the extended\nphase space is defined as\n\\beq\n\\label{i66}\n\\mathcal{Z} = \\int d\\mu \\exp_{* ( \\kappa)}\\left\\{\\frac{i}{ \\hbar}\\; W \\right\\}\n \\exp_{* ( -\\kappa)}\\left\\{\\frac{i}{\\hbar }\\; X \\right\\} =\n\\int d\\mu \\exp\\left\\{\\frac{i}{\\hbar }\\; A \\right\\},\n\\eeq\nwhere\n\\beq\n\\label{5.27}\nA = T_{ (\\kappa)} W + T_{( - \\kappa)}X, \n\\eeq\n\\beq\n\\Delta_{\\tau * ( \\kappa ) }\\left(\\exp_{*(\\kappa )}\\left\\{\\frac{ i}{ \\hbar}\\; W \\right\\}\n\\right) = 0,\\quad\n\\Delta_{\\tau * (- \\kappa )}\\left(\\exp_{*( - \\kappa )} \\left\\{\\frac{i}{\\hbar}\\; X \\right\\}\n\\right)= 0.\n\\eeq\nHere, $X$ satisfies the same equation as $W$ does, but with the formal replacement\n$\\kappa \\rightarrow - \\kappa$.\nThis replacement just provides for right transposition properties when\nintegrating by part.\n\nWe proceed in (\\ref{i66}) with the following integration measure\n\\beq\n\\label{i5.25}\nd \\mu = t^{-1} d t d\\theta d \\lambda_{\\theta} [ d Z ] [ d \\lambda ]. \n\\eeq\nThe transposed operator $A^{T}$ of the operator $A$ is defined via\n\\beq\n\\label{i5.26}\n\\int d\\mu ( A^{T} F ) G = (-1)^{\\varepsilon(A) \\varepsilon(F)} \\int d\n\\mu F ( A G). \n\\eeq\nOur main transposed operators are\n\\beq\n\\label{i5.27}\n\\Delta^{T} = \\Delta, \\quad N^{T} = - N, \\quad\n\\Delta^{T}_{\\tau} = \\Delta_{\\tau},\\quad N^{T}_{\\tau} = - N_{\\tau},\\quad\n\\Delta^{T}_{\\tau *( \\kappa )} = \\Delta_{\\tau * (- \\kappa )}. \n\\eeq\nLet us make in (\\ref{i66}) the variation of the form\n\\beq\n\\label{i5.28}\n\\delta \\exp_{*(-\\kappa)}\\left\\{\\frac{ i}{\\hbar }\\; X \\right\\} =\n\\Delta_{\\tau *(-\\kappa)} \\left( \\exp_{*(-\\kappa)}\\left\\{\\frac{i}{\\hbar\n}\\; X \\right\\} *_{(-\\kappa)} \\delta \\Psi \\right), \n\\eeq\nwith arbitrary infinitesimal Fermion $\\delta \\Psi$. The (\\ref{i5.28}) is consistent\nwith (\\ref{i63.4}) due to the quantum master equation for the $X$.\nThen, we deduce that the path\nintegral is independent of the gauge-fixing action $X$,\n\\beq\n\\nonumber\n&&\\delta_{X} \\mathcal{Z} = \\int d \\mu \\exp_{*( \\kappa )}\\left\\{\\frac{i}{\\hbar }\\;W \\right\\}\n\\delta \\exp_{* ( -\\kappa ) }\\left\\{\\frac{i}{\\hbar }\\; X \\right\\} =\\\\\n\\nonumber\n&&=\\int d \\mu \\exp_{* ( \\kappa ) }\\left\\{\\frac{i}{\\hbar}\\; W \\right\\}\\Delta_{\\tau *( -\n\\kappa )} \\left( \\exp_{ * ( - \\kappa ) }\\left\\{\\frac{i}{\\hbar }\\; X \\right\\}\n*_{ ( - \\kappa ) } \\delta \\Psi \\right) =\\\\\n\\label{i5.29}\n&&=\\int d \\mu \\left( \\Delta_{\\tau * ( \\kappa )}\n\\exp_{ * ( \\kappa ) }\\left\\{\\frac{i}{\\hbar }\\; W \\right\\} \\right)\n\\left(\\exp_{* ( - \\kappa) }\n\\left\\{\\frac{i}{\\hbar }\\; X \\right\\} *_{ ( - \\kappa ) } \\delta \\Psi \\right)\n= 0. \n\\eeq\n\nIt follows from (\\ref{i56}), (\\ref{i62}) and (\\ref{i64}) that\n\\beq\n\\Delta_{\\tau} \\exp\\left\\{\\frac{i}{\\hbar }\\; T_{ ( \\kappa) } W \\right\\} = 0.\n\\eeq\nFor similar reasons, it follows that\n\\beq\n\\Delta_{\\tau}\\exp\\left\\{\\frac{i}{\\hbar}\\;T_{( - \\kappa ) } X \\right\\} = 0.\n\\eeq\nThese equations tell us that in terms of the barred actions,\n\\beq\n\\bar{W} = T_{ (\\kappa) } W,\\quad \\bar{X} = T_{ ( -\\kappa ) } X,\n\\eeq\nthe path integral (\\ref{i66}) is just the standard $W - X$ version of the\nfield-antifield formalism. From the latter point of view, it is\nwell-known that the path integral (\\ref{i66}) is stable under the\ngauge variation\n\\beq\n\\delta{ \\bar{X} } = \\sigma_{\\tau}(\\bar{X}) \\bar{\\Psi },\n\\eeq\nwhere\n\\beq\n\\sigma_{\\tau}( \\bar{X} ) =-i\\hbar \\;\\Delta_{\\tau}+{\\rm ad}_{\\tau}( \\bar{X} ),\n\\eeq\nso that\n\\beq\n\\delta X = T^{-1}_{( - \\kappa ) } \\delta \\bar{X}.\n\\eeq\n If one identifies $\\bar{ \\Psi } =T_{ ( - \\kappa ) } \\Psi$ , then\n\\beq\n\\delta X = \\sigma_{ \\tau * ( -\\kappa ) }( X ) \\Psi ,\n\\eeq\nwhere\n\\beq\n\\sigma_{ \\tau * ( - \\kappa ) }( X )\n= -i\\hbar\\; \\Delta_{\\tau *( - \\kappa )}+{\\rm ad}_{\\tau *( -\\kappa)}( X ),\n\\eeq\nwhich is exactly the variation of $X$ generated by (\\ref{i5.28}).\n\n\n\\section{ Gauge-Fixing in the Classical Extended Nondeformed\\\\ Master\nEquation}\n\n\nHere we study, if the standard gauge-fixing procedure is capable to\neliminate the extra variable $t$, as applied\nto the classical $\\tau$ -extended nondeformed master equation,\n\\beq\n\\label{i6.1}\n( S, S )_{\\tau} = t^{2} ( S, S ) + 2 ( N_{\\tau} S ) ( \\pa_{\\theta} S ) = 0.\n\\eeq\nwhere we restrict ourselves to the simplest choice for $N_{\\tau}$,\n\\beq\n\\label{i6.2}\nN_{\\tau} = Z^{A} \\pa_{A} + t \\pa_{t}. \n\\eeq\nWe proceed with the following ansatz for S,\n\\beq\n\\label{i6.3}\nS = S ( Z, t, \\theta ) = S ( t^{-1} Z, \\theta ), \n\\eeq\nwhich implies\n\\beq\n\\label{i6.4}\nN_{\\tau} S = 0, \n\\eeq\nso that Eq. (\\ref{i6.1}) takes the usual form of the classical master\nequation,\n\\beq\n\\label{i6.5}\n( S, S ) = 0. \n\\eeq\nLet $\\mathcal{S}( \\phi )$ be an original action of original fields $\\phi^{i}$ ,\nand $R^{i}_{\\alpha}( \\phi )$ do satisfy the Noether identities,\n\\beq\n\\label{i6.6}\n\\mathcal{S} \\overleftarrow{\\pa}_{i} R^{i}_{\\alpha} = 0,\n\\quad \\pa_i=\\frac{\\pa}{\\pa \\phi^i}. \n\\eeq\nFor the sake of simplicity, let the generators $R^{i}_{\\alpha}$ be\nlinearly independent, so that the theory is irreducible. Let us expand\nthe ansatz (\\ref{i6.3}) in powers of antifields,\n\\beq\n\\nonumber\n&&S = \\mathcal{S}(t^{-1}\\phi) + t^{-1}\\phi^*_{i} R^{i}_{\\alpha}(t^{-1}\\phi)\nt^{-1} C^{\\alpha} + \\theta t^{-1} C^{t} +\n\\frac{1}{2}t^{-1} C^*_{\\gamma} U^{\\gamma}_{\\alpha \\beta}(t^{-1}\\phi) t^{-1}\nC^{\\beta} t^{-1} C^{\\alpha} (-1)^{\\varepsilon_{\\alpha}} +\\\\\n\\label{i6.7}\n&&+\nt^{-1} {\\bar C}^{*\\alpha} t^{-1} B_{\\alpha} + t^{-1} {\\bar C}^{*t} t^{-1}\nB_{t} + ... , \n\\eeq\nwhere the terms presented explicitly are enough for a rank one theory, while\nellipses mean terms nonlinear in antifields.\n\nNow, let us split $Z^{A}$ into fields and antifields,\n\\beq\n\\label{i6.8}\nZ^{A} = \\{\\Phi^{a}, \\Phi^*_{a}\\}. \n\\eeq\nThen the gauge-fixing Fermion allowed has the form\n\\beq\n\\label{i6.9}\n\\Psi= \\Psi( \\Phi, t ) = \\Psi( t^{-1} \\Phi, \\ln t ), \n\\eeq\nso that the antifields $\\Phi^*_{a}$ should be eliminated in (\\ref{i6.7}) by the\nconditions\n\\beq\n\\label{i6.10}\n\\Phi^*_{a} = t ^{2} \\Psi \\overleftarrow{\\pa}_{a}, \\quad \\pa_a=\\frac{\\pa}{\\pa \\Phi^a}, \n\\eeq\n\\beq\n\\label{i6.11}\n\\theta = N_{\\tau} \\Psi. \n\\eeq\nThese conditions do correspond to the following ansatz for the gauge-fixing\nmaster action X,\n\\beq\n\\label{i6.12.1}\nX = \\left( t^{-1} \\Phi^*_{a} - t \\Psi \\overleftarrow{\\pa}_{a} \\right) \\lambda^{a} + (\n\\theta - N_{\\tau} \\Psi ) \\lambda^{\\theta}, \n\\eeq\nwhere $\\lambda^{a}$ and $\\lambda^{\\theta}$ is the corresponding Lagrange\nmultiplier.\nIn fact, it is enough for our purposes to use the ansatz\n\\beq\n\\label{i6.12}\n\\Psi = t^{-1} \\bar{C}_{\\alpha} \\chi^{\\alpha}( t^{-1} \\phi ) + t^{-1}\n\\bar{C}_{t} \\chi^{t}( \\ln t ), \n\\eeq\nwith the following identification of fields\n\\beq\n\\label{i6.13}\n\\Phi^{a} = \\{\\phi^{i}, B_{\\alpha}, C^{\\alpha}, \\bar{C}_{\\alpha}, B_{t}, C^{t},\n\\bar{C}_{t}\\}. \n\\eeq\n\nThus, we arrive at the following complete gauge-fixed action\n\\beq\n\\nonumber\n&&S_{gauge-fixed} = \\mathcal{S}( t^{-1}\\phi ) +\n t^{-1} \\bar{C}_{\\alpha} \\chi( t^{-1} \\phi ) \\overleftarrow{\\pa}_{i}\nR^{i}_{\\alpha}( t^{-1} \\phi ) C^{\\alpha} +\nt^{-2} \\bar{C}_{t}( N_{\\tau} \\chi^{t} ) C^{t}+ \\\\\n\\label{i6.14}\n&&+ \\chi^{\\alpha}( t^{-1} \\phi\n) t^{-1} B_{\\alpha} + \\chi^{t}( \\ln t ) t^{-1} B_{t} + ... . \n\\eeq\nThat action has, in its terms presented explicitly, the standard structure\nof the Faddeev-Popov action, both\nin the sector of the usual gauge $\\chi^{\\alpha}$ and of the extra gauge\n$\\chi^{t}$. By choosing $\\chi^{t} = \\ln t$,\none removes the $t$-integration at the value $t = 1$. Thereby, it is shown that\nthe extra variable t is eliminated actually via the standard gauge-fixing procedure.\n\n\n\\section{ Gauge-Fixing in the Extended Trivially Deformed Classical\/Quantum Master\nEquation}\n\n\nLet us consider the extended trivially deformed classical master equation,\n\\beq\n\\label{i7.1}\n( S, S )_{\\tau{*}} = 0, \n\\eeq\nor in more detail,\n\\beq\n\\label{i7.2}\nt^{2} (S, S ) + 2 ( N_{\\tau} S )( \\pa_{\\theta} S ) +\n2 \\kappa ( N_{\\tau} S ) ( ( t^{2} \\Delta + N_{\\tau} \\pa_{\\theta} )\n( 1 - \\kappa N_{\\tau} )^{-1} S ) = 0. \n\\eeq\nThe same as in Sec. 6, we have chosen $N _{\\tau}$ in the\nsimplest form (\\ref{i6.2}).\nIn contrast to the previous section, we are not allowed to require for the\noperator $N_{\\tau}$\nto annihilate the $S$, as the deformation by itself would be eliminated\nimmediately in this way.\nHowever, if we do believe that a solution for $S$ does exist, we can try to\nrequire for $N_{\\tau}$ to\nannihilate the gauge-fixing part of $S$, at least. When doing that, we should\nprovide for the form\nof Eqs. (\\ref{i7.1}), or (\\ref{i7.2}), to be respected. Let us seek for $S$ in the\nform\n\\beq\n\\label{i7.3}\nS = S_{min} + t^{-2} ( \\bar{C}^{*{\\alpha}} B_{\\alpha} + \\bar{C}^{*{t}} B_{t} ), \n\\eeq\nwhere the minimal action $S_{min}$ depends on the minimal, gauge-algebra\ngenerating, set of variables, only,\n\\beq\n\\label{i7.4}\nS_{min} = S_{min}( \\phi, \\phi^*; t, \\theta; C, C^*; C^{t}, C^*_{t} ). \n\\eeq\nBy construction, the second and third term in (\\ref{i7.3}), those are just the\ngauge-fixing parts of $S$ , are\ncertainly annihilated by the $N_{\\tau}$, while the $S_{min}$ is not. In this\nway, one can see that\nthe $S_{min}$ by itself does satisfy exactly Eqs. (\\ref{i7.1}),\nor (\\ref{i7.2}). If\none chooses the gauge\nFermion $\\Psi$ in the simplest form (\\ref{i6.12}), then the antifields should be\neliminated by the conditions\n(\\ref{i6.10}), see also (\\ref{i6.13}). In turn, the second and third term\nin (\\ref{i7.3})\ntake exactly the form of fourth\nand fifth term in (\\ref{i6.14}), respectively. Thereby, it is shown that the extra\nvariable $t$ is eliminated\nactually via the standard gauge-fixing procedure.\n\nIn the same way, one can consider the extended trivially deformed quantum\nmaster equation, (\\ref{i65}),\n\\beq\n\\label{7.5}\nt^{2} ( W, W ) + 2 ( N_{\\tau} W ) ( \\pa_{\\theta} W ) +\n2( (\\kappa N_{\\tau} W) - i \\hbar) (( t^{2} \\Delta + N_{\\tau}\n\\pa_{\\theta} ) ( 1 - \\kappa N_{\\tau} )^{-1} W ) = 0. \n\\eeq\nAs Eq. (\\ref{i7.2}) is a classical limit to the quantum equation (\\ref{7.5}),\nall the above reasoning, as well as the final statement remains the same.\n\nFinally, let us notice the following. In Secs. 6 and 7, we have used the\nzero mode $t^{-1} Z^{A}$ of the\noperator (\\ref{i6.2}). Here, we mention in short how to deal with the general\noperator (\\ref{i3}), where $N^{A}$\nis defined by (\\ref{i20.7}) - (\\ref{i20.9}). Let $\\bar{Z}^{A}$\nbe the zero mode of the\noperator (\\ref{i3}). Then we have\nformally,\n\\beq\n\\label{i7.5}\n\\bar{Z}^{A} = \\exp\\{ - ( \\ln t ) N \\} Z^{A}, \\quad N = N^{A} \\pa_{A}.\n\\eeq\nIt follows from (\\ref{i26}) that\n\\beq\n\\label{i7.6.1}\n( \\bar{Z}^{A}, \\bar{Z}^{B} ) = t^{-2} E^{AB}, \\quad\n( \\bar{Z}^{A}, Z^{C} ) E_{CD} ( Z^{D}, \\bar{Z}^{B} )=( \\bar{Z}^{A},\\bar{Z}^{B} ). \n\\eeq\nIn turn, it follows from (\\ref{i7.6.1}) that the general solution for the\nleft off-diagonal block has the form,\n\\beq\n\\label{78}\n( \\bar{Z}^{A}, Z^{B} ) = t^{-1} S^{A}_{\\;\\;C}( t ) E^{CB}, \\quad\n\\bar{Z}^{A} = t^{-1} S^{A}_{\\;\\;B}( t ) Z^{B},\n\\eeq\nwhere $S^{A}_{\\;\\;B}(t)={\\rm const}(Z)$ is a $t$-dependent antisymplectic matrix,\n\\beq\n\\label{7.9}\nS^{A}_{\\;\\;C}( t ) E^{CD} S^{B}_{\\;\\;D}( t ) (-1)^{\\varepsilon_{D} ( \\varepsilon_{B}\n+ 1 ) } = E^{AB} , \n\\eeq\nsuch that\n\\beq\n\\label{7.10}\nS^{A}_{\\;\\;B}( t = 1 ) = \\delta^{A}_{\\;\\;B}. \n\\eeq\nOne can get the general solution for the right off-diagonal block via the\nsupertransposition in (\\ref{78}).\n\nNow, let us consider some explicit formulae concerning the modified\n$N$-operator. First, let us choose the quadratic Fermion $F$ entering (\\ref{i11})\nthat meets the condition (\\ref{i14}),\n\\beq\n\\label{7.11}\n2 F = Z^{A} F_{AB} Z^{B}, \\quad \\varepsilon (F) = 1, \n\\eeq\n\\beq\n\\label{7.12}\n\\varepsilon( F_{AB} ) = \\varepsilon_{A}+\\varepsilon_{B}+1, \\quad\nF_{AB} = F_{BA} (-1)^{\\varepsilon_{A} \\varepsilon_{B}} = {\\rm const}(Z). \n\\eeq\nWe have\n\\beq\n2\\Delta F = E^{AB} F_{BA} (-1)^{ \\varepsilon_{A} } = {\\rm const}( Z ),\n\\eeq\n\\beq\n\\label{7.14}\nN^{A} = Z^{A} + 2( F, Z^{A} ) = Z^{B} ( \\delta_{B}^{\\;\\;A} + 2F_{BC} E^{CA} ) =\n(\\delta^{A}_{\\;\\;B} - 2 E^{AC} F_{CB} ) Z^{B}. \n\\eeq\nOn the other hand, as the second in (\\ref{78}) is the zero mode of $N_{\\tau}$,\nwe have another expression for $N^{A}$,\n\\beq\n\\label{7.15}\nN^{A} = - ( S^{-1} )^{A}_{\\;\\;C}\\; t^{2} \\pa_{t} ( t^{-1} S^{C}_{\\;\\;B} ) Z^{B}.\n\\eeq\nIt follows from (\\ref{7.14}) and (\\ref{7.15}) that the Lie equation holds\n\\beq\n\\label{7.16}\nt \\pa_{t} S^{A}_{\\;\\;B} = 2 S^{A}_{\\;\\;C} E^{CD} F_{DB}, \\quad\nS^{A}_{\\;\\;B}( t = 1 ) =\\delta^{A}_{\\;\\;B}, \n\\eeq\nwhose formal matrix solution is\n\\beq\n\\label{7.17}\nS = S( t ) = \\exp\\{ 2\\ln( t ) E F \\}. \n\\eeq\nBy $t$-differentiating the formula (\\ref{7.9}), and then using the equation (\\ref{7.16}),\none confirms that the matrix (\\ref{7.17})\nby itself does satisfies exactly the antisymplicticity equation (\\ref{7.9}).\nThus, we see that the exponential (\\ref{7.17})\nprovides for the $t$-parametrization of a family of antisymplectic matrices,\nwith $( EF )$ being a generator.\n\n\nIf one splits the full set $Z^{A}$ into minimal sector $Z_{min}$ in (\\ref{i7.4}),\nexcept for $\\{ t, \\theta \\}$, and\nthe rest, $Z_{aux}$, $\\{Z\\} =\\{ Z_{min}\\}\\oplus \\{Z_{aux}\\}$, then, by choosing in (\\ref{i.V})\n$F=F(Z_{min})$, one has\n\\beq\n\\label{i7.6.2}\n\\bar{Z}_{aux} = t^{-1} Z_{aux}.\n\\eeq\nIn terms of (\\ref{i7.5}), the formula (\\ref{i6.3}) and (\\ref{i6.9}) takes the form,\n\\beq\n\\label{i7.6}\nS = S( \\bar{Z}, \\theta ), \n\\eeq\nand\n\\beq\n\\label{i7.7}\n\\Psi = \\Psi( \\bar{\\Phi}, \\ln t ), \n\\eeq respectively.\nIn turn, the formula (\\ref{i7.3}) in terms of\n(\\ref{i7.5}) preserves its\nform\n\\beq\n\\label{i7.8}\n S = S_{min} +\n(\\overline{\\bar{C}^{*\\alpha}B_{\\alpha}} +\n\\overline{\\bar{C}^{*t}B_t } )=\nS_{min} + t^{-2} ( \\bar{C}^{*{\\alpha}} B_{\\alpha} + \\bar{C}^{*{t}} B_{t} ), \n\\eeq\nwhere $S_{min}$ is given by (\\ref{i7.4}).\nFor the particular case $N^{A} = Z^{A}$, we reproduce from (\\ref{i7.5})\n$\\bar{Z}^{A} = t ^{-1} Z^{A}$.\n\nDue to (\\ref{i7.6.2}) and (\\ref{i7.8}), one has\n\\beq\n\\label{i7.11}\n( S_{min}, S_{aux} ) = 0, \\;\\; ( S_{aux}, S_{aux} ) = 0, \\;\\; N_{\\tau} S_{aux} =\n0, \\;\\; \\pa_{\\theta} S_{aux} = 0,\\;\\; S_{aux} = S - S_{min}, \n\\eeq\ntogether with\n\\beq\n\\label{i7.12}\n\\Delta S_{aux} = 0. \n\\eeq\nThe relations (\\ref{i7.11}) and (\\ref{i7.12}) allow one to preserve the form of the equation\n(\\ref{i7.2}) for the minimal action (\\ref{i7.4}) in the general case of (\\ref{i7.5}).\n\n\n\n\\section{ Generalized Darboux Coordinates \\cite{BB2}}\n\n\n\nThe $\\tau$ -extended trivially deformed classical\/quantum master\nequation takes its simplest form in the so-called generalized\nDarboux coordinates,\n\\beq\n\\label{i8.1}\n\\tau_{0}: Z^{A}_{0} = t^{-1} Z^{A},\\quad t_{0} = \\ln t,\\quad t^*_{0} = \\theta,\n\\eeq\nwith the following integration measure,\n\\beq\nd\\mu=d t_{0} d t^*_{0} d\\lambda_{t^*_{0}} [ d Z_{0} ] [ d \\lambda ].\n\\eeq\nWe have already used these coordinates partially when discussing\nthe gauge-fixing\nprocedure. In the case of the master equation, we have, with the use\nof (\\ref{i8.1}),\n\\beq\n\\label{i8.2}\n( W, W )_{\\tau_{0}{*}} = 2 i \\hbar \\Delta_{\\tau_{0}{*}} W, \n\\eeq\nwhere\n\\beq\n\\label{i8.3}\n( F, G )_{\\tau_{0}{*}} = ( F, G )_{\\tau_{0}} + ( K_{\\tau_{0}} F ) (\n\\Delta_{\\tau_{0}{*}} G ) +\n( \\Delta_{\\tau_{0}{*}} F ) ( K_{\\tau_{0}} G ) (-1)^{\n\\varepsilon_{F} },\n\\eeq\n\\beq\n\\label{i8.4}\n( F, G )_{\\tau_{0}} = F [ \\overleftarrow{\\pa}_{A 0} E^{AB}\n\\overrightarrow{\\pa}_{B 0} +\n\\overleftarrow{\\pa}_{t_{0}} \\overrightarrow{\\pa}_{t^*_{0}} -\n\\overleftarrow{\\pa}_{t^*_{0}} \\overrightarrow{\\pa}_{t_{0}} ] G,\\quad \\pa_{A 0} =\n\\frac{\\pa}{\\pa Z^{A}_{0}}, \n\\eeq\n\\beq\n\\label{i8.5}\nK_{\\tau_{0}} = \\kappa \\pa_{t_{0}}, \n\\eeq\n\\beq\n\\label{i8.6}\n\\Delta_{\\tau_{0}{*}} = \\Delta_{\\tau_{0}} ( 1 - K_{\\tau_{0}} )^{-1},\n\\eeq\n\\beq\n\\label{i8.7}\n\\Delta_{\\tau_{0}} = \\frac{1}{ 2 } (-1)^{\\varepsilon_{A}} \\pa_{A 0} E^{AB}\n\\pa_{B 0} + \\pa_{t_{0}} \\pa_{t^*_{0}}. \n\\eeq\nWe have the two main simplifications here. The first is the absence\nof the $t_{0}$ -dependent factors in the square bracket in (\\ref{i8.4}). The second\nis a very simple form of the operator (\\ref{i8.5}). The latter reduces to the $t_{0}$\n-derivative.\n\nThe antifields are eliminated by the conditions\n\\beq\n\\label{i8.8}\n\\Phi^*_{a 0} = \\Psi\\overleftarrow{\\pa}_{a 0}, \\quad t^*_{0} =\n\\Psi\\overleftarrow{\\pa}_{t_{0}}, \n\\eeq\nwhere\n\\beq\n\\label{i8.9}\nZ^{A}_{0} = \\{\\Phi^{a}_{0}, \\Phi^*_{a 0}\\},\\quad\n\\pa_{a 0} = \\frac{\\pa}{\\pa \\Phi^{a}_{0}}. \n\\eeq\n\nFinally, let us consider in short what happens if one ignores the\ngauge-fixing mechanism as to eliminate the extra variable $t$.\nIn that case, one assumes the $W$ to be $\\theta$ -independent,\n\\beq\n\\label{i8.10}\n\\pa_{\\theta} W = \\pa_{t^*_{0}} W = 0. \n\\eeq\nUnder the assumption (\\ref{i8.10}), the path integral (\\ref{i66}) becomes as\nrepresented in coordinates (\\ref{i8.1}),\n\\beq\n\\label{i8.11}\n\\mathcal{Z}= \\int d t_{0}\\int d \\Phi_{0}\\exp \\left\\{\\frac{i}{\\hbar}A \\right\\},\n\\eeq\n\\beq\n\\label{i8.12}\nA = W( \\Phi, \\Phi^*, t, \\kappa, \\hbar ) = W \\left(\\exp\\{ t _{0} \\} \\Phi_{0},\n\\exp\\{ t_{0} \\} \\left( \\Psi( \\Phi_{0} )\\frac{\\overleftarrow{\\pa}}{\\pa\\Phi_{0}} \\right),\n\\exp\\{t_{0}\\}, \\kappa, \\hbar \\right), \n\\eeq\n\\beq\n\\label{i8.13}\n( W, W )_{0} + 2 \\left( \\kappa ( \\pa_{t_{0}} W ) - i \\hbar )\n( \\Delta_{0} ( 1 - \\kappa \\pa_{ t_{0} } )^{-1} W \\right) = 0, \n\\eeq\n\\beq\n\\label{i8.14}\n( F, G )_{0} = F \\overleftarrow{\\pa}_{A 0} E^{AB}\\overrightarrow{\\pa}_{B 0} G,\n\\eeq\n\\beq\n\\label{i8.15}\n\\Delta_{0} =\\frac{1}{2 } (-1)^{\\varepsilon_{A}} \\pa_{A 0} E^{AB} \\pa_{B 0}.\n\\eeq\n\nIn Ref. \\cite{BB2}, it was suggested that the extra variable $t_{0}$, remaining in the\npath integral (\\ref{i8.11}), plays the role of the Schwinger proper time\nin the field-antifield formalism. It\nseems rather plausible that the variable $t$ has a non-perturbative status. If\none rescales in (\\ref{i8.12}): $W \\rightarrow \\exp\\{ - 2 t_{0} \\} W$,\nthen in (\\ref{i8.12}), (\\ref{i8.13}) one should\nsubstitute: $\\hbar \\rightarrow \\exp\\{ 2 t_{0} \\} \\hbar$,\n$ \\pa_{t_{0}}\\rightarrow \\pa_{t_0} - 2$.\n\n\n\n\\section*{Acknowledgments}\n\\noindent\n I. A. Batalin would like to thank Klaus Bering of Masaryk\nUniversity for interesting discussions. The work of I. A. Batalin is\nsupported in part by the RFBR grants 14-01-00489 and 14-02-01171.\n The work of P. M. Lavrov is supported in part by the Presidential grant\n 88.2014.2 for LRSS and by the RFBR grant 15-02-03594.\n\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe recent paper \\cite{BCMNO} characterises the automorphisms of the Higman-Thompson groups $G_{n,r}$ as a subgroup of the rational group $\\T{R}_{n,r}$ consisting of those elements which have the additional property of being bi-synchronizing. In this article, we extend the arguments of \\cite{BCMNO} and characterise the automorphism group of $T_{n,r}$, generalisations of Thompson's group $T$, as a subgroup of $\\aut{G}_{n,r}$.\n\n The automorphism group of $T_{2,1}$, or $T$, as well as the automorphism group of Thompson's group $F$, were studied in the paper \\cite{MBrin2}, which also demonstrates that $\\out{T_2}$ is isomorphic to the cyclic group of order $2$. The paper \\cite{JBurilloSClearyF} studies the metric properties of $\\aut{F}$ and gives a presentation for this group, whilst the follow up paper \\cite{MBrinFGuzman} studies the automorphisms of the groups $F_{n}$ and $T_{n,n-1}$, generalisations of Thompson's group $F$ and $T$ (the approach there gives no information about $\\aut{T_{n,r}}$ when $r \\ne n-1$). The paper \\cite{MBrinFGuzman}, amongst other things, demonstrates that $\\out{T_{n,n-1}}$ for $n\\ge3$ is infinite and contains an isomorphic copy of Thompson's group $F$. \n\nIn this paper we extend the results of \\cite{MBrinFGuzman} to the groups $\\Tnr$ for $1 \\le r < n-1$. Firstly, we prove the following,\n\n\\begin{Theorem}\nThe group $\\aut{T_{n,r}}$ consists of those elements of $ \\Rnr$ which can be represented by bi-synchronizing transducers such that the induced homeomorphisms on Cantor space respects the cyclic ordering. \n\\end{Theorem}\n\nOne immediately deduces the following corollary:\n\\begin{corollary}\n$\\aut{\\Tnr} < \\aut{G_{n,r}}$.\n\\end{corollary}\n\nWe now turn our attention to the quotient group $\\out{\\Tnr} = \\aut{\\Tnr}\/ \\inn{\\Tnr}$. The results below are equally true in the $\\Gnr$ context and so to avoid repetition we shall sometimes use the symbol $X$ to represent either $T$ or $G$, analogously the symbol $\\T{X}$ will represent either $\\T{TO}$ or $\\T{O}$.\n \n It is a result in the paper \\cite{BCMNO} that for $1 \\le r < n$ the group $\\out{\\Gnr}$ is isomorphic to a subgroup $\\Onr$ of a group $\\On$ consisting of non-initial, bi-synchronizing transducers. Furthermore, that paper also shows that $\\On = \\cup_{1\\le r \\le n-1} \\Onr$. More specifically, it is demonstrated in \\cite{BCMNO} that for $1 \\le i \\le j \\le n-1$ such that $i$ divides $j$ in the additive group $\\Z_{n-1}$, $\\Ons{i} \\le \\Ons{j}$. As a consequence of this, one may deduce that for all $1 \\le i \\le n-1$ $\\Ons{1}\\le \\Ons{i}$, $\\Ons{i} \\le \\Ons{n-1}$ (and so $\\On = \\Ons{n-1}$) and $\\Ons{i} = \\Ons{d}$ for $d = \\gcd(n-1, i)$. We extend these result to the group $\\out{\\Tnr}$. That is, we have:\n\n\\begin{Theorem}\nFor all $1 \\le r \\le n-1$, the group $\\out{\\Tnr}$ is isomorphic to a subgroup $\\TOnr$ of $\\Onr$. Moreover for all $1 \\le i \\le j \\le n-1$ such that $i$ divides $j$ in the additive group $\\Z_{n-1}$, we have $\\TOns{i} \\le \\TOns{j}$. \n\\end{Theorem} \n\n\\begin{corollary}\\label{Corollary:OniequalsOnjifgcdequal}\nFor $1 \\le 1 \\le n-1$, we have $\\TOns{1} \\le \\TOns{i}$, $\\TOns{i} \\le \\TOns{n-1}$ and $\\TOns{i} = \\TOns{d}$ for $d = \\gcd(n-1, i)$.\n\\end{corollary}\n\nWe shall subsequently use the symbol $\\TOn$ for the group $\\TOns{n-1}$.\n\nThe last phrase of Corollary~\\ref{Corollary:OniequalsOnjifgcdequal} is perhaps to be expected, certainly when $X = G$, as results of Higman (\\cite{GHigman}), Pardo (\\cite{EPardo}), and Dicks and Mart{\\' \\i}nez-P{\\' e}rez (\\cite{DicksPerez}) demonstrates that $\\Gnr \\cong \\Gmr{m}{s}$ if and only if $n=m$ and $\\gcd(n-1, r) = \\gcd(n-1,s)$. In fact it is a question in \\cite{BCMNO} whether or not $\\Onr \\cong \\Ons{s}$ if and only if $\\gcd(n-1,r) = \\gcd(n-1,s)$. We show that this question has a negative answer in both the $\\Gnr$ and $\\Tnr$ context. That is we prove the following:\n\n\\begin{Theorem}\nThere is a number $n \\in \\N$, $n >2$, and $1 \\le r, s \\le n-1$ such that, for $X = \\T{TO}, \\T{O}$, $\\XOnr = \\XOns{s}$ but $\\gcd(n-1, r) \\ne \\gcd(n-1, s)$.\n\\end{Theorem}\n\n\nOur next result demonstrates that the groups $\\XOns{r}$ for all $1 \\le r \\le n-1$ are normal subgroups of $\\XOn$.\n\n\\begin{Theorem}\nFor all $1 \\le r \\le n-1$ we have $\\XOns{r} \\unlhd \\XOn$.\n\\end{Theorem}\n\n \n \nWe next extend a result of Brin and Guzm{\\'a}n \\cite{MBrinFGuzman} for $\\TOns{n-1}$, and show that for $n \\ge 3$, $\\XOns{1}$ contains an isomorphic copy of R. Thompson's group $F$:\n\n\\begin{Theorem}\nLet $n \\ge 3$, then $\\out{X_{n,1}}$ contains a subgroup isomorphic to R. Thompson's group $F$. \n\\end{Theorem}\n\n From this and Corollary~\\ref{Corollary:OniequalsOnjifgcdequal}, we have the following:\n \\begin{corollary}\n For all $1 \\le r\\le n-1$, $\\XOns{r}$ contains a subgroup isomorphic to R. Thompson's group $F$.\n \\end{corollary}\n \n \n We further demonstrate (Section~\\ref{Section:nestingproperties2}), in the case $r=4$, that the set $\\TOms{4}{3}\\backslash \\TOms{4}{1}$ is non-empty. Notice that since $3$ is prime, for $ 1 \\le r <3$, $\\TOms{4}{r} = \\TOms{4}{1}$, thus this result indicates that, in general, the group $\\TOnr$ might depend on $r$.\n\n\n\n We also investigate in Section~\\ref{Section:nestingproperties2} the nesting properties of the groups $\\XOnr$ of $\\XOn$ for $1 \\le r \\le n-1$. We show that these groups from a lattice with the `meet' of $\\XOnr$ and $\\XOns{s}$ being the intersection of the two groups and the `join' of $\\XOnr$ and $\\XOns{s}$ being the smallest $t$, $1 \\le t \\le n-1$ such that $\\XOnr$ and $\\XOns{s}$ are subgroups of $\\XOns{t}$. We do not know if it is in fact the case that $\\XOns{t} = \\gen{\\XOns{s}, \\XOns{r}}$ (see Question~\\ref{Question:canjoinbereplacedwithubgroupgenerated}). To each element of $\\XOn$ we associate a numerical invariant which yields a group homomorphism from $\\XOn$ to the group of units of $\\Z_{n-1}$ (notice that for $n=2$, $\\Z_{n-1}$ is equal to its group of units), with kernel $\\XOns{1}$. That is, we prove the following:\n \n \\begin{Theorem}\\label{Theorem:homfromoutintounitsofZn}\n There is homomorphism from $\\XOn$ to the group of units of $\\Z_{n-1}$ with kernel $\\XOns{1}$. \n \\end{Theorem}\n \nWe should point out that the existence of this homomorphism is already a consequence of the fact that the dimension group (see \\cite{KriegerW1980} for the definition of the dimension group) of $\\XOnr$ is equal to the additive group $\\Z_{n-1}$ and automorphisms of $\\XOnr$ yield automorphisms of the dimension group of $\\XOnr$. The author is grateful to Prof. Nekrashevych for drawing these facts to his attention. Our proof of Theorem~\\ref{Theorem:homfromoutintounitsofZn} however does not rely on these observations, instead we explicitly construct the homomorphism by making use of the synchronizing condition to associate a numerical invariant to every element of $\\XOnr$. Our approach gives a means of resolving this question as we reduce it to one of constructing transducers which have certain properties. In particular, Theorem~\\ref{Theorem:homfromoutintounitsofZn} demonstrates that the homomorphism from $\\XOns{1}$ into the group of units of $\\Z_{n-1}$ is the trivial homomorphism. Though we are unable show in general that the homomorphism of Theorem~\\ref{Theorem:homfromoutintounitsofZn} is surjective, we show that under the assumption that it is surjective, then for $1 \\le s,r,t \\le n-1$ and $t$ the smallest element of $\\Z_{n}$ such that $\\XOms{n}{t}$ contains $\\XOms{n}{s}$ and $\\XOms{n}{r}$, $\\XOms{n}{t}= \\gen{\\XOms{n}{s}, \\XOms{n}{r}}$. The Theorem~\\ref{Theorem:surjectivityofrsigpartial} below, proven in Section~\\ref{Section:onsurjectivityofrsig}, indicates that the homomorphism of Theorem~\\ref{Theorem:homfromoutintounitsofZn} from $\\On$ to the group of units of $\\Z_{n-1}$ is surjective in many cases, indeed, elementary results in number theory indicate that there are infinitely many numbers $n$ which satisfying the hypothesis of the theorem. We do not know if the restriction to $\\TOn$ is also surjective.\n\n\n\\begin{Theorem}\\label{Theorem:surjectivityofrsigpartial}\nIf the divisiors of $n$ generate the group of units of $\\Z_{n-1}$ then the homomorphism of Theorem~\\ref{Theorem:rsigisahomomorphism} defined on $\\On$ is unto the group of units of $\\Z_{n-1}$.\n\\end{Theorem}\n\n \n \nWe conclude in Section~\\ref{Section:RftyTn} with the following result:\n\\begin{Theorem}\nThe groups $\\Tnr$ have the $\\rfty$ property.\n\\end{Theorem}\nWe recall that a group $G$ is said to have the $R_{\\infty}$ property if for every automorphism $\\varphi$ of $G$, the equivalence relation defined on $G$ by, for $x,y \\in G$, $x$ is equivalent to $y$ if there is an element $h$ in $G$ such that $h^{-1}x(h)\\varphi = y$, has infinitely many equivalence classes. The question of which groups have the $\\rfty$ property has received a lot of attention. This class of groups has been shown to include, for instance, all non-elementary Gromov hyperbolic groups (\\cite{Fel'shtyn2004}, \\cite{GLevittMLustig}), Baumslag-Solitar groups $BS(m,n)$ for $m,n \\in \\Z\\backslash\\{0\\}, (m,n) \\ne (1,1)$ (\\cite{AFelshtynDGonclavesBSG}), lamplighter groups $\\Z_{n} \\wr \\Z$ where $2|n$ or $3|n$ (\\cite{DGonclavesPWong}), the first Grigorchuk and the Gupta-Sidki groups (\\cite{AFelshtynLYuriyandETroitsky}), and R. Thompson's group $F$ (\\cite{BlkFelAlxGon}, \\cite{BMV}). It was shown independently by Burillo, Matucci, and Ventura (\\cite{BMV}), and Gon{\\c c}alves and Sankaran (\\cite{DGonclavesPSankaran}) that $R$. Thompson's group $T$ is in this class of groups. We thus extend this result to the family of groups $T_{n,r}$.\n\nInterlaced through out the text are several open questions. In work in preparation we apply the techniques in the current paper and in the paper \\cite{BCMNO} to the generalizations of Thompson's group $F$. \n\\section*{Acknowledgements}\nThe author is grateful to Collin Bleak and Matthew Brin for helpful discussions and their comments on early drafts of this article. This work was partially supported by Leverhulme Trust Research Project Grant RPG-2017-159.\n\\section{Some preliminaries and the groups \\texorpdfstring{$G_{n,r} \\mbox{ and } T_{n,r}$}{Lg}}\\label{Section:Preliminaries}\nWe begin by setting up some notation.\n\nLet $X$ be a topological space, we shall denote by $H(X)$ the group of homeomorphisms of $X$, and for $G \\le H(X)$, $N_{H(X)}(G)$ shall denote the normaliser of $G$ in $H(X)$. Let $S_r:= \\mathbb{R}\/r \\Z$ for $r \\in \\mathbb{R}$, the circle of length $r$ and for $n \\in \\N \\backslash \\{0\\}$, let $\\Z[1\/n] := \\{ a\/n : a \\in \\Z\\}$, the $n$-adic rationals. Though we will mainly think of $T_{n,r}$ as a subgroup of $G_{n,r}$, we shall at times consider it as a subgroup of $H(S_r)$ and we shall make it apparent at such times that we are doing so. We establish some further notation required to define the groups $G_{n,r}$ and $\\Tnr$.\n\nLet $\\dotr:= \\{\\dot{0},\\dot{1},\\ldots, \\dot{r-1}\\}$, and let $X_n := \\{0,1,\\ldots, n-1\\}$. We shall take the ordering $\\dot{0} < \\dot{1} < \\ldots < \\dot{r-1}$ and $0< 1 < \\ldots < n-1$ on $\\dotr$ and $\\Xn$ respectively. For $i \\in \\{0, n-1\\}$, if $i= 0$ then set $\\bar{i} = n-1$ otherwise set $\\bar{i} = 0$. Set $X_{n,r}^{\\ast} := \\{\\dot{a}w: \\dot{a} \\in \\dotr \\mbox{ and } w \\in X_n^{\\ast}\\} \\sqcup \\{\\epsilon\\}$ where $\\epsilon$ denotes the empty word, and $X_{n,r}^{+}:= \\{\\dot{a}w: \\dot{a} \\in \\dotr \\mbox{ and } w \\in X_n^{\\ast}\\}$. For $j \\in \\N$, let $\\Xnr^{j}$, respectively $\\Xn^{j}$, denote the set of all elements of $X_{n,r}^{*}$, respectively $X_n^{*}$, of length $j$. Given two words $u$ and $v$ in $X_n^{*}$ of $X_{n,r}^{*}$, we say that $u$ is a prefix of $v$ and denote this by $u \\le v$. We write $u < v$ if $u$ is a proper prefix of $v$. For two words $u, v$ in $X_{n}^{*}$ or $X_{n,r}^{*}$ if $u \\nleq v$ and $v \\nleq u$ then we say that $u$ is incomparable to $v$ an write $u \\perp v$. If instead $ v = u \\nu$ for some $\\nu \\in \\Xns \\sqcup \\Xnrs$ then we set $v-u := \\nu$. The relation $\\le$ is a partial order on the set $X_{n,r}^{*}$ and $X_n^{*}$.\n\nFor two incomparable words $\\nu, \\eta$ in $\\Xnp$ and $\\Xnrp$ we say \\emph{$\\nu$ is less than $\\eta$ in the lexicographic ordering}, denoted $\\nu \\lelex \\eta$, if there are $wi, wj \\in \\Xnrp$ or $wi, wj \\in \\Xnp$ prefixes of $\\nu$ and $\\eta$ respectively with $i < j$ in the ordering on $\\Xn$ or $\\dotr$. If $i \\le j$ then we write $\\nu \\leqlex \\eta$.\n\n We may now defined a total order of $\\Xnrs$ and $\\Xns$ as follows: for $\\nu, \\eta \\in \\Xnrs$ or $\\nu, \\eta \\in \\Xns$ we say that \\emph{$\\nu$ is less than $\\eta$ in the short-lex ordering} if either $|\\nu| \\le |\\eta|$ or if $|\\nu| = |\\eta|$ then $\\nu \\leqlex \\eta$, with strict inequalities if $\\nu$ is strictly less than $\\eta$. We write $\\nu \\leqslex \\eta$ if $\\nu$ is less than $\\eta$ in the short-lex ordering, and $\\nu \\leslex \\eta$ is the $\\nu$ is strictly less than $\\eta$ in the short-lex ordering. \n\nNow let $\\CCn := X_n^{\\omega}$ a Cantor space with the usual topology, and let $\\CCnr:= \\{\\dot{a}x | \\dot{a} \\in \\dotr \\mbox{ and } x \\in \\CCn \\}$, the disjoint union of $r$ copies of $\\CCn$. For a word $\\nu \\in X_{n,r}^{+}$, let $U_{\\nu}:= \\{\\nu\\delta \\mid \\delta \\in \\CCn\\}$, and $U_{\\epsilon}:= \\CCnr$. For $\\nu \\in X_n^{\\ast}$, set $U_{\\nu}:= \\{ \\nu \\delta: \\delta \\in \\CCn\\}$. Let $\\Banr:= \\{U_{\\nu}\\mid \\nu \\in X_{n,r}^{\\ast} \\}$ and let $\\Ban:= \\{U_{\\nu} \\mid \\nu \\in \\Xn \\}$, then $\\Banr$ and $\\Ban$ are a basis for the topology on $\\CCnr$ and $\\CCnr$ respectively. For a subset $Z \\subseteq \\Xns \\sqcup \\Xnrs$ we shall denote by $U(Z)$ the set $\\{ U_{z} \\mid z \\in Z \\}$.\n\n\nThe lexicographic extends to a total order on $\\CCn$ and $\\CCnr$ in the natural way, thus we extend the meanings of the symbols $\\leqlex$ and $\\lelex$ to this sets also. We further extend this notation to subsets of $\\CCn$ and $\\CCnr$. Given two subsets $V, W \\subset \\CCn$ or $V, W \\subset \\CCnr$ we write $V \\leqlex W$ if every element of $U$ is less than every element of $W$ in the lexicographic ordering. The strict inequality $V \\lelex W$ is analogously defined.\n\nLet $\\nu, \\eta \\in X_{n,r}^{+}$, then we shall denote by $g_{\\nu,\\eta}$ the map from $U_{\\nu}$ to $U_{\\eta}$ that replaces the prefix $\\nu$ with $\\eta$.\n\nA finite set $\\overline{u}:= \\{u_0, \\ldots, u_l\\}$ for $u_i \\in X_{n,r}^{+}$ and some $1\\le l \\in \\N$ is called a \\emph{complete antichain}, if for any pair $u_i, u_j \\in \\overline{u}$, $u_i \\perp u_j$ and for any word $v \\in X_{n,r}^{\\ast}$, there is some $u_i \\in \\overline{u}$ such that $v \\le u_i$ or $u_i \\le v$. We shall assume through out that any given antichain $\\overline{u}$ is ordered according to the lexicographic ordering. If we have an antichain $\\ac{u}:= \\{u_0, \\ldots, u_l\\}$ then we may form another antichain $\\ac{u}' = \\{u_0, \\ldots, u_i0, u_i1, \\ldots u_in-1, u_{i+1}, \\ldots, u_l\\}$ where we have replaced $u_i$ with $u_i0, \\ldots, u_in-1$. We call $\\ac{u}'$ a subdivision of $\\ac{u}$. Notice that each time we make a subdivision the length of the antichain increases by $n-1$.\n\nWe now define the Higman-Thompson group $G_{n,r}$ and the subgroup $T_{n,r}$.\n\nGiven two complete antichains $\\overline{u} = \\{u_0,\\ldots,u_{l-1} \\}$ and $\\overline{v} = \\{v_0, \\ldots, v_{l-1}\\}$ of equal length, we can define a homeomorphism from of $\\CCnr$ as follows. Let $\\pi$ be a bijection from $\\overline{u}$ to $\\overline{v}$, define a map $g : \\CCnr \\to \\CCnr$ by $ u_ix \\mapsto (u_i)\\pi x$ for $u_i \\in \\overline{u}$ and $x \\in X_n^{\\omega}$. Since $u$ is a complete antichain this map is well-defined on all of $\\CCnr$. The group $G_{n,r}$ consists of all homeomorphisms of $\\CCnr$ which can be defined in this way. The group $T_{n,r}$ is the subgroup of $G_{n,r}$, consisting of those element $g$ that are derived from complete antichains $\\overline{u}$ and $\\ac{v}$ of the same length $l$, and a bijection $\\pi$ which maps $u_i$ to $v_{i+j \\mod l}$ for some fixed $j \\in \\{0,1 \\ldots, l-1\\}$. \n\nDefine an equivalence relation on $\\CCnr$ by $x \\sim_{t} y$ if and only if there are words $u$ and $v$ in $X_{n,r}^{+}$ such that $x = u z$ and $y = v z$ for some $z \\in \\CCn$. We write $[x]$ to denote the equivalence class of $x$. Let $H_{n,r,\\sim_{t}}$ be the subgroup of $H(\\CCnr)$ consisting of those elements of $H(\\CCnr)$ that preserve $\\sim_{t}$.\n\nWe shall need the following transitivity result for $T_{n,r}$\n\n\\begin{lemma}\\label{Lemma: Tnr preserves tail classes and acts transitively on each tail class}\nThe group $T_{n,r}$ fixes the equivalence classes of $\\sim_{t}$ and acts transitively on each equivalence class.\n\\end{lemma}\n\\begin{proof}\nThe first statement follows from the definition of $T_{n,r}$ as a subgroup of $G_{n,r}$. The second part of the lemma follows by the following observation: if $x = \\mu z$ and $y = \\nu z$ for some $\\mu, \\nu \\in X_{n,r}^{+}$ and $z \\in \\CCn$, then there is an element of $T_{n,r}$ mapping $x$ to $y$. To see this observe that the lengths of the complete antichains $\\ac{u}:= X_{n,r}^{|\\mu|}$ and $\\ac{v} := X_{n,r}^{|\\nu|}$ are congruent modulo $n-1$. We may assume that the length of $\\ac{u}$ is smaller than the length of $\\ac{v}$. Therefore there is a complete antichain $\\ac{u}'$ such that $\\mu$ is an element of $\\ac{u}'$, $\\ac{u}'$ is a repeated subdivision of $\\ac{u}$ and the length of $\\ac{u}'$ is equal to the length of $\\ac{v}$. It is now straightforward to see that, since $|\\ac{v}_1'| = |\\ac{v}_2$, there is an element of $T_{n,r}$ that sends any element $\\mu z'$ to the element $\\nu z'$ for $z' \\in \\CCn$. In particular such a map sends $x$ to $y$.\n\\end{proof} \n\nIn the next Section, we see that elements of $T_{n,r}$ induce homeomorphisms of the circle $\\mathbb{R}\/r\\Z$ by thinking of the circle as a quotient of Cantor space.\n\n\\section{From Cantor Space to the Circle}\\label{Section:fromcantorspacetothecircle}\n\n \n \\begin{Definition}\n We say that a homeomorphisms $h$ of $\\CCnr$ is \\emph{orientation preserving} if whenever $x,y \\in \\CCnr$ and $x \\lelex y$ then $(x)h \\lelex (y)h$. We say that $h$ is \\emph{orientation reversing} if whenever $x\\lelex y$ then $(y)h \\lelex (x)h$. We say that $h$ is \\emph{locally orientation preserving\/ reversing} if there is a neighbourhood of $\\CCnr$ on which $h$ is orientation preserving\/reversing.\n \\end{Definition}\n \nWe identify $S_r$ with the interval $[0, r]$ with the end points identified. Now, every point in $[0,r]$ can be written as $\\dot{a}x$ for $x \\in \\CCn$ and $\\dot{a} \\in \\dotr$ in $n$-ary expansion. However this representation is not unique. In particular two elements $x, y \\in \\CCnr$ represents the same element of $[0,r]$ if and only if there is some integer $i$, $0< i \\le n-1$ or $\\dot{0}< i \\le \\dot{r-1}$, and some $w \\in \\Xnr^{*}$ such that $x = wi00\\ldots$ and $y= w(i-1)n-1n-1\\ldots$ (i.e only elements of $(0,r)\\cap \\Z[1\/n]$ have non-unique $n$-ary representations ). Let $\\simeq$ be the equivalence relation on $\\CCnr$ defined by $x \\simeq y$ if and only if there is some $0< i \\le n-1$ or $\\dot{0} < i \\le \\dot{r-1}$ and some $w \\in \\Xnr^{+}$ such that $x = wi00\\ldots$ and $y= w(i-1)n-1n-1\\ldots$ or $x= \\dot{0}00\\ldots$ and $y = \\dot{r}n-1 n-1 \\ldots$ then $\\CCnr\/\\simeq$ is homeomorphic to $S_{r}$. Let $\\simeq_{{\\bf{I}}}$ be the relation on $\\CCn$ given by $x \\simeq_{{\\bf{I}}} y$ if and only if there is some $0< i \\le n-1$ or $\\dot{0} < i \\le \\dot{r-1}$, and some $w \\in \\Xns$ such that $x = wi00\\ldots$ and $y= w(i-1)n-1 n-1\\ldots$ then $\\CCn\/\\simeq_{{\\bf{I}}}$ is homeomorphic to the interval $[0,r]$.\n\n\n\n\nLet $N$ be the subgroup of $H(\\CCnr)$ consisting of those elements $h$ which preserve $\\simeq$ and which satisfy the following: for all points $t \\in \\CCnr$ there is a neighbourhood of $t$ in $\\CCnr$ such that $h$ is orientation preserving ($h$ is orientation reversing), and there is some $w \\in \\Xnrs$, $0< i \\le n-1$ or $\\dot{0} < i \\le \\dot{r-1}$, and points $x= wi00\\ldots$ and $y= wi-1n-1n-1\\ldots$ so that $(y)h = \\dot{r}n-1n-1\\ldots$ and $(x)h = \\dot{0}00\\ldots$ ($(y)h = \\dot{0}00\\ldots$ and $(x)h = \\dot{r}n-1n-1\\ldots$). Observe that all elements of $N$ induce homeomorphisms of $S_{r}$, and $N$ contains $T_{n,r}$. Our aim shall be to show that $N_{H(\\CCnr)}(\\Tnr) \\le N$.\n\n\nWe shall need the following results from \\cite{MBrinFGuzman}. The first follows by Rubin's Theorem and the transitivity of $T_{n,r}$ on the circle $S_{r}$ and the second follows by studying the germs of elements of $T_{n,r}$ at a fixed point.\n\n\n\\begin{Theorem}\nLet $r < n \\in \\mathbb{\\N}$ and let $n \\ge 2$, then $\\aut{T_{n,r}} \\cong N_{H(S_{r})}(T_{n,r})$.\n\\end{Theorem}\n\n\\begin{lemma}\\label{Lemma: normalisers map n-adic rationals to n-adic rationals}\nIf $h \\in N_{H(S_{r})}(\\Tnr)$, then $ ( [0,r] \\cap \\Z[1\/n])h = [0,r] \\cap \\Z[1\/n]$. \n\\end{lemma}\n\nNow let $g \\in N_{H(S_r)}(\\Tnr)$. Define $h \\in H(\\CCnr)$ as follows. For all $t \\in S_{r}$ such that $t \\notin [0,r] \\cap \\Z[1\/n]$ (notice that this also means $(t)g \\notin [0,r] \\cap \\Z[1\/n]$ by Lemma~\\ref{Lemma: normalisers map n-adic rationals to n-adic rationals}), let $x$ be the unique $n$-ary expansion of $t$, and $y$ be the unique $n$-ary expansion of $(t)g$, set $(x)h = y$. For $t \\in [0,r] \\cap \\Z[1\/n]$ let $x$ and $x'$ be the $n$-ary expansions of $t$ such that $x \\lelex x'$ in the lexicographic ordering of $\\CCnr$ (if $t =0$, then chose $x$ and $x'$, satisfying $x \\lelex x'$, from the set $\\{\\dot{0}00\\ldots, \\dot{r-1}n-1n-1\\ldots\\}$). Also let $y$ and $y'$ be the $n$-ary expansions of $(t)g$ in $\\CCnr$ such that $y \\lelex y'$ (if $(t)g = 0$ then take $y= \\dot{0}00\\ldots$ and $y' = \\dot{r}n-1n-1\\ldots$). If $g$ is an orientation preserving homeomorphism of $S_{r}$, set $(x)h = y$ and $(x')h = y'$, otherwise $g$ is orientation reversing and we set $(x)h = y'$ and $(x')h = y$. Thus $h$ is now defines a bijection from $\\CCnr$ to itself. Moreover it is easy to see that since $g$ is continuous, $h$ is also continuous on $\\CCnr$. Furthermore if $h'$ is the homeomorphism obtained from $g^{-1}$ in the same way, it is easy to see that $hh' = h'h = \\id \\in H(\\CCnr)$. Therefore $h \\in N$. Let $\\phi: N_{S_r}(\\Tnr) \\to N$ such that an element $g \\in N_{S_r}(\\Tnr)$ maps to the element $h \\in H(\\CCnr)$, constructed as above, which induces the map $g$ on $S_r$. It follows that $\\phi$ is an injective homomorphism. \n\nLet $N(\\Tnr)$ denote the image of $\\phi$. We now show that $N_{H(\\CCnr)}(\\Tnr) = N(\\Tnr)$. Observe that as $N(\\Tnr) \\subseteq N_{H(\\CCnr)}(\\Tnr)$ it suffices to show only that $N(\\Tnr) \\supseteq N_{H(\\CCnr)}(\\Tnr)$. \n\nLet $\\tau, \\eta \\in X_{n,r}^{+}$ such that $\\tau \\perp \\eta$. We further assume that the points $x=\\tau00\\ldots$, $y=\\eta n-1 n-1\\ldots $, $z= \\tau n-1 n-1 \\ldots$ and $t = \\eta 00\\ldots$ of $\\CCnr$ satisfy $x \\not\\simeq y$ or $z \\not\\simeq t$. \nLet $\\mathscr{G}$ denote the set of incomparable pairs $(\\tau, \\eta)$ satisfying the conditions above where $\\tau \\lelex \\eta$. For a pair $(\\tau, \\eta)$ in $\\mathscr{G}$ let $\\ac{u} := \\{u_1, \\ldots, u_l\\}$ be any complete finite antichain of $\\Xnrs$ containing $\\tau$ and $\\eta$ (since $\\tau \\perp \\eta$ such an antichain exists). Let $u_{l_1} = \\tau$ and $u_{l_2} = \\eta$. Let $j = l_1-l_2 -1$. Then we say that $j$ is the \\emph{node distance between $\\tau$ and $\\eta$ in $\\ac{u}$} i.e $j$ is the number of elements of $\\ac{u}$ which are strictly between $u_{l_1}$ and $u_{l_2}$. Notice that $j$ is necessarily non-zero by assumption. Moreover we have, for any $v \\in \\{0\\}^{+}$ and $w \\in \\{n-1\\}^{+}$, that the pair $(\\dot{0}v, \\dot{r-1}w)$ is not in $\\mathscr{G}$.\n\nIt is straight-forward to see that if $(\\tau, \\eta) \\in \\mathscr{G}$ have node distance $i$ in some complete antichain $\\ac{u}_{1}$, then for any other complete antichain $\\ac{u}_2$ containing $\\tau$ and $\\eta$ the node distance between $\\tau$ and $\\eta $ in $\\ac{u}_{2}$ is congruent to $i$ modulo $n-1$. Moreover, for any $\\chi \\in X_n^{*}$, $(\\tau \\chi, \\eta \\chi) \\in \\mathscr{G}$ and there is a complete antichain $\\ac{v}$ containing $\\tau\\chi$ and $\\eta\\chi$ such that the node distance between $\\tau\\chi$ and $\\eta \\chi$ in $\\ac{v}$ is congruent to $i$ modulo $n-1$. Since modulo $n-1$ the node distance between a pair $(\\tau, \\eta)$ in $\\mathscr{G}$ in a given complete antichain containing $\\tau$ and $\\eta$ is independent of the complete antichain, we define the \\emph{reduced node distance} between $\\tau$ and $\\eta$ to be $i \\in \\{0,1, \\ldots, n-2\\}$ such that for any complete antichain $\\ac{u}$ containing $\\tau$ and $\\eta$ the node distance between $\\tau$ and $\\eta$ in $\\ac{u}$ is congruent to $i$ modulo $n-1$.\n\nWe require the following transitivity result of $\\Tnr$.\n\n\\begin{lemma}\\label{Lemma: can swap things with same node distance}\nLet $(\\nu_1, \\nu_2), (\\eta_1, \\eta_2) \\in \\mathscr{G}$ be such that the reduced node distance between $\\nu_1$ and $\\nu_2$ is equal to the reduced node distance between $\\eta_1$ and $\\eta_2$. Then there is a $g \\in \\Tnr$ such that $g\\restriction_{U_{\\nu_1}}= g_{\\nu_1, \\eta_1}$ and $g\\restriction_{U_{\\nu_2}} = g_{\\nu_2, \\eta_2}$. \n\\end{lemma}\n\\begin{proof}\nLet $\\ac{u}$ be a complete antichain containing $\\nu_1$ and $\\nu_2$, and $\\ac{v}$ be a complete antichain containing $\\eta_1$ and $\\eta_2$. Let $\\ac{u}= \\{u_1, \\ldots, u_{l}\\}$ and $\\ac{v} = \\{v_1, \\ldots, v_{m}\\}$. Let $u_{l_1} = \\nu_1$ and $u_{l_2} = \\nu_2$ and $v_{m_1} = \\eta_1$ and $v_{m_2} = \\eta_2$ where $l_1 < l_2$ and $m_1 < m_2$. By assumption $i:= l_2 - l_1 -1 $ and $j:=m_2 - m_1 -1$ are both non-zero and congruent modulo $n-1$. Without loss of generality we may assume $i< j$, moreover there is an element $u_{l_1+1} \\in \\ac{u}$ between $\\nu_1$ or $\\nu_2$. By replacing $u_{l_1 + 1} \\in \\ac{u}$ with $\\{u_{l_1 +1}0, \\ldots, u_{l_1 +1} n-1 \\}$ we obtain a new complete antichain $\\ac{u}'$ containing $\\nu_1$ and $\\nu_2$ such that the node distance between $\\nu_1$ and $\\nu_2$ is equal to $i + n-1$. Replace the antichain $\\ac{u}$ with $\\ac{u}'$\n\nBy repeatedly performing this operation we may assume that the complete antichain $\\ac{u}$ containing $\\nu_1$ and $\\nu_2$ is such that the node distance between $\\nu_1$ and $\\nu_2$ in $\\ac{u}$ is equal to $j$.\n\nObserve that as $\\ac{u}$ and $\\ac{v}$ are complete antichains the difference $| m-l|$ is congruent to $0$ modulo $n-1$. Without loss of generality (as we may relabel to achieve this) we assume that $l < m$.\n\nNow observe that $u_1$ must be equal to $\\dot{0}w_1$ for some $w_1 \\in \\{0\\}^{*}$ and $u_{l} = \\dot{r-1}w_2$ for some $w_2 \\in \\{n-1\\}^{*}$. Moreover, since the pair $(\\dot{0}v, \\dot{r-1}w)$ for $v \\in \\{0\\}^{+}$ and $w \\in \\{n-1\\}^{+}$ is not in $\\mathscr{G}$, it follows that either $\\nu_{l_1} \\ne u_{l} = \\dot{0}w_1$ or $\\nu_{l_2} \\ne \\dot{r-1}w_2$. Suppose that $\\nu_{l_1} \\ne u_{l} = \\dot{1}w_1$ (the other case is handled similarly). Then by repeatedly expanding along the node $\\dot{1}w_1$, we obtain a complete antichain $\\ac{u}''$ of length $m$ containing $\\nu_1$ and $\\nu_2$ such that the node distance between $\\nu_1$ and $\\nu_2$ is $j$.\n\nSince the complete antichain $\\ac{u}''$ and $\\ac{v}$ have the same lengths and the node distance between $\\nu_1$ and $\\nu_2$ in $\\ac{u}''$ is equal to the node distance between $\\eta_1$ and $\\eta_2$ in $\\ac{v}$, it is easy to construct an element $g$ of $\\Tnr$ such that $g\\restriction_{U_{\\nu_1}}= g_{\\nu_1, \\eta_1}$ and $g\\restriction_{U_{\\nu_2}} = g_{\\nu_2, \\eta_2}$. \n\\end{proof}\n\nUsing this transitivity result we show below that $N(\\Tnr)$ is equal to $N_{H(\\CCnr)}(\\Tnr)$.\n\n\\begin{lemma}\nLet $h \\in H(\\CCnr)$ be such that $h \\in N_{H(\\CCnr)}(\\Tnr)$. Then $h$ (and so $h^{-1}$) preserves the equivalence relation $\\simeq$.\n\\end{lemma}\n\\begin{proof}\nSuppose there are $x, y \\in \\CCnr$ such that $x \\simeq y$ but $(x)h \\not\\simeq (y)h$. By relabelling if necessary we may assume that $(x)h < (y)h$. Since $(x)h \\not \\simeq (y)h$ there are $(\\tau, \\eta) \\in \\mathscr{G}$ such that $(x)h \\in U_{\\tau}$ and $(y)h \\in U_{\\eta}$. Let $j \\in \\{0,1 , \\ldots, n-2\\}$ be the reduced node distance between $\\tau$ and $\\eta$\n\nLet $(\\mu, \\nu) \\in \\mathscr{G}$ and consider the clopen sets $(U_{\\mu})h$ and $(U_{\\nu})h$. It is straight-forward to see that there are $\\mu'$ and $\\nu'$ such that $(\\mu', \\nu') \\in \\mathscr{G}$, the reduced node distance between $\\mu'$ and $\\nu'$ is equal to $j$, and $U_{\\mu'} \\subset (U_{\\mu})h$ and $U_{\\nu'} \\subset (U_{\\nu})h$ (the case where $U_{\\mu'} \\subset (U_{\\nu})h$ and $U_{\\nu'} \\subset (U_{\\mu})h$ is analogous). By Lemma~\\ref{Lemma: can swap things with same node distance} there is an element $g \\in \\Tnr$ such that $g\\restriction_{U_{\\tau}} = g_{\\tau,\\mu'}$ and $g \\restriction_{U_{\\eta}} = g_{\\eta, \\nu'}$.\n\nNow consider the product $h g h^{-1}$. Notice that $(x) h g h^{-1}$ is contained in the clopen set $U_{\\mu}$ since $(x)h \\in U_{\\tau}$ and $(U_{\\tau})g \\in U_{\\mu'} \\subset (U_{\\mu})h$, likewise $(y)h g h^{-1} \\in U_{\\nu}$. Therefore $(x)hgh^{-1} \\not\\simeq (y)hgh^{-1}$, which is a contradiction since $T_{n,r}$ preserves $\\simeq$.\n\\end{proof}\n\n\\begin{lemma}\nLet $h \\in H(\\CCnr)$ be such that $h \\in N_{H(\\CCnr)}(\\Tnr)$, then $h \\in N(\\Tnr)$.\n\\end{lemma}\n\\begin{proof}\nSince $h \\in N_{H(\\CCnr)}(\\Tnr)$, then by the previous lemma $h$ preserves $\\simeq$. Therefore $h$ induces a continuous function $g$ on $S_{r}$. However since $h^{-1}$ is also in $N_{(H(\\CCnr))}$, $g$ must be a homeomorphism and $(g)\\phi = h$, therefore $h \\in N(\\Tnr)$.\n\\end{proof}\n\nThus we have now proved that $N_{H(\\CCnr)}(\\Tnr) = N(\\Tnr) \\cong \n\\aut{\\Tnr}$. \n\n\\section{Automorphisms of \\texorpdfstring{$T_{n,r}$}{Lg}}\\label{Section:AutTnr}\n In what follows we adapt the results of \n\\cite{BCMNO} to show that elements of \n$N_{H(\\CCnr)}(\\Tnr)$ can be represented by bi-synchronizing \ntransducers. However we shall need to define the group $\\T{R}_{n,r}$ introduced in \\cite{BCMNO} which is a slight modification of the Rational group $\\mathcal{R}_{n}$ in \\cite{GriNekSus}. The exposition in this section will largely mirror that found in \\cite{BCMNO}.\n\nFirst we introduce transducers generally then we introduce transducers over $\\CCnr$.\n\nLet $X_{I}$ and $X_{O}$ be finite sets of symbol. A transducer over the alphabet $X_{I}$ is a tuple $T = \\gen{X_{I},X_O, Q_{T} \\pi_{T}, \\lambda_{T}}$ such that:\n\n\\begin{enumerate}[label = (\\roman{*})]\n\\item $X_{I}$ is the \\emph{input alphabet} and $X_{O}$ is the \\emph{output alphabet}.\n\\item $Q_T$ is the set of states of $T$.\n\\item $\\pi_{T}: X_I \\times Q_{T} \\to Q_{T}$ is the \\emph{transition function} and,\n\\item $\\lambda_{T}: X_I \\times Q_{T} \\to X_{I}^{*}$ is the \\emph{output function}.\n\\end{enumerate}\n\n\nIf $|Q_T| < \\infty$ then we say that $T$ is a finite transducer. If $X_{I} = X_{O} = X$ then we shall write $T = \\gen{X, Q_{T}, \\pi_{T}, \\lambda_{T}}$. If we fix a state $q \\in Q_{T}$ from which we begin processing inputs then we say that $T$ is \\emph{initialised at $q$} and we denote this by $T_{q}$ and we call $T_{q}$ an \\emph{initial transducer}. Given an initial transducer $T_{q_0}$ we shall write $T$ for the underlying transducer with no initialised states.\n\nWe inductively extend the domain of the transition and rewrite function to $X_{I}^{*} \\times Q_T $ by the following rules:\nfor a word $w \\in X_{I}^{*}$, $i \\in X_{I}$ and any state $q \\in Q_T$ we have $\\pi_T(wi, q ) = \\pi_T(i, \\pi_T(w, q))$ and $\\lambda_{T}(wi, q) = \\lambda_{T}(w, q)\\lambda_{T}(i, \\pi_{T}(w, q)) $. We then extend the domain of $\\pi_{T}$ and $\\lambda_{T}$ to $X_{I}^{\\omega} \\times Q_{T}$.\n\nIn this paper we shall insist that for $\\delta \\in X_{I}^{\\omega}$ and any state $q \\in Q_T$ we have $\\lambda_{T}(\\delta, q) \\in X_{O}^{\\omega}$. This means that for a state $q \\in Q_T$ the initial transducer $T_{q}$ induces a continuous function $h_{T_{q}}$ ($h_{q}$ if it is clear from the context that $q$ is a state of $T$) from $X_I^{\\omega}$ to $X_O^{\\omega}$. For $q \\in Q_{T}$ we denote by $\\im(q)$ the image of the map $h_{q}$; if $h_{q}$ is a homeomorphism form $X_{I}^{\\omega} \\to X_{O}^{\\omega}$, then we call $q$ a \\emph{homeomorphism state}.\n\nTwo states $q_1$ and $q_2$ of $T$ are called $\\omega$-equivalent if $h_{q_1} = h_{q_2}$. A state $q$ of $T$ is called a state of incomplete response if for some $i \\in X_{I}$ $\\lambda_{T}(i, q)$ is not equal to the greatest common prefix of the set $\\{(i \\delta)h_{q} \\mid \\delta \\in X_{I}^{\\omega}\\}$. If $T$ is an initial transducer with initial state $q_0$, then $q$ is called \\emph{accessible} if there is a word $w \\in X_I^*$ such that $\\pi_{T}(w, q_0) = q$. If all the states of $T_{q_0}$ are accessible then $T_{q_0}$ is called \\emph{accessible}.\n\nAn initial transducer $T_{q_0}$ is called \\emph{minimal} if $T_{q_0}$ is accessible, has no states of incomplete response and no pair of $\\omega$-equivalent states. The initial transducer $T_{q_0}$ is also called invertible if the state $q_0$ is a homeomorphism state. Given an initial transducer $T_{q_0}$ there is a unique minimal transducer $S_{p_0}$ $\\omega$-equivalent to $T_{q_0}$ (\\cite{GriNekSus}).\n\nWe give below the method given in \\cite{GriNekSus} for constructing the inverse of an invertible, minimal transducer $T_{q_0}$. We first define the following function.\n\n\\begin{Definition}\nLet $T_{q_0} = \\gen{X_{I}, X_{O}, Q_{T}, \\pi_{T}, \\lambda_{T}}$ be an invertible minimal transducer and $q$ a state of $T_{q_0}$. Define a function $L_{q}: X_{O}^{+} \\to X_{I}^{\\ast}$ by $(\\nu)L_{q} = \\varphi$ where $\\varphi$ is the greatest common prefix of the set $(U_{\\nu})h_{q}^{-1}$.\n\\end{Definition}\n\nObserve that since $T_{q_0}$ is minimal and invertible, then each state $q$ of $T_{q_0}$ induces an injective function from $X_{I}^{\\omega}$ to $X_{O}^{\\omega}$ with clopen image. From this one can deduce that for each state $q$ of $T_{q_0}$ the set of words $w \\in X_{O}^{\\ast}$ such that $(w)L_{q} = \\epsilon$ and $U_{w} \\subset \\im(q)$ is finite (see \\cite{GriNekSus}).\n\nWe now form a transducer $T_{(\\epsilon, q_0)} = \\gen{X_{O}, X_{I}, Q'_{T}, \\pi'_{T}, \\lambda'_{T}}$ where $Q'_{T} = \\{ (w,q) \\mid q \\in Q_{T}, (w)L_{q} = \\epsilon, U_{w} \\subset \\im(q) \\}$ and $\\pi'_{T}$ and $\\lambda'_{T}$ are defined, for all $i \\in X_{O}$ and $(w,q) \\in Q'_{T}$, by the rules:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\pi'_{T}(i, (w,q)) = (wi - \\lambda_{T}((wi)L_{q},q), \\pi_{T}((wi)L_{q}, q)$ and, \n\\item $\\lambda'_{T}(i, (w,q)) = (wi)L_{q}$.\n\\end{enumerate} \n\nThe following proposition is a result in \\cite{GriNekSus}:\n\n\\begin{proposition}\\cite{GriNekSus}\nFor $T_{q_0}$ a minimal invertible transducer, the transducer $T_{(\\epsilon, q_0)}$ is well-defined, has no states of incomplete response, and satisfies $h_{(\\epsilon, q_0)} = h_{q_0}^{-1}$.\n\\end{proposition}\n\nWe observe that given a minimal invertible transducer $T_{q_0}$, the transducer $T_{(\\epsilon,q_0)}$ is accessible if $T_{q_0}$. This is because if for any word $\\Gamma \\in \\Xnp$, $(\\Gamma)L_{q_0} = \\epsilon$, then $\\pi'_{T}(\\Gamma, (\\epsilon, q_0)) = (\\Gamma, q_0)$. Furthermore, for any word $w \\in \\Xnp$ and state $q \\in Q_{T}$, such that $U_{w} \\subset \\im(q)$ and $(w)L_{q} = \\epsilon$, picking a word $\\Gamma \\in \\Xnp$ such that $\\pi_{T}(\\Gamma,q_0) = q$, we observe that $(\\lambda_{T}(\\Gamma, q_0)w)L_{q_0} = \\Gamma$ and so $\\pi'_{A}(\\lambda_{T}(\\Gamma, q_0)w, (\\epsilon,q_0)) = (w,q)$. Note that given a minimal invertible transducer $T_{q_0}$, the transducer $T_{(\\epsilon, q_0)}$, even though it has no states of incomplete response and is accessible, might not be minimal. \n\nGiven two transducers $A = \\gen{X, Q_{A}, \\pi_{A}, \\lambda_{A}}$ and $B = \\gen{X, Q_B, \\pi_B, \\lambda_B}$ then the product $A*B = \\gen{X, Q_{A*B}, \\pi_{A*B}, \\lambda_{A*B}}$ is the transducer defined as follows. The set of states $Q_{A*B}$ of $A*B$ is equal to the cartesian product $Q_{A} \\times Q_{B}$. The transition and output function of $A*B$ are given by the following rules: for states $q \\in Q_A$ , $p \\in Q_B$ and $i \\in X$ we have $\\pi_{A*B}(i, (q,p)) = (\\pi_{A}(i,q),\\pi_B(\\lambda_{A}(i,q), p) )$ and $\\lambda_{A*B}(i, (q,p)) = \\lambda_{B}(\\lambda_{A}(i, q), p)$. For two initial transducers $A_{q_0}$ and $B_{p_0}$, where $A$ and $B$ are the resulting transducers with no initialised states, the product of the initial transducer $A_{q_0} * B_{q_0}$, is the initial transducer $(A*B)_{(q_0, p_0)}$. It is straightforward to see that $h_{(A*B)_{(q_0, p_0)}} = h_{A_{q_0}}\\circ h_{B_{p_0}}$.\n\nA transducer (initial or non-initial) $T = \\gen{X_I, X_O, Q_T, \\pi_T, \\lambda_T}$ is said to be \\emph{synchronizing at level $k$} if there is a natural number $k \\in \\N$ and a map $\\mathfrak{s}: X_I^{k} \\to Q_T$ such that for a word $\\Gamma \\in X_I^{{k}}$ and for any state $q \\in Q_T$ we have $\\pi_{T}(\\Gamma, q) = (\\Gamma)\\mathfrak{s}$. We will denote by $\\core(T)$ the sub-transducer of $T$ induced by the states in the image of $\\mathfrak{s}$. We call this sub-transducer the \\emph{core of $T$}. If $T$ is equal to its core then we say that $T$ \\emph{core}. Viewed as a graph $\\core(T)$ is a strongly connected transducer. If $T$ is an initial transducer $T_{q_0}$ which is invertible, then we say that $T_{q_0}$ is bi-synchronizing if both $T_{q_0}$ and its inverse are synchronizing. Note that when $T$ is synchronous, then we shall say $T$ is bi-synchronizing if $T$ and its inverse are synchronizing.\n\nGiven a transducer $T$ and states $q_1, q_2$ of $T$ we shall sometimes use the phrase $q_1$ and $q_2$ \\emph{transition identically on a subset $W \\subset \\Xns$} to mean that the functions $\\pi_{T}(\\centerdot, q_1): W \\to Q_{T}$ and $\\pi_{T}(\\centerdot, q_2): W \\to Q_{T}$ are identical. We might also say that $q_1$ and $q_2$ \\emph{read all elements of $W$ to the same location}.\n\nNow we introduce transducers over $\\CCnr$. An initial transducer for $\\CCnr$ is a tuple $A_{q_0}=(\\dotr, X_n, R, S, \\pi, \\lambda, q_0)$ such that:\n\n\\begin{enumerate}[label = (\\roman*)]\n\\item $R$ is a finite set, and the set $Q$ of states of $A$ is the disjoint union $R \\sqcup S$. The state $q_0 \\in R$ is the initial state\n\\item $\\pi: \\dotr \\times \\{q_0\\} \\sqcup X_n \\times Q \\to Q\\backslash \\{q_0\\}$ and $\\lambda: \\dotr \\times q_0 \\sqcup X_n \\times Q \\to X_{n,r}^{\\ast} \\sqcup X_n^{\\ast}$\n\\end{enumerate}\n\nNotice that a letter from $\\dotr$ can only be read from the initial state $q_0$ and we can never return to $q_0$ after leaving $q_0$. We Inductively extend the domain of $\\pi$ and $\\lambda$ to $ \\dotr \\times \\{q_0\\} \\sqcup X_n^{*} \\times Q$ by the following rules:\n\n$\\pi(wx, q) = \\pi(x, \\pi(w,q))$ and $\\lambda(wx, q) = \\lambda(w, q)\\lambda(x, \\pi(w,q))$. Where $w \\in X_{n,r}^{+} \\sqcup X_n^{+}$ and $x \\in X_n$ (if $w \\in X_{n,r}^{+}$ then $q = q_0$). We also take the convention that $\\pi(\\epsilon, q) = q$ and $\\lambda(\\epsilon, q) = \\epsilon$ for any state $q$ of $A$. By transfinite induction we may further extend the domains of $\\pi$ and $\\lambda$ to $ \\dotr \\times \\{q_0\\} \\sqcup X_n^{\\omega} \\times Q$.\n\nWe impose the following rules on $\\pi$ and $\\lambda$:\n\n\\begin{enumerate}[label = (\\arabic*)] \n\\item For a state $r \\in R $ and for $i$ in $\\dotr \\sqcup X_n$ such that $\\pi(i, r)$ is defined, if $\\pi(i, r) \\in R$ then $\\lambda(i, r) = \\epsilon$, otherwise $\\lambda(i, r) \\in X_{n,r}^{*}$. \\label{List: conditions of pi and lambda 1}\n\\item For $x \\in X_n$ and $q \\in S$, $\\lambda(x, q) \\in X_{n}^{*}$ and $\\pi(x, q) \\in S$. \\label{List: conditions of pi and lambda 2}\n\\item For a state $s \\in S$ and $\\delta \\in \\CCn$ we have that $\\lambda(\\delta, s) \\in \\CCn$. \\label{List: conditions of pi and lambda 3}\n\\item If there is a word $w \\in X_n^{+}$ and a state $q \\in Q$ such that $\\pi(w, q) = q$ then $q \\in S$. \\label{List: conditions of pi and lambda 4}\n\\end{enumerate}\n\nThese rules serve the purpose of ensuring that whenever an element of $\\CCnr$ is is processed through $A_{q_0}$ the output is also in $\\CCnr$.\n\nLet $A_{q_0}$ be an initial transducer on $\\CCnr$ as above and let $q$ be a state of $A_{q_0}$. Let $A_q$ denote the initial transducer $A_{q_0}$ where we process inputs from the state $q$. Observe that $A_{q_0}$ induces a continuous function $h_{A_{q_0}}$ (or $h_{q_0}$ if it clear that $q_0$ is the initial state of $A$) from $\\CCnr$ to itself. Furthermore every non-initial state $q$ of $A_{q_0}$ which is also an element of $R$ induces a continuous function $h_{q}$ from $\\CCn$ to $\\CCnr$, otherwise it induces a continuous function from $\\CCn$ to itself. Once again we denote by $\\im(q)$ the image of $q$ and call $q$ a homeomorphism state if $h_{q}$ is a homeomorphism from its domain to its range.\n\nWe extend, in the natural way, the definition of accessibility, accessible transducers,and states of incomplete response given in the general setting to the specific setting of transducers over $\\CCnr$. We also extend the function $L_{q}$ for a minimal invertible transducer $T_{q_0}$ over $\\CCnr$ and $q \\in Q_{T}$. Having done this, we may thus define, given $T_{q_0}$ a minimal, invertible transducer, the transducer $T_{(\\epsilon, q_0)}$ such that $h_{(\\epsilon, q_0)} = h_{q_0}^{-1}$.\n\nWe say that a transducer $A_{q_0}$ over $\\CCnr$ is synchronizing if there is a $k \\in \\mathbb{N}$ such that given any word $\\Gamma$ of length $k$ in $\\Xnrs \\sqcup \\Xns$ the active state of $A_{q_0}$ when $\\Gamma$ is processed from any \\emph{appropriate} state of $A_{q_0}$ is completely determined by $\\Gamma$. Thus we may also extend the notions of `core' for synchronizing transducers over $\\CCn$. We now introduce the notion of $\\omega$-equivalence and minimality for transducers over $\\CCnr$. Two initial automata with the same domain and range are said to be \\emph{$\\omega$-equivalent} if they induce the same continuous function from their domain to their range. \n\nAn initial transducer $A_{q_0}$ is called \\emph{minimal} if $A_{q_0}$ is accessible, no states of $A$ are states of incomplete response and for any distinct pair $q_1, q_2$ of states of $A_{q_0}$, $A_{q_1}$ and $A_{q_2}$ are not $\\omega$-equivalent. In \\cite{BCMNO} the authors show, by slight modifications of arguments in \\cite{GriNekSus}, that for an initial transducer $A_{q_0}$ on $\\CCnr$ there is a unique minimal transducer under $\\omega$-equivalence.\n\n\nThe product $(A*B)_{(q_0,p_0)} = \\gen{\\dotr, \\Xn, R_{A} \\times R_{B} \\sqcup S_{A} \\times R_{B}, S_A \\times S_B, \\pi_{A*B}, \\lambda_{A*B}}$ of the initial transducers $A_{q_0}$ and $B_{p_0}$ over $\\CCnr$ is defined as follows. The set of states $Q_{A*B} = R_{A} \\times R_{B} \\sqcup S_{A} \\times R_{B} \\sqcup S_A \\times S_B$, and the state $(q_0, p_0) \\in R_{A} \\times R_{B}$ is the initial state. The transition and output functions are defined as follows. First for $a \\in \\dotr$ we have $\\pi_{A*B}(a, (q_0, p_0)) = (\\pi_{A}(a, q_0), \\pi_{B}(\\lambda_{A}(a, q_0), p_0 )$, and $\\lambda_{A*B}(a, (p_0, q_0)) = \\lambda_{B}(\\lambda_{A}(a, q_0), p_0)$. Now for any pair $(q, p) \\in Q_{A,B}$ and for any $i \\in X_n$ we have $\\pi_{A*B}(i, (q,p)) = (\\pi_{A}(i, q), \\pi_B(\\lambda_{A}(1, q),p)$ and $\\lambda_{A*B}(i, (q,p)) = \\lambda_{B}(\\lambda_{A}(i,q),p)$. Now observe that as $A$ and $B$ satisfy condition \\ref{List: conditions of pi and lambda 1} to \\ref{List: conditions of pi and lambda 4} above then so does the product $(A*B)_{(q_0, p_0)}$. Furthermore, as before, it is a straightforward observation that $h_{(A*B)_{(q_0,p_0)}} = h_{A_{q_0}}\\circ h_{B_{p_0}}$.\n\nBelow we outline a procedure given in \\cite{BCMNO} for constructing from a homeomorphism $h$ of $\\CCnr$, an initial transducer $A_{q_0}$ of $\\CCnr$ such that $h_{q_0} = h$. \n\n\nWe first need to define local actions.\n\nFor an arbitrary homeomorphism $g: \\CCnr \\to \\CCnr$, fix the notation $P_{g} \\subset X_{n,r}^{*}$ for the unique maximal subset of $X_{n,r}^{*}$ set satisfying: $(U_{\\nu})g \\subseteq U_{\\dot{a}}$ for $\\nu \\in P_{h}(a)$, and for any proper prefix $\\mu$ of $\\nu$ there are elements $\\delta_1, \\delta_2 \\in \\CCn$ and $\\dot{a}_1, \\dot{a}_{2} \\in \\dotr$ such that $(\\mu\\delta_1)g \\in U_{\\dot{a}_{1}}$ and $(\\mu\\delta_2)g \\in U_{\\dot{a}_{2}}$. Observe that since $g$ is a homeomorphism and since $\\sqcup_{\\dot{a} \\in \\dotr} (U_{\\dot{a}})g^{-1}$ is clopen, $P_{g}$ exists. Moreover maximality of $P_{g}$ implies that $P_{g}$ is a complete antichain for $\\Xnrs$.\n\n\nLet $h : \\CCnr \\to \\CCnr$ be a homeomorphism. Define $\\theta_{h}: X_{n,r}^{*} \\to X_{n,r}^{*}$ as follows: for $\\mu \\in X_{n,r}^{*}$ if $\\mu$ is a prefix of some $\\nu \\in P_{h}$ set $(\\mu)\\theta_{h}:= \\epsilon$, otherwise $\\mu = \\nu \\chi$ for some $\\nu \\in P_{h}$ and $\\chi \\in X_{n,r}^{*}$, in this case set $(\\mu)\\theta_{h}$ to be the greatest common prefix of the set $(U_{\\mu})h$, since $h$ is a homeomorphism and by choice of $P_{h}$, $(U_{\\mu})h \\in X_{n,r}^{+}$.\n\nNow for $\\mu \\in X_{n,r}^{*}$ we define a map $h_{\\mu}$ on $\\CCn$ by $(\\delta)h_{\\mu} = (\\mu\\delta)h_{\\mu} - (\\mu)\\theta_{h}$. We call $h_{\\mu}$ the \\emph{local action of $h$ at $\\mu$}. Observe that if $\\mu$ is a prefix of an element of $P_{h}$ then the range of $h_{\\mu}$ is $\\CCnr$ otherwise the range of $\\mu$ is $\\CCn$, in either case $h_{\\mu}$ is continuous. The following fact is straightforward, let $\\mu, \\nu \\in X_{n,r}^{*}$ then $(\\mu \\nu)\\theta_{h} = (\\mu)\\theta_{h}(\\nu)\\theta_{h_\\mu}$. \n\n\nUsing the function $\\theta_{h}$ we now construct an initial transducer $A_{q_0}$ of $\\CCnr$ such that $h_{q_0} = h$.\n\nLet $h: \\CCnr \\to \\CCnr$ be a homeomorphism. Form a transducer \n$A_{\\epsilon} = (\\dot{r}, X_n, R_{A}, S_{A}, \\pi_A, \\lambda_{A}, \\epsilon \n)$. The set $Q_{A}$ of states of $A$ is precisely \n$X_{n,r}^{*}$ and $R_A \\subset Q_{A}$ is the set of proper \nprefixes of elements of $P_{g}$ and $S_A := \\Xns \\backslash R$. The \ntransition of function and output functions $\\pi_A$ and \n$\\lambda_{A}$ are defined as follows. For $\\dot{a} \\in \\dotr$ we \nhave that $\\pi_{A}(\\dot{a}, \\epsilon) = \\dot{a}$ and $\\lambda_A \n(\\dot{a}, \\epsilon) = (\\dot{a})\\theta_{h}$; for $\\nu \\in \\Xnrp$ \nand $i \\in X_n$ we have $\\pi_{A}(i, \\nu) =\\nu i$ and \n$\\lambda_{A}(i, \\nu) = (\\nu i) \\theta_{h} - (\\nu)\\theta_{h}$.\n\nObserve that the transducer $A_{\\epsilon}$ satisfies the conditions \\ref{List: conditions of pi and lambda 1} to \\ref{List: conditions of pi and lambda 4}. The following claim is straightforward to prove:\n\n\\begin{claim}\\label{Claim:tranducerarisingfromhomeo}\nLet $h: \\CCnr \\to \\CCnr$ be a homeomorphism, and $A_{\\epsilon} = (\\dot{r}, X_n, R_{A}, S_{A}, \\pi_A, \\lambda_{A}, \\epsilon \n)$ be the transducer constructed form $h$ as above. Let $\\nu \\in \\Xnrs$ then for all $w \\in \\Xns$ if $\\nu \\ne \\epsilon$ or $w \\in \\Xnrp$ if $\\nu = \\epsilon$ we have that $(w)\\theta_{h_{\\nu}} = \\lambda_{A}(w, \\nu)$. \n\\end{claim} \n\\begin{proof}\nFirst suppose $\\nu \\ne \\epsilon$ and let $x \\in X_n$. Then by \ndefinition we have $\\lambda_{A}(i,\\nu) = (\\nu i) \\theta_{h} - \n(\\nu)\\theta_{h}$, however by an observation above we have $(\\nu \ni) \\theta_{h} - (\\nu)\\theta_{h}) = (i)\\theta_{h_{\\nu}}$. Now \nassume by that for all $w \\in \\Xnp$ we have that $\\lambda(w, \\nu \n) = (w)\\theta_{h_{\\nu}}$. Let $i \\in \\Xn$ and consider \n$\\lambda_{A}(w i, \\nu)$. We may write $\\lambda_{A}(w i, \\nu) = \n\\lambda_{A}(w, \\nu) \\lambda_{A}(i, \\nu w)$ (since $\\pi_A(w, \\nu) \n= \\nu w$). Therefore $\\lambda_{A}(w i, \\nu) = (w)\\theta_{h_{\\nu}} \n(\\nu w i)\\theta_{h} - (\\nu w)\\theta_{h}$. Observe that $(\\nu w \ni)\\theta_{h} - (\\nu w)\\theta_{h} = (i)\\theta_{h_{\\nu w}}$, \ntherefore $\\lambda_{A}(w i, \\nu) = (w)\\theta_{h_{\\nu}} \n(i)\\theta_{h_{\\nu w}}$ however, $(w)\\theta_{h_{\\nu}} \n(i)\\theta_{h_{\\nu w}} = (wi)\\theta_{h_{\\nu}}$.\n\nNow suppose that $\\nu = \\epsilon$. Let $w \\in \\Xnrp$ and suppose $w = \\dot{a}v$ for some $v \\in \\Xns$ and $\\dot{a} \\in \\dotr$. Consider $\\lambda_{A}(w, \\epsilon)$, this can be broken up into $\\lambda_{A}(\\dot{a}w, \\epsilon) = \\lambda_{A}(\\dot{a}, \\epsilon)\\lambda_{A}(v, \\dot{a})$. Now $\\lambda_{A}(\\dot{a}, \\epsilon) = (\\dot{a})\\theta_{h}$ by definition, and $\\lambda_{A}(v, \\dot{a}) = (v)\\theta_{h_{\\dot{a}}}$ by the previous paragraph. Observe that $(\\dot{a}v)\\theta_{h} = (\\dot{a})\\theta_{h} (v)\\theta_{h_{\\dot{a}}}$ therefore $\\lambda_{A}(\\dot{a}w, \\epsilon) = (\\dot{a}v)\\theta_{h}$ as required.\n\\end{proof}\n\n\n\\begin{Remark}\\label{Remark: finitely many local actions implies finite transducer}\nFrom the claim we deduce that for a homeomorphism $h: \\CCnr \\to \\CCnr$ and for $A_{\\epsilon} = (\\dotr, \\Xn, R_A, S_A, \\pi_A, \\lambda_A, \\epsilon)$ the transducer constructed from $h$, we have that $h_{A_{\\epsilon}} = h$, moreover for any $\\nu \\in \\Xnrs$ and any local action $h_{\\nu}$ of $h$ we have that $h_{\\nu} = h_{A_{\\nu}}$. Therefore if $h$ has finitely many local actions, then it follows that the minimal transducer on $\\CCnr$, under $\\omega$-equivalence, representing $h$ has finitely many states. Moreover, since by Claim~\\ref{Claim:tranducerarisingfromhomeo}, $A_{\\epsilon}$ has no states of incomplete response, it follows that, if $B_{q_0}$ is the minimal transducer representing $A_{\\epsilon}$, then for all $\\nu \\in \\Xnr$ $h_{\\nu} = h_{q}$ for some state $q$ of $B$.\n\\end{Remark}\n\nIn what follows we show that all homeomorphisms of $\\CCnr$ in the group $N(\\Tnr)$ can be represented by a minimal finite transducer. The previous paragraph demonstrates that it suffices to show that such homeomorphisms have finitely many local actions.\n\nOur approach shall essentially mirror that taken in \\cite{BCMNO}, the arguments differ only in so far as we need to make modifications to allow for the fact that the action of $\\Tnr$ on $\\CCnr$ is not as transitive as the action $\\Gnr$ on $\\CCnr$.\n\nFirst we need the following result:\n\n\\begin{lemma}\\label{Lemma: normalisers in H-tilde}\nLet $X$ be a topological space and let $\\sim$ be any equivalence relation on $X$. Let $H_{\\sim}$ be the subgroup of $H(X)$ consisting of those elements of $H(X)$ which respect $\\sim$. Let $G \\le H(X)$ be a subgroup which fixes each equivalence class of $\\sim$ and acts transitively on each class. Then $N_{H(X)}(G) \\le H_{\\sim}$.\n\\end{lemma}\n\nObserve that $T_{n,r} \\le H(\\CCnr)$, by Lemma ~\\ref{Lemma: Tnr preserves tail classes and acts transitively on each tail class}, preserves $\\sim_{t}$ and acts transitively on each equivalence class of $\\sim_{t}$. Therefore, as corollary of the lemma above, we have $N(\\Tnr) \\le H_{\\sim_{t}}$. \n\nThe following definitions and lemmas appear in \\cite{BCMNO} and introduce crucial notions and ideas in understanding local actions of homeomorphisms of $\\CCnr$.\n\n\\begin{Definition}\nLet $V \\subseteq \\CCnr$ be a clopen set. Let $B \\subset \\Xnrs$ be the minimal antichain such that $U_{B}:=\\{U_{\\nu} \\mid \\nu \\in B \\}$ and for all $\\mu$ a prefix of some element of $B$ we have $U_{\\mu} \\not \\subset V$. We call $U_{B}$ the \\emph{decomposition of $V$} and denote it by $\\dec(V)$. \n\\end{Definition}\n\n\n\\begin{Definition}\\label{Def: acting in the same fashion}\nLet $h \\in H(\\CCnr)$, and $U_{\\nu}$ and $U_{\\eta}$ be elements of $\\Banr$ for $\\nu, \\eta \\in \\Xnrs$. Then we say that $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in \\emph{the same fashion} if $h_{\\nu} = h_{\\eta}$. If $V,W \\subset \\CCnr$ are a clopen subsets then we say that $h$ acts on $V$ and $W$ \\emph{in the same fashion} if for any $U_{\\nu} \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$, $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion.\n\\end{Definition}\n\n\\begin{Definition}\\label{Def: almost in the same fashion}\nLet $h \\in H(\\CCnr)$ and let $V, W \\subset \\CCnr$ be clopen subsets. Then we say that $h$ acts on $V$ and $W$ \\emph{almost in the same fashion} if there is some $k \\in \\mathbb{N}$ such that for $U_{\\nu} \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$, and for any $\\chi \\in \\Xn^{k}$ $h$ acts on $U_{\\nu \\chi}$ and $U_{\\eta\\chi}$ in the same fashion. We call the minimal $k$ satisfying this condition the \\emph{critical level of $V$ and $W$} and denote it by $\\crit_{h}(V, W)$.\n\\end{Definition}\n\n\\begin{Definition}\\label{Def: (almost) in the same fashion uniformly}\nLet $h \\in H(\\CCnr)$ and let $V, W \\subset \\CCnr$ be clopen subsets. We say that $h$ acts on $V$ and $W$ \\emph{in the same fashion uniformly} if for any $\\chi \\in \\Xns$ $h$ and $U_{\\nu} \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$, $h$ acts on $U_{\\nu}$ and $U_{\\eta \\chi}$ in the same fashion. We say that $h$ acts \\emph{almost in the same fashion uniformly} on $U$ and $W$ if there is some $k \\in \\N$ such that for $U_{\\nu} \\in \\dec(V)$ and $U_{\\eta} \\in \\dec(W)$ and for any $\\chi, \\xi \\in \\Xn^{k}$, $h$ acts on $U_{\\nu\\chi}$ and $U_{\\eta \\xi}$ in the same fashion.\n\\end{Definition}\n\n\\begin{Remark}\\label{Remark: Tnr acts almost in the same fashion uniformly}\nLet $V$ and $W$ be clopen subsets of cantor space and $g \\in \\Tnr$ then $g$ acts on $V$ and $W$ almost in the same fashion uniformly.\n\\end{Remark}\n\nThe following lemmas are crucial in our understanding of local actions of elements of $N(\\Tnr)$ and are taken from \\cite{BCMNO}:\n\n\\begin{lemma}\\label{Lemma: g acts in same fashion h acts in same fashion gh acts in same fashion}\n Let $g,h \\in H(\\CCnr)$ and let $U_{\\nu}, U_{\\eta} \\in \\Banr$. Suppose there are $\\nu', \\eta' \\in \\Xnrp$ such that $(U_{\\nu})g = U_{\\nu'}$ and $(U_{\\eta})g = U_{\\eta'}$, $g$ acts in the same fashion on $U_{\\nu}$ and $U_{\\eta}$ and $h$ acts in the same fashion on $U_{\\nu'}$ and $U_{\\eta'}$, then $gh$ acts in the same fashion on $U_{\\nu}$ and $U_{\\eta}$. \n\\end{lemma}\n\n\\begin{lemma}\\label{Lemma: g acts in same fashion, h acts in almost same fashion uniformly, then gh acts in almost the same fashion}\nLet $g,h \\in H(\\CCnr)$ and let $U_{\\nu}, U_{\\eta} \\in \\Banr$. Suppose $g$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion and $h$ acts on $(U_{\\nu})g$ and $(U_{\\eta})g$ in almost the same fashion uniformly, then $gh$ acts in almost the same fashion on $U_{\\nu}$ and $U_{\\eta}$. \n\\end{lemma}\n\n \nWe shall also need the following proposition from the same source:\n\n\\begin{proposition}\\label{Prop: conjugators have same local action at pair of nodes}\nLet $h \\in H(\\CCnr)$ and let $U_{\\tau}$ and $U_{\\eta}$ be elements of $\\Banr$. Suppose that $h_{\\tau \\chi} \\ne h_{\\eta \\chi}$ holds for every $\\chi \\in X_n^{\\ast}$. Then $h \\notin H_{n,r, \\sim_{t}}$.\n\\end{proposition}\n\nWe have the following corollary:\n\n\\begin{corollary}\\label{Cor:conjugators have same local action at pair of nodes with any node distance}\nLet $h \\in H_{n,r \\sim_{t}}$ and let $i \\in \\{0, 1, \\ldots, n-2\\}$. Then there exists $\\tau, \\eta \\in X_{n,r}$ such that $U_{\\tau} \\cup U_{\\eta} \\ne \\CCnr$, $(\\tau, \\eta) \\in \\mathscr{G}$ the reduced node distance between $\\tau$ and $\\eta$ is $i$ modulo $n-1$, and $h_{\\tau} = h_{\\eta}$.\n\\end{corollary}\n\\begin{proof}\nThis is a direct consequence of Proposition~\\ref{Prop: conjugators have same local action at pair of nodes}, since we may chose a pair $(\\tau, \\eta) \\in \\mathscr{G}$ such that there is a complete antichain $\\ac{u}$ containing $\\tau$ and $\\eta$ and the node distance in $\\ac{u}$ between $\\tau$ and $\\eta$ is congruent to $i$ modulo $n-1$. Moreover, by observations above, for any $\\chi \\in X_n^{*}$, $(\\tau \\chi, \\eta \\chi) \\in \\mathscr{G}$ and any complete antichain $\\ac{v}$ containing $\\tau\\chi$ and $\\eta\\chi$, the node distance between $\\tau\\chi$ and $\\eta \\chi$ in $\\ac{v}$ is congruent to $i$ modulo $n-1$. \n\\end{proof}\n\n\nWe have the following lemma:\n\n\n\\begin{lemma} \\label{Lemma: automorphisms have few local actions}\nLet $h \\in H(\\CCnr)$ such that $h^{-1}T_{n,r}h \\subseteq T_{n,r}$. Then for every $U_{\\nu}, U_{\\eta} \\in \\Banr$ such that $U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$, the map $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ almost in the same fashion.\n\\end{lemma}\n\\begin{proof}\n\nFirst observe that, by Lemma~\\ref{Lemma: normalisers in H-tilde}, \n$h \\in H_{\\sim_{t}}$. Moreover by Corollary~\\ref{Cor:conjugators \nhave same local action at pair of nodes with any node distance} \nfor any $i \\in \\{0,1,\\ldots, n-2\\}$ there are elements $\\nu$ and \n$\\eta$ of $X_{n,r}^{+}$ such that $(\\nu, \\eta) \\in \\mathscr{G}$, \nthe reduced node distance between $\\nu$ and $\\eta$ is $i$, \n$U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$, and $h_{\\nu} = h_{\\eta}$. Fix \n$i \\in \\{0,1, \\ldots, n-2\\}$ and let $(\\nu, \\eta) \\in \\mathscr{G}$ \nbe such that the reduced node distance between $\\nu$ and $\\eta$ \nis $i$ and $h_{\\nu} = h_{\\eta}$.\n\nLet $\\nu'$ and $\\eta'$ be elements of $X_{n,r}^{+}$ be such that $(\\nu', \\eta') \\in \\mathscr{G}$ and the reduced node distance between $\\nu'$ and $\\eta'$ is also equal to $i$. By Lemma ~\\ref{Lemma: can swap things with same node distance} there is an element $g \\in \\Tnr$ such that $g\\restriction_{U_{\\nu'}} = g_{\\nu', \\nu}$ and $g\\restriction_{U_{\\eta'}} = g_{\\eta', \\eta}$.\n\nBy Lemma~\\ref{Lemma: g acts in same fashion h acts in same fashion gh acts in same fashion}, since $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion, and $g$ acts on $U_{\\nu'}$ and $U_{\\eta'}$ in the same fashion, then $gh$ acts on $U_{\\nu}$ and $U_{\\eta}$ in the same fashion. Let $f = h^{-1}gh$. By assumption $f \\in T_{n,r}$ and so $f$ and $f^{-1}$ act on any pair of clopen sets of $\\CCnr$ in almost the same fashion. Therefore, since $h = ghf^{-1}$, by Lemma~\\ref{Lemma: g acts in same fashion, h acts in almost same fashion uniformly, then gh acts in almost the same fashion} it follows that $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ in almost the same fashion.\n\nSince $i$ was arbitrarily chosen, we conclude that for any pair $(\\tau, \\mu) \\in \\mathscr{G}$ with reduced node distance equal to $i$, $h$ acts on $U_{\\tau}$ and $U_{\\eta}$ in almost the same fashion.\n\nNow let $\\nu'$ and $\\eta'$ be arbitrary elements of $X_{n,r}^{*}$ such that $U_{\\nu'} \\cup U_{\\eta'} \\ne \\CCnr$ . Observe that $\\nu', \\eta' \\in X_{n,r}^{+}$ otherwise $U_{\\nu} = \\CCnr$ or $U_{\\eta} = \\CCnr$. Since $U_{\\nu'} \\cup U_{\\eta'} \\ne \\CCnr$ there exists an element $\\beta \\in X_{n,r}^{+}$ such that $\\beta \\perp \\nu'$ and $\\beta \\perp \\eta'$ and such that $(\\nu', \\beta) \\in \\mathscr{G}$ and $(\\eta', \\beta) \\in \\mathscr{G}$. By arguments in the previous paragraph we therefore have that $h$ acts on $U_{\\nu'}$ and $U_{\\beta}$ in almost the same fashion, and $h$ acts on $U_{\\eta'}$ and $U_{\\beta}$ in almost the same fashion. Let $k_1 = \\crit_{h}(U_{\\eta'}, U_{\\beta} )$ and $k_2 = \\crit_{h}(U_{\\nu'}, U_{\\beta} )$. Let $k = \\max\\{k_1, k_2\\}$. Then for any $\\chi \\in X_n^{k}$ we have that $h_{\\nu'\\chi} = h_{\\beta\\chi} = h_{\\eta' \\chi}$. Therefore we conclude that $h$ acts on $U_{\\eta'}$ and $U_{\\nu'}$ in almost the same fashion.\n\\end{proof}\n\nThe Corollary below demonstrates that if $h \\in H(\\CCnr)$ is such that $h^{-1}\\Tnr h \\subseteq \\Tnr$, then $h$ uses only finitely many types of local action. This Corollary should be compared with Corollary 6.16 of \\Cite{BCMNO} and is proved almost identically; we reproduce the proof here for completeness. This is a key result in demonstrating that automorphisms of $\\Tnr$ can be represented by bi-synchronizing transducers.\n\n\\begin{corollary}\\label{Corollary: Automorphisms have finite local actions}\nLet $h \\in H(\\CCnr)$ be such that $h^{-1}\\Tnr h \\subseteq \\Tnr$, then $h$ uses only finitely many types of local action.\n\\end{corollary}\n\\begin{proof}\nLet $A$ be a complete antichain for $\\Xnrs$ having at least 3 elements. For instance we may take $A = X_{n,r}^{3}$. Notice that for any pair of element $\\nu$ and $\\eta$ in $A$ we have by Lemma ~\\ref{Lemma: automorphisms have few local actions} that $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ almost in the same fashion, and for $i \\in X_n$ $h$ acts on $U_{\\nu}$ and $U_{\\nu i}$ almost in the same fashion. Set \n\n$$ k= \\max \\{ \\crit_{h}(U_{\\nu},U_{\\eta}) \\mid \\nu, \\eta \\in A, \\mbox{ or } \\nu \\in A \\mbox{ and } \\eta= \\nu i \\mbox{ for some } i \\in \\Xn \\}$$\n\nWe now demonstrate that for any $ m\\ge k+3$, $m \\in \\N$ and for any word $\\eta \\in X_{n,r}^{m}$ we have that $h_{\\eta} = h_{\\nu \\xi}$ for some $\\nu \\in A$ and $\\xi \\in X_n^{k}$. We proceed by induction on $m$.\n\nThe base case is trivially satisfied. Let $m \\in \\N$ be strictly greater than $k+3$, and assume that for all $k+3< l < m$ we have that for any word $\\eta \\in \\Xnr^{l}$ $H_{\\eta} = h_{\\nu \\xi}$ for some $\\nu \\in A$ and $\\xi \\in \\Xn^{k}$. \n\nNow let $\\tau \\in \\Xnr^{m}$. We may write $\\tau = \\nu i \\xi$ for $\\nu \\in A$, some $i \\in \\Xn$ and $\\xi \\in X_n^{\\ast}$ such that $|\\xi| \\ge k$ (since $m$ is greater than $k+3$).\n\nObserve that as $h_{\\nu} = h_{\\nu i}$ we have that $h_{\\tau} = h_{\\nu \\xi}$. Now as $|\\nu \\xi| < m$, by the inductive assumption we have that there is some $\\chi \\in X_n^{k}$ and $\\mu \\in A$ such that $h_{\\nu \\xi} = h_{\\mu \\chi}$ and the conclusion follows.\n\\end{proof}\n\n\\begin{Remark}\nThe above corollary together with Remark~\\ref{Remark: finitely many local actions implies finite transducer} means that $N(\\Tnr)$ is a subgroup of the group $\\Rnr$ of those homeomorphisms of $\\CCnr$ that can be represented by minimal, finite, initial transducers. In what follows we shall demonstrate the element of $N(\\Tnr)$ can be represented by minimal, bi-synchronizing transducers in $\\Rnr$. \n\\end{Remark}\n\n\\section{The group \\texorpdfstring{$N(\\Tnr)$}{Lg} is a subgroup of \\texorpdfstring{$\\Bnr$}{Lg}} \\label{Section: N(Tnr) is a subgroup of Bnr}\nFollowing \\cite{BCMNO} we shall use the notation $\\Bnr$ for the group consisting of those elements of $\\Rnr$ which can be represented by bi-synchronizing transducers. In this section we demonstrate that $N(\\Tnr)$ is a subgroup $\\T{T}\\Bnr$ of $\\Bnr$.\n\nThe lemma below is straight-forward and connects the property of a homeomorphism acting in the almost the same fashion (Definition~\\ref{Def: almost in the same fashion}), to the synchronizing property of a transducer representing the homeomorphism.\n\n\\begin{lemma}\\label{Lemma:automorphisms have synch transducers}\nLet $h \\in H(\\CCnr)$ satisfy the following two conditions,\n\\begin{enumerate}[label = (\\alph*)]\n\\item $h$ has finitely many local actions,\\label{Lemma:automorphisms have synch transducers assumption 1}\n\\item for every pair $U_{\\nu}, U_{\\eta} \\in \\Banr$ such that $U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$ $h$ acts on $U_{\\nu}$ and $U_{\\eta}$ almost in the same fashion.\\label{Lemma:automorphisms have synch transducers assumption 2}\n\\end{enumerate}\nThen the minimal transducer representing $h$ is synchronizing.\n\\end{lemma}\n\\begin{proof}\nLet $A_{q_0}$ be the minimal transducer representing $h$. As $h$ has finitely many local actions, by Remark~\\ref{Remark: finitely many local actions implies finite transducer}, $A_{q_0}$ is a finite transducer. Applying Remark~\\ref{Remark: finitely many local actions implies finite transducer}, there is an $m \\in \\N$ such that if $q \\in Q_{A}$ is accessible from $q_0$ by a word of length at least $m$, then $h_{q} = h_{\\eta}$ for some $\\eta \\in \\Xnrs$ of length at least $m$. Of course by the definition of $A_{q_0}$, it is the case that $h_{\\nu} = h_{p}$, for $p = \\pi_{A}(\\nu, q_0)$ and $\\nu \\in \\Xnrs$. We may also choose $m$ bigger than $1$ and long enough, so that for any pair $\\nu, \\eta \\in \\Xnrs$ of length $m$, $U_{\\nu} \\cup U_{\\eta} \\ne \\CCnr$. Fix such an $m \\in \\N$.\n\nBy assumption~\\ref{Lemma:automorphisms have synch transducers assumption 2} above and Definition~\\ref{Def: almost in the same fashion}, it follows that for every pair $\\nu, \\eta \\in \\Xnrs$ of length $m$, there is a minimal number $k_{\\nu, \\eta} \\in \\N$ such that for all words $\\xi \\in \\Xn^{k_{\\nu, \\eta}}$, $h_{\\nu \\xi} = h_{\\eta \\xi}$. Set $k:= \\max_{\\nu,\\eta \\in \\Xn^{m}}\\{k_{\\nu, \\eta}\\}$. It therefore follows that for any pair $\\tau, \\mu \\in \\Xnrs$ of length $m$, and any word $\\chi \\in \\Xn^{k}$, $h_{\\tau \\chi} = h_{\\mu \\chi}$.\n\nNow since any state $q$ of $A_{q_0}$ accessible from $q_0$ by a word of length at least $m$ satisfies $h_{q} = h_{\\eta}$ for some $\\eta \\in \\Xnrs$ of length $m$. It is therefore the case, by minimality of $A_{q_0}$, that for any $\\chi \\in \\Xn^{k}$ and any pair $\\nu, \\eta \\in \\Xnrs$ of length $m$, we have $\\pi_{A}(\\nu\\chi, q_0) = \\pi_{A}{\\eta\\chi, q_0}$. Thus we have that if $Q_{A,\\ge m} = \\{\\pi_{A}(\\nu, q_0) \\mid \\nu \\in \\Xnrs, |\\nu| \\ge m \\}$, then for any $\\xi \\in \\Xn^{k}$, $|\\{\\pi_{A}(\\chi,p) \\mid p \\in Q_{A, \\ge m} \\}| = 1$. From this we deduce that $A_{q_0}$ is synchronizing at level at least $k+m$.\n\n\n\\end{proof}\n\n\\begin{comment}\nThe lemma below is proved in \\cite{BCMNO} and gives essentially one direction of the main result. We need to make the following definitions in order to state the lemma.\n\n\\begin{Definition}\nLet $A_{q_0} = \\gen{X_A, Q_A, \\pi_{A}, \\lambda_{A}, q_0}$ be an initial transducer. A state $p$ of $A$ is called \\emph{non-trivially accessible} if there is a word $\\Gamma \\in X_A^{*}$ such that $\\pi_{A}(\\Gamma, q_0) = p$ and $\\lambda(\\Gamma, q_0) \\ne \\epsilon$.\n\\end{Definition}\n\n\\begin{lemma}\\label{Lemma: automorphisms have synch transducers}\nLet $A_{q_0} = \\gen{\\dotr, X_n,Q_A,\\pi_A,\\lambda_A,q_0}$ be a minimal, initial, finite, invertible transducer. Let $q_0^{-1}$ represent the initial state of the minimal transducer $B_{q_0^{-1}}= \\gen{\\dotr, X_n,Q_B,\\pi_B,\\lambda_B,q_0^{-1}}$ which is the inverse transducer of $A_{q_0}$ (so, in particular, $X_A=X_B$). Suppose that for all non-trivially accessible states $p^{-1} \\in Q_B$ and for all $q \\in Q_A$ we have the initial transducer $B_{p^{-1}}A_q$ admits an $\\iota \\in \\N$ so that for each $\\gamma \\in \\Xn^{\\iota}$ there is some $\\delta \\in \\Xn^{*} $ with $(\\gamma \\Gamma)B_{p^{-1}}A_{q}= \\delta \\Gamma$, for all $\\Gamma \\in \\Xn^{\\omega}$ . Then $A_{q_0}$ is synchronizing\n\\end{lemma}\n\\end{comment}\nAs a corollary we have the following:\n\n\\begin{corollary}\\label{Cor: Transducers representing automorphims are synchronizing}\nLet $h \\in H(\\CCnr)$ be such that $h^{-1}T_{n,r}h \\subseteq \\Tnr$ and $A_{q_0} = \\gen{\\dotr, \\Xn, Q_A, \\pi_A, \\lambda_A, q_0}$ be an initial transducer such that $h_{q_0} = h$. Then $A_{q_0}$ is synchronizing.\n\\end{corollary}\n\\begin{proof}\nBy Corollary~\\ref{Corollary: Automorphisms have finite local actions} and Lemma~\\ref{Lemma: automorphisms have few local actions} it follows that $h$ and $h^{-1}$ satisfy the conditions of Lemma~\\ref{Lemma:automorphisms have synch transducers}. By definition of bi-synchronicity, the minimal transducer $A_{q_0}$ representing $h$ is bi-synchronizing.\n\\begin{comment}\nLet $B_{q_0^{-1}}= \\gen{\\dotr,\\Xn,Q_B,\\pi_B,\\lambda_B,q_0^{-1}}$ be the minimal initial transducer representing the inverse of $A_{q_0}$. It suffices to show that $A_{q_0}$ and $B_{q_0^{-1}}$ satisfy the conditions of Lemma~\\ref{Lemma: automorphisms have synch transducers}.\n\nLet $q_b$ be any non-trivially accessible state of $B_{q_0^{-1}}$. Let $\\Gamma \\in \\Xnr^{+}$ be such that $\\pi_{B}(\\Gamma, q_0^{-1}) = q_b$ and $\\lambda_{B}(\\Gamma, q_0^{-1}) \\ne \\epsilon$. Let $\\Delta \\in \\Xnr^{+}$ be such that $\\Delta = \\lambda_{B}(\\Gamma, q_0^{-1})$. Let $q_a$ be any state of $A_{q_0}$ and let $\\chi \\in \\Xnr^{+}$ be such that $\\pi_A(\\chi, q_0) = q_a$. Observe that there is an element $g \\in \\Tnr$ such that $g\\restriction_{U_{\\Delta}} = g_{\\Delta, \\chi}$.\n \nLet $C = \\gen{\\dotr, \\Xn, \\pi_C, \\lambda_C, p_0}$ be a minimal, finite initial transducer such that $h_{p_0} = g$. Observe that $C$ is a synchronizing transducers whose core is the single state identity transducer. Denote this state by $\\id$. Moreover since $C$ has no states of incomplete response (by minimality) it follows that $\\lambda_{C}(\\Delta,p_0) = \\chi$ and $\\pi(\\Delta, p_0) = \\id$.\n\nNow consider the product $D_{(q_0^{-1}, q_0, p_0)} :=B_{q_0^{-1}} \\ast C_{p_0} \\ast A_{q_0}$. Now after reading $\\Gamma$ through this product, the active state of the transducer is $(q_b, \\id, q_a)$. Now observe that as $h_{q_0^{-1}} g h_{q_0} \\in \\Tnr$, there is an $l \\in \\mathbb{N}$ such that for an word $\\gamma \\in \\Xnr^{l}$ and any $\\delta \\in \\CCn$ we have $(\\gamma \\delta)h_{q_0^{-1}}g h_{q_0} = \\rho \\delta$, for some $\\rho \\in \\Xnr^{+}$. In particular for any $\\xi \\in \\Xn^{l}$ and for any $\\delta \\in \\CCn$ we have $(\\chi\\xi\\delta)h_{q_0^{-1}}gh_{q_0} = \\varphi \\delta$ for some $\\varphi \\in \\Xnr^{+}$. Now since the active state of $B_{q_0^{-1}} C_{p_0} A_{q_0}$ after processing $\\chi$ is $(q_b, \\id, q_a)$ we therefore conclude that $\\lambda_{D}(\\xi\\delta, (q_b, \\id, q_a)) = \\varphi'\\delta$ for some (possibly empty) suffix $\\varphi'$ of $\\varphi$. Therefore $(\\xi\\delta) B_{q_b}A_{q_a} = \\varphi' \\delta$. Hence for any $\\xi \\in \\Xnr^{l}$ there is some $\\varphi' \\in \\Xn^{*}\\sqcup \\Xnr^{\\ast}$ such that $(\\xi\\delta) B_{q_b}A_{q_a} = \\varphi' \\delta$ for any $\\delta \\in \\CCn$.\n\nSince $q_b$ was an arbitrary non-trivially accessible state of $B_{q_0^{-1}}$ and since $q_a$ was an arbitrary state of $A_{q_0}$, we see that $B_{q_0^{-1}}$ and $A_{q_0}$ satisfy the hypotheses of Lemma~\\ref{Lemma: automorphisms have synch transducers}.\n\\end{comment}\n\\end{proof}\n\nNow we may prove the main result.\n\n\\begin{Theorem}\nLet $h \\in H(\\CCnr)$ then $h \\in N(\\Tnr)$ if and only if there is a bi-synchronizing transducer $A_{q_0}$ such that $h_{q_0} = h$ and preserves $\\simeq$.\n\\end{Theorem}\n\\begin{proof}\nIt follows by Corollary~\\ref{Cor: Transducers representing automorphims are synchronizing} that if $A_{q_0}$ is a transducer such that $h_{q_0} \\in N(\\Tnr)$ then $A_{q_0}$ is synchronizing. Since $N(\\Tnr)$ is a group it follows that $A_{q_0}$ is bi-synchronizing. This proves the forward implication.\n\nFor the reverse implication, suppose that $A_{q_0} = \\gen{\\dotr, \\Xn, Q_A, \\pi_A, \\lambda_A, q_0}$ is a bi-synchronising transducer such that $h_{q_0}$ preserves $\\simeq$. Let $T \\in T_{n,r}$ observe that since $h_{q_0}$ preserves $\\simeq$ it follows that $h^{-1}_{q_0}T h_{q_0}$ also preserves $\\simeq$. Let $C_{p_0} = \\gen{\\dotr, \\Xn, Q_{C}, \\pi_{C}, \\lambda_{C}, p_0}$ be an initial transducer such that $h_{p_0} = T$. Observe that $C_{p_0}$ is synchronizing and $\\core(C_{p_0})$ is the single state identity transducer, let us denote this state by $\\iota$. Let $B_{q_0^{-1}} = \\gen{\\dotr, \\Xn, Q_B, \\pi_B, \\lambda_B, q_0^{-1}}$ be a minimal initial transducer representing the inverse of $A_{q_0}$. \n\nObserve that since $A_{q_0}$ is bi-synchronizing, that $B_{q_0}^{-1}$ is also bi-synchronizing. Let $k_1 \\in \\N$ be minimal so that $A_{q_0}$ and $B_{q_0}^{-1}$ are synchronizing at level $k_1$. let $k_2 \\in \\N$ be minimal so that for any word $\\Gamma \\in \\Xnr^{k_2}$ we have $\\pi_{C}(\\Gamma, p_0) = \\iota$. Let $k = \\max\\{k_1, k_2\\}$. Let $j \\in \\N$ be the minimal number greater than $k$ such that for any word $\\Delta \\in \\Xnr^{j}$ we have that $|\\lambda_{B}(\\Delta, q_0^{-1})|> 2k$. Therefore after processing any word of length $j$, the active state of the product transducer $B_{q_0}^{-1} \\ast C_{p_0} A_{q_0}$ is of the for $(r, \\iota, q )$ for some state $r$ of $\\core(B_{q_0^{-1}})$ and some state $q$ of $A_{q_0}$. Notice that as $\\iota$ is the single state identity transducer we may identify $(r, \\iota, q)$ with the state $(r, q)$ of $B_{q_0^{-1}} \\ast A_{q_0}$.\n\nNow let $\\Lambda \\in \\Xnr^{*}$ be a word such that $\\pi_{B}(\\Lambda, q_0^{-1}) = r$ and let $\\Delta \\in \\Xn$ be such that $\\lambda_{B}(\\Delta, r)$ is greater than $k$ (recall that states of $\\core(B_{q_0^{-1}}$ only process elements from $\\Xn$). Let $q'$ be the state of $A_{q_0}$ forced by $\\lambda_{B}(\\Delta,r)$ and let $r'$ be the state $\\pi_{B}(\\Delta, r)$. Then observe that after processing the word $\\Lambda\\Delta$ the active state of $B_{q_0^{-1}} \\ast A_{q_0}$ is $(r', q')$. However after processing $\\Delta$ from the state $(r,q)$ the active state is also $(r', q')$. However since $B_{q_0^{-1}}\\ast A_{q_0}$ is $\\omega$-equivalent to the single state identity transducer, we see that $(r',q')$ which is a state of $\\core(B_{q_0^{-1}}\\ast A_{q_0})$ is $\\omega$-equivalent to a map which produces some finite initial prefix and then acts as the identity.\n\nFrom this we may conclude that there some $N \\in \\N$ such that $B_{q_{0}^{-1}} \\ast C_{p_0} \\ast A_{q_0}$ acts as the identity after processing a word of length $N$. This means that since $h_{q_0}^{-1} T h_{q_0}$ preserves $\\simeq$ and is a homeomorphism that $h_{q_0}^{-1} T h_{q_0} \\in \\Tnr$.\n\\end{proof}\n\nThus we have the following:\n\n\\begin{Theorem}\n The group $\\aut{\\Tnr}$ is isomorphic to the subgroup $\\mathcal{T}\\Bnr \\le \\Bnr$ consisting of those elements of $\\Rnr$ which may be represented by finite, initial, bi-synchronizing transducers which preserve $\\simeq$. \n\\end{Theorem}\n\n\\begin{Remark}\\label{Remark:elements of Tnr correspond to bisynch transducers with trivial cores}\nNotice that since all elements of $\\Tnr$ eventually act as the identity, then a minimal transducer $T_{q_0}$ representing an element $t \\in \\Tnr$ satisfies $\\core(T_{q_0})$ is the single states identity transducer over $\\CCn$ which we denote $\\id$. Furthermore any element of $\\TBnr$ which may be represented by a bi-synchronizing transducer with trivial core, is an element of $\\Tnr$. This is because $\\Tnr \\le \\TBnr$ is uniquely defined by the fact that all elements act as the identity after modifying an finite initial prefix.\n\\end{Remark}\n\n\\section{\\texorpdfstring{$\\out{\\Tnr}$}{Lg}}\\label{Section: out(Tnr)}\n\nThe paper \\cite{BCMNO} introduces a group $\\On$ which contains the group $\\Onr\\cong \\out{G_{n,r}}$ for all valid $r$. In this section we shall demonstrate that there is a corresponding subgroup $\\TOn \\le \\On$ containing the groups $\\TOnr:= \\out{\\Tnr}$ for all valid $r$. \n\nLet $\\widetilde{\\Bnr}$ be the set of minimal transducer $A_{q_0}$ such that $h_{q_0} \\in \\Bnr$. The set $\\widetilde{\\Bnr}$ of transducers becomes a group isomorphic to $\\Bnr$ with the product defined by for elements $A_{q_0}, B_{p_0} \\in \\widetilde{\\Bnr}$, $AB_{(q_0,p_0)}$ is the minimal initial transducer representing the product $A_{q_0} \\ast B_{p_0}$. Notice that since $h_{(q_0, p_0)} = h_{q_0} \\circ h_{p_0}$ then we have that $AB_{(q_0,p_0)} \\in \\widetilde{\\Bnr}$. To simplify the discussion below we shall identify $\\Bnr$ with the group $\\widetilde{\\Bnr}$ of transducers. In particular we shall no longer distinguish between an element $h \\in \\Bnr$ and the minimal transducer $A_{q_0} \\in \\widetilde{\\Bnr}$ representing $h$, thus we shall use the symbol $\\Bnr$ for both $\\widetilde{\\Bnr}$ and $\\Bnr$. Likewise we shall longer distinguish between elements of $\\TBnr$ and the minimal initial transducers representing them.\n\n \n\\begin{lemma}\nLet $A_{q_0}, B_{p_0} \\in \\TBnr$ such that $\\core(A_{q_0}) = \\core(B_{p_0})$. Then $A_{q_0}B_{p_0}^{-1} \\in \\Tnr$.\n\\end{lemma}\n\\begin{proof}\nBy Remark~\\ref{Remark:elements of Tnr correspond to bisynch transducers with trivial cores} it suffices to show that $\\core(A_{q_0}B_{p_0}^{-1})$ is equal to the single state identity transducer $\\id$.\nHowever it is a consequence of a result in \\cite{BCMNO}, that $\\core(A_{q_0}B_{p_0}^{-1})$ is trivial.\n\\end{proof}\n\nAs a corollary we have that an element of $\\out{\\Tnr}$ corresponds to a subset of elements of $\\TBnr$ which all have the same core. Therefore we may identify an element $[A_{q_0}] \\in \\out{\\Tnr}$, for $A_{q_0}\\in \\TBnr$, with the element $\\core(A_{q_0})$. \n\nLet $\\Onr:= \\{ \\core(A_{q_0}) \\mid A_{q_0} \\in \\Bnr \\}$. This set inherits a product from $\\Bnr$ as follows. For $g_1, g_2 \\in \\Onr$, let $A_{q_0}, B_{p_0} \\in \\Bnr$ be elements such that $\\core(A_{q_0}) = g_1$ and $\\core(B_{q_0}) = g_2$, and set $g_1g_2 := \\core( AB_{(q_0, p_0)})$. Now observe that $\\core( AB_{(q_0, p_0)})$ depends only on the states of $\\core(g_1 \\ast g_2)$ by a result in \\cite{BCMNO} which states that $\\core (A_{q_0}* B_{q_0}) = \\core(\\core(A_{q_0})* \\core(B_{q_0}))$ and since removing states of incomplete response and identifying $\\omega$-equivalent depends only on $\\core(g_1 \\ast g_2)$ (see \\cite{GriNekSus}). Therefore it follows that for any other element $A'_{q'_0}$ and $B'_{p'_0}$ with cores $g_1$ and $g_2$ respectively we have $\\core(A'B'_{(q'_{0}p'_{0})}) = \\core(AB_{(q_0,p_0)})$. Thus setting the product $g_1g_2 := \\core( AB_{(q_0, p_0)})$ for elements $A_{q_0}, B_{q_0} \\in \\Bnr$ with cores $g_1$ and $g_2$ respectively, results in a well-defined product.\n\nThe authors of \\cite{BCMNO} show that this product is equivalent to the following. Let $g, h \\in \\Onr$, observe that $g$ and $h$ are core non-initial transducers. Fix a state $q_1$ and $q_2$ of $g$ and $h$ respectively. Let $gh := \\core( (gh)_{(q_{1},q_{2})})$ where $(gh)_{(q_1,q_2)}$ is the minimal transducer representing the product ${g}_{q_1} \\ast {h}_{q_2}$. The set $\\Onr$ is isomorphic to $\\out{G_{n,r}}$ under this product.\n\nThe inverse of an element of $\\Onr$ can likewise be defined in two equivalent ways as demonstrated in \\cite{BCMNO}. The first approach is as follows. Given an element $g \\in \\Onr$ let $A_{q_0} \\in \\Bnr$ be such that $\\core(A_{q_0}) = g$. Let $B_{p_0}$ be the minimal transducer representing the inverse of $A_{q_0} $ then set $g^{-1} = \\core(B_{q_0})$. It turns out (see \\cite{BCMNO}) that this inverse is independent of the the choice of transducer $A_{q_0} \\in \\Bnr$ with core equal to $g$ and so is well-defined. The second approach makes use only of the states of $g$ to construct the inverse of $g$, thus removing the need of finding an element of $\\Bnr$ with core equal to $g$.\n\nThus having removed the need to find elements of $\\Bnr$ with cores equal to the relevant transducers in order to compute products and inverses, the paper \\cite{BCMNO} introduces the group $\\On:= \\cup_{1 \\le r < n} \\Onr$ with product of two elements constructed as in the second approach, likewise for inverses. The groups $\\Onr$ are moreover subgroups of $\\On$. Another way of defining the group $\\On$ (see \\cite{BCMNO}) is as the group of all bi-synchronizing transducers (note that as we have a method for computing inverses, our use of the word bi-synchronizing makes sense) all of whose states induce injective maps of $\\CCn$ with clopen images. \n\nLet $\\TOnr := \\{\\core(A_{q_0}) \\mid A_{q_0} \\in \\TBnr \\} \\subset \\Onr$. Then $\\TOnr$ is a subgroup of $ \\Onr$, under the product inherited from $\\On$, moreover $\\TOnr \\cong \\out{\\Tnr}$ under this product.\n\nThe following lemma essentially solves one half of the membership problem of $\\TOnr$ in $\\Onr$.\n\n\\begin{lemma}\\label{Lemma: states of TBnr induces continuous functions from the interval to itsel }\nLet $A_{q_0} \\in \\TBnr$, then for all states $q$ of $\\core(A_{q_0})$ the maps $h_{q} : \\CCn \\to \\CCn$ either all preserve or all reverse the lexicographic ordering of $\\CCn$ and preserve the relation $\\simeq_{{\\bf{I}}}$ on $\\CCn$.\n\\end{lemma}\n\\begin{proof}\nWe consider only elements $A_{q_0} \\in \\TBnr$ such that $h_{q_0}$ induces an orientation preserving homeomorphism of $S_{r}$. The proof is similar in the other case.\n\nObserve that since $A_{q_0} \\in \\TBnr$ this means that $h_{q_0}: \\CCnr \\to \\CCnr$ is a homeomorphism which preserve $\\simeq$. Hence there is a $\\nu \\in \\Xnrs$ such that $h_{q_0}$ preserve the lexicographic ordering on $U_{\\nu}$. By restricting to a small enough open set contained in $\\nu$ we may assume that assume that $\\nu$ is longer than the synchronizing level of $A_{q_0}$. Let $\\gamma \\in \\Xns$ be longer than the synchronizing level of $A_{q_0}$ such that the state of $A_{q_0}$ forced by $\\gamma$ is a state $p \\in \\core(A_{q_0})$. Let $q = \\pi_{A}(\\nu, q_0) \\in \\core(A_{q_0})$. Observe that as $h_{q_0}$ preserves the lexicographic ordering of $U_{\\nu}$, it must be the case that $h_{q}: \\CCn \\to \\CCn$ preserves the lexicographic ordering on $U_{\\nu}$. Thus $h_{p}$ must also preserve the lexicographic ordering on $\\CCn$ otherwise $h_{q}$ does not preserve the lexicographic ordering. Moreover since $h_{q_0}$ preserves $\\simeq$ we must also have that $h_{p}$ preserves $\\simeq_{{\\bf{I}}}$ otherwise $h_{q_0}$ would not preserve $\\simeq$. Now since $A_{q_0}$ is synchronizing for all states $p'$ of $\\core(A_{q_0})$ there is a word $\\gamma'$ in $\\Xns$ longer than the synchronizing length of $A_{q_0}$ such that the state of $A_{q_0}$ forced by $\\gamma'$ is $p'$. This concludes the proof.\n\\end{proof}\n\nThis prompts the following definition:\n\n\\begin{Definition}\\label{Def: noninitial transducers preserving relations}\nLet $A$ be a non-initial transducer of over $\\CCn$, and let $P$ be some relation on $\\CCn$. Then we say that \\emph{$A$ preserves $P$} if for all states $q$ of $A$ the map $h_{q}$ preserves the relation $P$.\n\\end{Definition}\n\n\\begin{lemma}\\label{Lemma: images of states of core of elements of TBnr are connected}\nLet $A_{q_0} \\in \\TBnr$ then for all states $q \\in \\core(A_{q_0})$ there is a finite antichain $\\ac{u} = \\{u_1, u_2, \\ldots, u_{m}\\}$ such that $\\im(q) = \\cup_{1 \\le i \\le m}U_{u_i}$. Moreover for $1 \\le i \\le m-1$ we have that $u_in-1 n-1\\ldots \\simeqI u_{i+1}00\\ldots$.\n\\end{lemma}\n\\begin{proof}\nObserve that by Lemma \\ref{Lemma: states of TBnr induces continuous functions from the interval to itsel } it follows that for every state $q \\in \\core(A_{q_0}$ $h_{q}$ induces a continuous function from the interval to itself. However since the interval is connected, and the continuous image of a connected topological space is connected, it follows that $\\im(q)\/\\simeqI$ is a connected. The lemma is now an easy consequence of this.\n\\end{proof}\n\n The result below was demonstrated first by Brin in his seminal paper \\cite{MBrin2} describing the automorphisms of R. Thompson's groups $F$ and $T$ and we have simple translated it into the language of transducers.\n\n\n\\begin{Theorem}\nLet $n = 2$, and let $A_{q_0} \\in \\TBnr$, then $\\core(A_{q_0})$ is either the single state identity transducer or the single state transducer mapping induce the transposition $(0 \\ 1)$ on $X_{2}$. More specifically we have $\\TOn \\cong C_2$.\n\\end{Theorem} \n\nIn order to show the subgroup $\\TOnr \\le \\Onr$ is completely characterised by the fact that all of its states preserve the relation $\\simeqI$ and either all preserve or all reverse the lexicographic ordering of $\\CCn$ we need the following notions from \\cite{BCMNO}. Recall that $\\On$ is the group of core, bi-synchronizing transducers all of whose states induce injective maps of $\\CCn$ with clopen images. \n\n\\begin{Definition}[\\cite{BCMNO}]\\label{Def: Viable combination}\nLet $g \\in \\On$. A \\emph{viable combination} $\\mathfrak{v}_{j}$ for $g$ is a pair of tuples $$((\\rho_1, \\ldots, \\rho_j), (p_1, \\ldots, p_j))$$ satisfying \n\\begin{enumerate}\n\\item $\\bigcup_{1\\le i \\le j} \\rho_i\\im(p_i) = \\CCn$ \\label{Def: Viable combination union fills cantor space}\n\\item for $1 \\le i < l \\le j$ we have $\\rho_i \\im(p_i) \\cap \\rho_l \\im(p_l) = \\emptyset$ \\label{Def: Viable combination unions are disjoint}.\n\\end{enumerate}\n where the $\\rho_{i} \\in \\Xns$, $1 \\le i \\le j$ are not necessarily incomparable or distinct, and $p_i$, $1 \\le i \\le j$ are not necessarily distinct states of $g$.\n\\end{Definition} \n\n\\begin{Definition}[Single expansions of viable combinations \\cite{BCMNO}] \\label{Def: expansions of viable combinations}\nLet $T = (X_n, Q, \\pi, \\lambda) \\in \\On$ and let \n$\\mathfrak{v}_{j} = ((\\rho_1, \\ldots, \\rho_j), (p_1, \\ldots, \np_j))$ be a viable combination for $T$. Fix an $i$ such that $1 \n\\le i \\le j$, and let $\\rho_{i,l} := \\rho_i\\lambda(l, \np_i)$ and $p_{i,l} := \\pi(l, p_i)$ for $l \\in X_n$, then \n$$\\mathfrak{v}_{j+n-1} := ((\\rho_1, \\ldots, \\rho_{i-1}, \n\\rho_{i,0}, \\ldots,\\rho_{i,(n-1)}, \\rho_{i+1}, \\ldots, \\rho_{j} \n), (p_1, \\ldots, p_{i-1}, p_{i,0}, \\ldots, p_{i, (n-1)}, \np_{i+1},\\ldots, p_{i,k}))$$ is called a \\emph{single expansion of \n$\\mathfrak{v}_{j}$}. \n\\end{Definition}\n\n\n\\begin{Remark}\\label{Remark: sequence of expansions of viable combinations give rise to viable combinations}\nClearly a single expansion of a viable combination for an element $T \\in \\On$ results in a new viable combination for $T$. Therefore a sequence of single expansions applied to a viable combination for $T$ also results in a new viable combination for $T$. See \\cite{BCMNO} for more detail.\n\\end{Remark}\n\nThe following lemma is proved in \\cite{BCMNO}:\n\n\\begin{lemma}\\label{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties}\nLet $g \\in \\On$ and let $\\mathfrak{v}_{g}$ denote the set of all viable combinations of $g$. Let $1 \\le r \\le n-1$. There is a transducer $A_{q_0} \\in \\Bnr$ whose core is equal to $g$ if and only if there is a sequence $\\mathfrak{v}_{j_1}, \\mathfrak{v}_{j_2} \\ldots, \\mathfrak{v}_{j_m}$ of elements of $\\mathfrak{v}_{g}$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_i \\equiv m \\mod{n-1}$.\n\\end{lemma}\n\nNow we have the following lemma which solves the other half of the membership problem of $\\TOnr \\in \\Onr$. However we make the following definition first.\n\n\\begin{Definition}\\label{Def: lexicographic viable combinations}\nLet $g \\in \\On$. A \\emph{lexicographic viable combination} $\\mathfrak{v}_{j}$ for $g$ is a viable combination $$\\mathfrak{v}_{j}:=((\\rho_1, \\ldots, \\rho_j), (p_1, \\ldots, p_j))$$ satisfying: for $1 \\le a < b \\le j$ $\\rho_{a}\\im(p_a) \\lelex \\rho_{b}\\im(p_b)$.\n\\end{Definition}\n\n\\begin{lemma}\\label{Lemma: g in Onr induces map on the line iff it has lexicographic viable combination}\nLet $g \\in \\Onr$ be such that $g$ preserves $\\simeqI$ and preserves (reverses) the lexicographic ordering on $\\CCn$ then any viable combination of $g$ may be re-ordered to get a lexicographic viable combination.\n\\end{lemma}\n\\begin{proof}\n\nSince $g \\in \\Onr$ there is an element $A_{q_0} \\in \\TBnr$ such that $\\core(A_{q_0}) = g$. Now by Lemma~\\ref{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties} it follows that there are viable combinations $\\mathfrak{v}_{j_1}, \\mathfrak{v}_{j_2}, \\ldots, \\mathfrak{v}_{j_{m}}$ of $g$. Thus the set $\\mathfrak{v}_{g}$ of viable combinations of $g$ is non-empty. Let $\\mathfrak{v}_{l} \\in \\mathfrak{v}_{g}$. We shall now make use of Lemma~\\ref{Lemma: images of states of core of elements of TBnr are connected} to re-order $\\mathfrak{v}_{l}$ to obtain a lexicographic viable combination of $g$. \n\nSuppose $\\mathfrak{v}_{l} = ((\\rho_1, \\ldots, \\rho_l),(p_1, \\ldots, p_l))$. Let $R_{\\mathfrak{v}_{l}} := \\{ \\rho_k \\mid 1 \\le k \\le l \\mbox{ such that }\\rho_{t} \\not\\leq \\rho_{k} \\forall 1\\le t \\le l \\}$. Observe that $R_{\\mathfrak{v}_{l}}$ is an antichain by construction, thus we may assume that the set $R_{\\mathfrak{v}_{l}}$ is totally ordered in the lexicographic order. Fix $\\rho_{k} \\in R_{\\mathfrak{v}_{l}}$. Let $D(\\rho_{k}) = \\{ \\rho_{t} \\mid \\rho_{k} \\le \\rho_{t} \\}$. We assume that $D(\\rho_{k})$ is ordered according to the short-lex ordering. Observe that $\\rho_{k} \\in D(\\rho_{k})$ and in particular is the smallest element of $\\rho_{k}$. Let $d_{\\rho_{k}} = | D(\\rho_{k})|$. Now for each $\\rho_{t} \\in D(\\rho_{k})$ let $\\mu_{\\rho_{t}}$ be the number of times it occurs in the tuple $(\\rho_1, \\ldots, \\rho_k)$. Let $\\bar{l}:= |R_{\\mathfrak{v}_{l}}|$. \n\nGiven a tuple $(y_1, y_2, \\ldots, y_t)$ we shall use the notation $(y_1, \\ldots, y_s^{p}, \\ldots, y_t)$ to mean that $y_s, \\ldots, y_{s+p-1}$ are all equal to $y_s$. Now let $\\xi_1, \\ldots \\xi_{\\bar{l}}$ be the elements of $R_{\\mathfrak{v}_{l}}$ in lexicographic ordering. For $1 \\le s \\le \\bar{l}$ let $\\xi_{(s,1)}, \\ldots \\xi_{(s, d_{\\xi_{s}})}$ be the elements of $D(\\xi_{s})$ in short-lex ordering. Then we have \n\\begin{equation}\\label{Equation: first rearrangement of tuple}\n\\left(\\xi_1^{\\mu_{\\xi_1}}, \\xi_{(1,1)}^{\\mu_{\\xi_{(1,1)}}}, \\ldots, \\xi_{(1, d_{\\xi_1})}^{\\mu_{\\xi_{(1, d_{\\xi_1})}}}, \\ldots, \\xi_{\\bar{l}}^{\\mu_{\\xi_{\\bar{l}}}}, \\xi_{(\\bar{l},1)}^{\\mu_{\\xi_{(\\bar{l},1)}}}, \\ldots, \\xi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}^{\\mu_{\\xi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}}}\\right)\n\\end{equation}\nis a reordering of $(\\rho_1, \\ldots, \\rho_l)$. Let $(q_1, \\ldots, q_l)$ be the induced reordering on the tuple of states $(p_1,\\ldots, p_l)$.\n\nNow fix $1 \\le s \\le \\bar{l}$ and consider the sequence \n$\\xi_s^{\\mu_{\\xi_s}}, \\xi_{(s,1)}^{\\mu_{\\xi_{(s,1)}}}, \\ldots, \n\\xi_{(s, d_{\\xi_1})}^{\\mu_{\\xi_{(s, d_{\\xi_s})}}}$ let \n$$q_{(s_0,1)}, \\ldots, q_{(s_0, \\mu_{\\xi_s})}, \\ldots, \nq_{(s_{d_{\\xi_s}},1)} ,\\ldots , q_{(s_{d_{\\xi_s}}, \\mu_{\\xi_{(s, \nd_{\\xi_{s}})}})}$$ be the corresponding states. Then observe that $$ \n\\xi_{s}\\im(q_{(s_0,1)} \\cup \\ldots \\cup \\xi_s \\im(q_{(s_0, \\mu_{\\xi_s})}) \\cup \\bigcup_{1 \\le k \\le \nd_{\\xi_s}} \\xi_{(s,k)} \\im(q_{(s_{k},1)}) \\cup \\ldots \\cup \\xi_{(s,k)} \\im(q_{(s_{k}, \\mu_{\\xi_{(s, k)})}}) = \nU_{\\xi_{1}}$$ by construction and the definition of viable \ncombinations. Now let $0 \\le k < d_{\\xi_s}$ and let $k \\le k' \\le d_{\\xi_s}$, for $1\\le t_1 \\le \\mu_{\\xi_{(s,l)}}$ and $1 \\le t_2 \\le \\mu_{\\xi_{(s, k')}}$ consider $\\xi_{(s,k)} \\im(q_{(s_{k}, t_1)})$ and $\\xi_{(s,k')} \\im(q_{(s_{k'},t_2)})$ (note that we set $\\xi_{(s,0)} \\im(q_{(s_{0},t_1)}):= \\xi_{s}\\im(q_{s_{0}, t_1})$). Observe that by the ordering of the set $D(\\xi_{s})$, by Lemma~\\ref{Lemma: images of states of core of elements of TBnr are connected} and by Definition~\\ref{Def: Viable combination} part~\\ref{Def: Viable combination unions are disjoint} of viable combinations, we must have that exactly one of the following holds: \n\\begin{enumerate}[label = (\\roman*)]\n\\item $\\xi_{(s,k)} \\im(q_{(s_{k}, t_1)}) \\lelex \\xi_{(s,k')} \\im(q_{(s_{k'},t_2)})$ or,\n\\item $\\xi_{(s,k')} \\im(q_{(s_{k'},t_2)}) \\lelex \\xi_{(s,k)} \\im(q_{(s_{k}, t_1)})$ or,\n\\item $k = k'$ and $t_1 = t_2$.\n\\end{enumerate}\n\nThus we may reorder the tuple \\eqref{Equation: first rearrangement of tuple} to obtain a new tuple\n\n\\begin{equation}\\label{Equation: second rearrangement of tuple}\n\\left(\\chi_1^{\\mu_{\\chi_1}}, \\chi_{(1,1)}^{\\mu_{\\chi_{(1,1)}}}, \\ldots, \\chi_{(1, d_{\\xi_1})}^{\\mu_{\\chi_{(1, d_{\\xi_1})}}}, \\ldots, \\chi_{\\bar{l}}^{\\mu_{\\chi_{\\bar{l}}}}, \\chi_{(\\bar{l},1)}^{\\mu_{\\chi_{(\\bar{l},1)}}}, \\ldots, \\chi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}^{\\mu_{\\chi_{(\\bar{l},d_{\\chi_{\\bar{l}}})}}}\\right)\n\\end{equation} \n\nsatisfying the following conditions. Let $1 \\le s \\le \\bar{l}$ and consider the subsequence $$\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n\\chi_{(s, d_{\\xi_s})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$$ then:\n\\begin{enumerate}[label = (\\roman*)]\n\\item Each $\\chi_{(s, r)}^{\\mu_{\\chi_{(s,r)}}}$ for $0 \\le r \\le d_{\\xi_{s}}$ (where $\\chi_{(s,0)} = \\chi_{s}$) corresponds to precisely one element $\\xi_{(s,r')}^{\\mu_{\\xi_{(s,r')}}}$ for $0 \\le r' \\le d_{\\xi_{s}}$ (where $\\xi_{(s,0)} = \\xi_{s}$). \\label{List: rearrangement 2 cond 1}\n\\item The subsequence $\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n\\chi_{(s, d_{\\xi_s})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$ is ordered so that the following is true. If $q_{(s_0,1)}, \\ldots, q_{(s_0, \\mu_{\\chi_s})}, \\ldots, \nq_{(s_{d_{\\xi_s}},1)} ,\\ldots , q_{(s_{d_{\\xi_s}}, \\mu_{\\chi_{(s, \nd_{\\xi_{s}})}})}$ are the set of states corresponding to the sub-sequence $\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n\\chi_{(s, d_{\\xi_1})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$ then for $0 \\le k < d_{\\xi_{s}}$, for $k \\le k' \\le d_{\\xi_{s}}$, for $1 \\le t_1 \\le \\mu_{\\chi_{(s,k)}}$ and $1 \\le t_2 \\le \\mu_{\\chi_{(s,k')}}$ such that if $k = k'$ and $\\mu_{\\chi_{(s,k)}} >1$ then $t_1 \\ne t_2$, we have $\\chi_{(s,k)} \\im(q_{(s_{k}, t_1)}) \\lelex \\chi_{(s,k')} \\im(q_{(s_{k'},t_2)})$. \n\\end{enumerate}\n \n Notice condition~\\ref{List: rearrangement 2 cond 1} above means all elements of the subsequence $\\chi_s^{\\mu_{\\chi_s}}, \\chi_{(s,1)}^{\\mu_{\\chi_{(s,1)}}}, \\ldots, \n \\chi_{(s, d_{\\xi_s})}^{\\mu_{\\chi_{(s, d_{\\xi_s})}}}$ have prefix $\\xi_{s}$. \n\nNow observe that for $1 \\le s < s' \\le \\bar{l}$, for $ 0 \\le k \\le d_{\\xi_{s}}$, for $0 \\le k' \\le d_{\\xi_{s'}}$, for $1 \\le t \\le \\mu_{\\chi_{(s, k)}}$ and for $1 \\le t' \\le \\mu_{\\chi_{(s', k')}}$ so that if the $t$\\textsuperscript{th} copy of $\\chi_{(s, k)}$ ($\\chi_{s}$ if $k=0$) and the $t'$\\textsuperscript{th} copy of $\\chi_{(s', k')}$ ($\\chi_{s'}$ if $k'=0$) in the sequence \\eqref{Equation: second rearrangement of tuple} correspond to the states $q_{(s_{k}, t_1)}$ and $q_{(s'_{k}, t_2)}$ respectively, then $\\chi_{(s,k)} \\im(q_{(s_{k}, t_1)}) \\lelex \\chi_{(s',k')} \\im(q_{(s'_{k'},t_2)})$ in the lexicographic ordering on $\\CCn$. This follows from the observation in the previous paragraph and because $\\xi_{s} \\lelex \\xi_{s'}$.\n\nLet $$(\\zeta_1, \\ldots, \\zeta_{l}) := \\left(\\chi_1^{\\mu_{\\chi_1}}, \\chi_{(1,1)}^{\\mu_{\\chi_{(1,1)}}}, \\ldots, \\chi_{(1, d_{\\xi_1})}^{\\mu_{\\chi_{(1, d_{\\xi_1})}}}, \\ldots, \\chi_{\\bar{l}}^{\\mu_{\\chi_{\\bar{l}}}}, \\chi_{(\\bar{l},1)}^{\\mu_{\\chi_{(\\bar{l},1)}}}, \\ldots, \\chi_{(\\bar{l},d_{\\xi_{\\bar{l}}})}^{\\mu_{\\chi_{(\\bar{l},d_{\\chi_{\\bar{l}}})}}}\\right)$$ and let \n\\begin{IEEEeqnarray*}{rCl}\n(q_1', \\ldots q_l') := (& q_{(1_0,1)}& , \\ldots, q_{(1_0, \\mu_{\\chi_{1}})}, \\ldots, q_{(1_{d_{\\xi_{1}}},1)},\\ldots, q_{(1_{d_{\\xi_{1}}}, \\mu_{\\chi_{(1, d_{\\xi_1})}})}, \\ldots, \\\\\n & q_{(\\bar{l}_0,1)},& \\ldots, q_{(\\bar{l}_0, \n \\mu_{\\chi_{\\bar{l}}})}, \\ldots, \n q_{(\\bar{l}_{d_{\\xi_{1}}},1)},\\ldots, \n q_{(\\bar{l}_{d_{\\xi_{\\bar{l}}}}, \\mu_{\\chi_{(\\bar{l}, \n d_{\\xi_{\\bar{l}}})}})})\n\\end{IEEEeqnarray*}\n\n\nObserve that $((\\zeta_1, \\ldots, \\zeta_l), (q_1', \\ldots, q_l'))$ is a viable combination for $g$. Moreover, by discussion above, for $1 \\le a < b \\le l$ it satisfies $\\zeta_{a} \\im(q_a) \\lelex \\zeta_{b}\\im(q_b)$ in the lexicographic ordering of $\\CCn$ i.e $((\\zeta_1, \\ldots, \\zeta_l), (q_1', \\ldots, q_l'))$ is a lexicographic viable combination of $g$. \n\\end{proof}\n\nLet $ \\pi_{R,n} \\in \\On$ be the single state transducer which induces the permutation $i \\mapsto n-1- i$ on $\\Xn$. Then $\\pi_{R,n}$ has order $2$, preserves $\\simeqI$, and reverses the lexicographic ordering on $\\CCn$. It is also not hard to see that $\\pi_{R,n} \\in \\TOnr$ for all $1 \\le r < n-1$. For instance the map $ f_{\\pi_{R,n}}: \\CCnr \\to \\CCnr$ given by $ \\dot{a} \\delta \\mapsto \\dot{r-a +1} (\\delta)h_{\\pi_{R,n}}$ is induces by an element of $\\TBnr$ with core equal to $\\pi_{R,n}$ since it replaces some finite prefix before acting by $\\pi_{R,n}$.\n\nThe following lemma follows immediately from the fact that $\\pi_{R,n}$ has order $2$ and reverses the lexicographic ordering on $\\CCn$.\n\n\\begin{lemma}\\label{Lemma: multiplication by piR bijection from orientation preserving to reversing and vice versa}\nMultiplication by $\\pi_{R,n}$ induces a bijection from the subset of $\\Onr$ preserving (reversing) the lexicographic ordering, to the subset of $\\Onr$ reversing (preserving) the lexicographic ordering.\n\\end{lemma}\n\n\n\n\\begin{lemma}\\label{Lemma: g in Onr is in TOnr if g prserve simeqI and lex order}\nLet $g \\in \\Onr$ be such that $g$ preserves $\\simeqI$ and preserves (reverses) the lexicographic ordering on $\\CCn$ then $g \\in \\TOnr$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma~\\ref{Lemma: multiplication by piR bijection from orientation preserving to reversing and vice versa} it suffices to prove this for elements $g \\in \\Onr$ which preserves $\\simeqI$ and the lexicographic ordering on $\\CCn$. Since if $g \\pi \\in \\TOnr$, then as $\\TOnr$ is a group containing $\\pi_{R,n}$ by an observation above, then $g \\pi_{R,n} \\pi_{R,n} = g \\in \\TOnr$ also. Thus fix $g \\in \\Onr$ an element which preserves $\\simeqI$ and the lexicographic ordering on $\\CCn$.\n\nSince $g \\in \\Onr$ by Lemma~\\ref{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties} there are viable combinations $\\viable{j_1}, \\viable{j_2},\\ldots,\\viable{j_m}$ of $g$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_{i} \\equiv m \\mod{n-1}$. Let $j= \\sum_{1 \\le i \\le m} j_i$. By Remark \\ref{Remark: sequence of expansions of viable combinations give rise to viable combinations} we may assume that $j_i > 1$ for $1 \\le i \\le m$. Now by Lemma~\\ref{Lemma: g in Onr induces map on the line iff it has lexicographic viable combination} we may assume that all the $\\viable{j_{i}}$ for $1\\le i \\le m$ are lexicographic viable combinations for $g$.\n\nLet $1 \\le i \\le m$, and consider the viable combination $\\viable{j_i}$ of $g$. Suppose $\\viable{j_{i}} = ((\\rho_1, \\ldots, \\rho_{j_i}), (p_1, \\ldots, p_{j_i}))$. Let $\\ac{u}= \\{ u_1, u_2, \\ldots, u_{j_i}\\}$ be an antichain of $\\Xnrs$ of length $j_i$, and let $w \\in \\Xnrp$. Recall that all antichains are assumed to be ordered in the lexicographic ordering. Now if $g$ preserves the lexicographic ordering of $\\CCn$ then define a map $f_{\\mathfrak{v}_{j_i},w}: U_{u_1} \\sqcup \\ldots \\sqcup U_{u_{j_i}} \\to U_{w}$ by $u_a \\delta \\mapsto w\\rho_a (\\delta)h_{p_{a}}$ for $1 \\le a \\le j_i$ and $\\delta \\in \\CCn$. O Then $f_{\\mathfrak{v}_{j_i},w}$ is homeomorphism from its domain to its range (by the definition of a viable combination), moreover $f_{\\mathfrak{v}_{j_i},w}$ preserves the lexicographic ordering on $U_{u_1} \\sqcup \\ldots \\sqcup U_{u_l}$ since $\\viable{j_i}$ is a lexicographic viable combination and for all states $q$ of $g$ $h_{q}$ preserves the lexicographic ordering on $\\CCn$ and the relation $\\simeqI$ (Lemma~\\ref{Lemma: states of TBnr induces continuous functions from the interval to itsel }). \n\nNow let $\\ac{v}$ be a complete antichain of $\\Xnrs$ of length $j$ and let $\\ac{w}$ be a complete antichain of $\\Xnrs$ of length $m$. These antichains exist since $j \\equiv m \\equiv r \\mod{n-1}$. Recall that as $ 1 \\le r < n-1$ we must have that $j$ and $m$ are both non-zero. Let $\\ac{v}_1, \\ldots, \\ac{v}_m$ be disjoint subsets of $\\ac{v}$ such that each $\\ac{v}_i$, $1 \\le i \\le m$ is an antichain (ordered lexicographically) of length $j_i$ and for $1 \\le a < b \\le m$ we have $\\ac{v}_{a} \\lelex \\ac{v}_{b}$. Note that the stipulation that $|\\ac{v}_{i}| = j_i$ for $1 \\le i \\le m$ means that $\\sqcup_{1 \\le i \\le m} \\ac{v}_{i} = \\ac{v}$. Suppose that $\\ac{w} = \\{ w_1, w_2, \\ldots, w_m\\}$. \n\nRecall that for a subset $Z \\subseteq \\Xns \\sqcup \\Xnrs$ we denote by $U(Z)$ the set $\\{ U_{z} \\mid z \\in Z \\}$. Let $f: \\CCnr \\to \\CCnr$ be defined such that $f \\restriction _{U(\\ac{v}_{i})} = f_{\\ac{v}_i,w_i}$ for $1 \\le i \\le m$. Then clearly $f$ is a homeomorphism, and since each $f_{\\ac{v}_i,w_i}$, $1 \\le i \\le m$, preserves the lexicographic ordering of $U(\\ac{v_i})$ and $\\simeqI$ then, $f$ preserves the lexicographic ordering and the relation $\\simeqI$ on $\\CCnr$. Moreover since $f$ replaces some initial prefix before acting as a state of $g$, it follows that there is a transducer $A_{q_0} \\in \\TBnr$ with $h_{q_0} = f$.\n\n\\end{proof}\n\n\\begin{Remark}\\label{Remark: direct proof that g in Onr preseving simeq and reversing lex is in TOnr}\nOne may also prove Lemma~\\ref{Lemma: g in Onr is in TOnr if g prserve simeqI and lex order} above for $g \\in \\Onr$ which preserves $\\simeqI$ and reverses the lexicographic ordering by directly constructing a transducer $A_{q_0} \\in \\TBnr$ which induce an orientation reversing homeomorphism of the line. As in the proof above, we make use of the lexicographic viable combinations to induce maps on $\\CCnr$ which preserve $\\simeqI$ and reverse the lexicographic ordering.\n\\end{Remark}\n\n\\begin{lemma}\\label{Lemma:preservinglexsuffices}\nLet $g \\in \\Onr$ be such that $g$ preserves or reverses the lexicographic ordering, then $g$ also preserves $\\simeqI$.\n\\end{lemma}\n\\begin{proof}\nFix $q$ any state of $g$. We observe that, by definition $q$ induces a continuous injection function from $\\CCn$ to itself with clopen image. Let $x, y \\in \\CCn$ be such that $x \\ne y$, $x \\lelex y$, $x \\simeqI y$ and there is a word $\\nu \\in \\Xnrp$ such that $(x)h, (y)h \\in U_{\\nu}$. As there is no point $y' \\in \\CCn$ not equal to $x$ or $y$ satisfying $x \\lelex y' \\lelex y$, then it must be the case that $(x)h \\simeq (y')h$. Now as $(U_{\\nu})h_{q} ^{-1}$ is open, there is a $\\mu \\in \\Xnrp$ such that $(U_{\\mu})h_{q} = U_{\\nu}$ and so $h_{q}$ preserves the relation $\\simeqI$ on $U_{\\mu}$. Finally, observe that as $g$ is synchronizing for any other state $p$ of $g$, there is a word $\\Gamma \\in \\Xnrp$ such that $\\pi_{g}(\\Gamma, q) = p$. Therefore, we deduce that all states of $g$ preserve $\\simeqI$ as required.\n\\end{proof}\n\n\n\nPutting together Lemmas~\\ref{Lemma: g is in Onr if and only if there is a viable combination with the right modulo arithmetic properties}, \\ref{Lemma: g in Onr induces map on the line iff it has lexicographic viable combination}, Lemma~\\ref{Lemma:preservinglexsuffices}, and \\ref{Lemma: g in Onr is in TOnr if g prserve simeqI and lex order}, we obtain the following result.\n\n\\begin{Theorem}\\label{Thm: Equivalent conditions for an element of $On$ to belong to TOnr}\nLet $g \\in \\On$. The following are equivalent:\n\\begin{enumerate}[label =(\\alph*)]\n\\item $g \\in \\TOnr$\n\\item $g$ preserves or reverses the lexicographic ordering, and there are lexicographic viable combinations $\\viable{{j_1}}, \\ldots, \\viable{{j_m}}$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_{i} \\equiv m \\mod{n-1}$.\n\\item $g$ preserves or reverses the lexicographic ordering, and there are viable combinations $\\viable{{j_1}}, \\ldots, \\viable{{j_m}}$ such that $r \\equiv \\sum_{1 \\le i \\le m} j_{i} \\equiv m \\mod{n-1}$.\n\\item $g \\in \\Onr$ and preserves or reverses the lexicographic ordering on $\\CCn$.\n\\end{enumerate}\n\n\\end{Theorem}\n\n\n\\section{ The enveloping group $\\TOn$}\\label{Section:nestingproperties1}\n\nWe make the following definition.\n\n\\begin{Definition}\\label{Def: External description of TOn.}\nLet $\\TOn \\subset \\On$ consists of those element which reverse or preserve the lexicographic ordering on $\\CCn$.\n\\end{Definition}\n\n\\begin{Remark}\\label{Remark: TOn is the union of the TOnr's}By an observation in \\cite{BCMNO}, for an element $g \\in \\On$ the set $\\viable{g}$ of viable combinations of $g$ is non-empty. From this it follows that $\\TOn = \\cup_{1\\le r \\le n-1} \\TOnr$.\n\\end{Remark}\n\n\\begin{proposition} \\label{Proposition:TOn is a subgroup of On}\nThe set $\\TOn$ is a subgroup of $\\On$.\n\\end{proposition}\n\\begin{proof}\nIt suffices to show that $\\TOn$ is closed under inverses and products. It is clear that the single state identity transducer is an element of $\\TOn$.\n\n\nLet $g \\in \\TOn$. Then there is an $r \\in \\{1,2, \\ldots, n-1\\}$ such that there is an $A_{q_0} \\in \\TBnr$ with $\\core(A_{q_0}) = g$. Let $B_{p_0}$ be the minimal transducer representing the inverse of $A_{q_0}$, then $B_{p_0} \\in \\TBnr$ and $g^{-1} = \\core(B_{p_0}) \\in \\TOnr \\subset \\TOn$.\n\nNow let $h \\in \\TOn$. Let $q$ be a state of $g$ and $p$ be a state of $h$. Let $gh_{(p,q)}$ be the minimal transducer representing the product $g_{p} \\ast h_{q}$. Then observe that $gh_{(p,q)}$ induces the function $g \\circ h : \\CCn \\to \\CCn$. Thus since $g$ and $h$ either preserve or reverse the lexicographic ordering on $\\CCn$ it follows that the states of $gh_{(p,q)}$ either all reverse or all preserve the lexicographic ordering on $\\CCn$. Therefore $gh = \\core(gh_{(p,q)}) \\in \\TOn$.\n\\end{proof}\n\n\\begin{Definition}\nLet $\\widetilde{\\TOn}$ be the subset of $\\TOn$ consisting of all those elements which preserve the lexicographic ordering on $\\CCn$. Set $\\WTOnr:= \\WTOn \\cap \\TOnr$. Then we call elements of $\\WTOn$ \\emph{orientation preserving}, and elements of $\\TOn \\backslash \\WTOn$ \\emph{orientation reversing}. Likewise set $\\widetilde{\\TBnr}$ to be those elements of $\\TBnr$ with core in $\\widetilde{\\TOn}$, then $\\widetilde{\\TBnr}$ are precisely those elements of $\\TBnr$ which induce orientation preserving maps of $S_{r}$. We will also call elements of $\\widetilde{\\TBnr}$ \\emph{orientation preserving} and the elements of $\\TBnr \\backslash \\widetilde{\\TBnr}$ \\emph{orientation reversing}.\n\\end{Definition}\n\nThe following proposition is straightforward and so we omit its proof.\n\n\\begin{proposition}\nThe set $\\WTOn$ is an index 2 subgroup of $\\TOn$ and so a normal subgroup of $\\TOn$. The set $\\WTOnr$ is an index 2 subgroup of $\\TOnr$ and so a normal subgroup of $\\TOnr$. \n\\end{proposition}\n\nWe now investigate how the groups $\\TOn$ intersect each other. For this we require the following definition, which in fact applies to the elements of the group $\\On$.\n\n\\begin{Definition}[Signature]\\label{Definition:signature}\nLet $T \\in \\On$, for each state $q \\in Q_{T}$ let $m_q$ be the size of the smallest subset $V$ of $\\Xns$ such that $U(V) = \\{ U_v \\mid v \\in V\\}$ is a clopen cover of $\\im(q)$ and $U_{v} \\subset \\im(q)$ for all $v \\in V$. Let $k \\in \\N$ be the minimal synchronizing level of $T$ an order the elements of the set $\\Xn^{k}$ lexicographically as follows: $x_1 < x_2 < \\ldots < x_{n^{k}}$. Let $(q_{x_1}, q_{x_2}, \\ldots, q_{x_{n^{k}}}) \\in Q_{T}^{n^{k}}$ be such that, for all $1 \\le i \\le n^{k}$, $q_{x_i}$ is the unique state of $T$ forced by $x_i$. Set $(T)\\sig = \\sum_{1 \\le i \\le n^{k}} m_{q_{x_i}}$, we call $(T)\\sig$ the \\emph{signature} of $T$; set $(T)\\rsig = (T)\\sig \\mod{n-1}$, we call $(T)\\rsig$ the \\emph{reduced signature of $T$}.\n\\end{Definition}\n\nWe have the following proposition. We prove the proposition below for the group $\\TOn$ noting that a similar result holds, with almost identical proof, in the group $\\On$.\n\n\\begin{proposition}\\label{Proposition:sigdeterminesmembership}\nLet $T \\in \\TOn [T \\in \\On]$, and $1 \\le r < n$, then $T \\in \\TOnr [T \\in \\Onr]$ if and only if $r (T)\\sig \\equiv r \\mod{n-1}$ (equivalently $r((T)\\sig-1) \\equiv 0 \\mod {n-1}$.\n\\end{proposition}\n\\begin{proof}\nWe begin with the forward implication. First suppose that $T \\in \\TOnr$ and $T$ has minimal synchronizing level $k$. Since $T \\in \\TOnr$, there is an element $A_{q_0} \\in \\TBnr$ with $\\core(A_{q_0}) = T$. Let $j \\in \\N$ be minimal such that after reading a word of length $j$ from the state $q_0$ of $A$, the resulting state is a state of $T$. Let $\\{\\mu_i \\mid 1 \\le i \\le rn^{k+j-1}\\}$ be the set of all words of length $j+k$ in $\\Xnrs$ ordered lexicographically. For $1 \\le i \\le rn^{k+j-1}$, set $\\nu_i = \\lambda_{A}(\\mu_i, q_0)$ and $q_{\\mu_i}$ to be the state of $T$ forced by $\\mu_i$. Observe that the state $q_{\\mu_i}$ depends only on the last $k$ letters of $\\mu_i$, hence if the elements of $\\Xn^{k}$ are ordered lexicographically as $x_1 < x_2< \\ldots < x_{n^{k}}$, the sequence ${(q_{\\mu_i})}_{1 \\le i \\le rn^{j+k-1}}$, where $q_{x_i}$, $1 \\le i \\le n^{k}$, is the state of $T$ forced by $x_i$, is precisely the sequence $q_{x_1}, \\ldots, q_{x_{n^{k}}}$ repeated $rn^{j-1}$ times. Since $\\bigcup_{1 \\le i \\le rn^{j+k-1}} \\nu_i \\im(q_{\\mu_i}) = \\CCnr$, it must therefore be the case that $rn^{j-1}(\\sum_{1\\le i \\le n^{k}} m_{q_{x_i}}) \\equiv r \\mod n-1$. This is because if, for each $q_{\\mu_i}$, $1 \\le i \\le rn^{k+j-1}$, $V_{q_{\\mu_i}}$ is the smallest subset of $\\Xns$ such that $U(V_{q_{\\mu_i}})$ is a cover of $\\im(q)$ and $U_{v} \\subset \\im(q)$ for all $v \\in V_{q_{\\mu_i}}$, then $ \\bigcup_{1 \\le i \\le rn^{j+k-1}}\\{ \\nu_i v \\mid v \\in V(q_{\\mu_i}) \\}$ must be a complete antichain of $\\CCnr$ (otherwise $A_{q_0}$ is not a homeomorphism). Therefore, setting $m_{q_{x_i}} = |V_{q_{x_i}}|$ we have, $$\\left| \\bigcup_{1 \\le i \\le rn^{j+k-1}}\\{ \\nu_i v \\mid v \\in V(q_{\\mu_i}) \\}\\right | = rn^{j-1}(\\sum_{1\\le i \\le n^{k}} m_{q_{x_i}}) \\equiv r \\mod n-1.$$\nSince $n \\equiv 1 \\mod{n-1}$ we therefore have that $r(T)\\sig \\equiv r \\mod n-1$ as required.\n\nFor the reverse implication let $T \\in \\TOn$ and $1 \\le r < n-1$ be such that $r (T)\\sig \\equiv r \\mod{n-1}$. Let $k \\in \\N$ be such that $T$ is synchronizing at level $k$, once more assume that the set $\\Xn^{k}$ is ordered lexicographically as $x_1 < x_2< \\ldots < x_{n^{k}}$. For $1 \\le i \\le n^{k}$, let $q_{x_i}$ be the state of $T$ forced by $x_i$. For each $1 \\le i \\le n^{k}$, let $V_{q_{x_i}}$ be the smallest subset of $\\Xns$, with size $m_{q_{x_i}}$, such that $U(V_{q_{x_i}})$ is a clopen cover of $\\im(q)$ consisting of clopen subsets of $\\im(q)$; let $M_{i} = \\max\\{\\left|v\\right| \\mid v \\in V_{q_{x_i}}\\}$ and set $M = \\max_{1 \\le i \\le n^{k}} M_{i}$. Let $j \\in \\N$ be minimal such that for any word $\\Gamma \\in \\Xn^{j}$ and any state $q$ of $T$, $|\\lambda_{T}(\\Gamma, q)|> M$. Order the set $\\Xn^{j}$ lexicographically as $y_1 < y_2 < \\ldots < y_{n^{j}}$. For each state $q_{x_i}$, $1 \\le i \\le n^{k}$, order the set $V_{q_{x_i}}$ lexicographically as $\\nu_{i,1} < \\nu_{i,2}< \\ldots < \\nu_{i,m_{q_{x_i}}}$, and for all $1 \\le l \\le n^{j}$ let $\\mu_{i,l}\\varphi_{i,l} = \\lambda_{T}(y_l, q_{x_i})$, for some $\\mu_{i,l} \\in V_{q_{x_i}}$, $\\varphi_{i,l} \\in \\Xnp$, and $p_{i,l} = \\pi_{T}(y_l, q_{x_i})$. Now since, for $1 \\le i \\le n^{k}$, $|V_{q_{x_i}}| = m_{q_{x_i}}$, each $\\mu_{i,l} = \\nu_{i,a}$ for some $1 \\le a \\le m_{q_{x_i}}$ and we may write, for all $1 \\le l \\le n^{j}$, $\\mu_{i,l}\\varphi_{i,l} = \\nu_{i,a}\\rho_{i,l_a}$ where $\\nu_{i,a} = \\mu_{i,l}$ and $\\rho_{i,l_a} = \\varphi_{i,l}$ for some $1 \\le a \\le m_{q_{x_i}}$, we also adopt the same notation for the set of $p_{i,l}$ and write $p_{i,l_a}$, for $1 \\le a \\le m_{q_{x_i}}$, where $\\nu_{i,a} \\rho_{i,l_a} = \\mu_{i,l}\\varphi_{i,l}$.\n\nNow let $\\ac{u}$ be a maximal antichain of $\\CCnr$ of length $r n^{j+k}$ and let $\\ac{v}$ be a maximal antichain of $\\CCnr$ of length $r (T)\\sig$. Write $\\ac{u} = \\cup_{1 \\le t \\le r}\\ac{u}_t$ where each $\\ac{u}_{t}$ is ordered lexicographically, $| \\ac{u}_t| = n^{j+k}$ and for $1 \\le t_1 < t_2 \\le r$ all elements of $\\ac{u}_{t_1}$ are strictly less than all elements of $\\ac{u}_{t_2}$ in the lexicographic ordering on $\\Xns$. Let $\\ac{v} = \\cup_{1 \\le t \\le r}\\ac{v}_t$ where each $\\ac{v}_{t}$ is ordered lexicographically, $| \\ac{v}_{t}| = (T)\\sig$ and for $1 \\le t_1 < t_2 \\le r$ all elements of $\\ac{v}_{t_1}$ are strictly less than all elements of $\\ac{v}_{t_2}$ in the lexicographic ordering on $\\Xns$.\n\nFix $1 \\le t \\le r$ and consider $\\ac{u}_{t}$. Write $\\ac{u}_{t}= \\{u_{t,i,l} \\mid 1 \\le i \\le n^{k}, 1 \\le l \\le n^{j}\\}$. We further assume that, for a fixed $1 \\le i \\le n^{k}$, the set $\\{u_{t,i,l} \\mid 1 \\le l \\le n^{j}\\}$ is ordered lexicographically and, for $1 \\le i_1 < i_2 \\le n^{k}$ and for all $1 \\le l_1, l_2 \\le n^{j}$, $u_{t,i_1, l_1} \\lelex u_{t,i_2, l_2}$. Likewise write $\\ac{v}_{t} = \\{ v_{t,i,a} \\mid 1 \\le i \\le n^{k}, 1 \\le a \\le m_{q_{x_i}} \\}$. We further assume that, for a fixed $1 \\le i \\le n^{k}$, the set $\\{v_{t,i,a} \\mid 1 \\le a \\le m_{q_{x_i}}\\}$ is ordered lexicographically and, for $1 \\le i_1 < i_2 \\le n^{k}$, $v_{t,i_1,a} \\lelex v_{t,i_2,b}$ for all $1 \\le a \\le m_{q_{x_{i_1}}}$ and $1 \\le b \\le m_{q_{x_{i_2}}}$. Furthermore, whenever, for $1 \\le a \\le m_{q_{x_i}}$ and $1 \\le l \\le n^{j}$, we have $\\nu_{i,a}\\rho_{i,l_a} = \\mu_{i,l}\\varphi_{i,l}$ we set $\\eta_{t,i,l} := v_{t,i,a}$.\n\nDefine a map $f$ from $\\CCnr$ to itself as follows. For $1 \\le t \\le r$, $f$ acts on elements with prefix in the set $\\ac{u}_{t}$ as follows. For $1 \\le i \\le n^{k}$ and $1 \\le l \\le n^{j}$, and $\\Gamma \\in \\CCn$, $u_{t,i,l}\\Gamma \\mapsto \\eta_{t,i,l}\\varphi_{i,l}(\\Gamma)p_{i,l}$. Observe that for a fixed $1 \\le i \\le n^{k}$, and a fixed $1 \\le a \\le m_{q_{x_i}}$, since we have $\\cup \\{ \\mu_{i,l}\\varphi_{i,l}\\im(p_{i,l}) \\mid \\mu_{i,l} = \\nu_{i,a} \\} = U_{\\nu_{i,a}}$ then it is the case that, for a fixed $1 \\le t \\le r$, $\\cup \\{ \\eta_{t,i,l}\\varphi_{i,l}\\im(p_{i,l}) \\mid \\eta_{i,l} = v_{t,i,a} \\} = U_{v_{t,i,a}}$. Now since, for a given $1 \\le i \\le n^{k}$, $\\bigcup_{1 \\le l \\le n^{j}} \\mu_{i,l} \\varphi_{i,l} \\im(p_{i,l}) = \\im(q)$ and as $q$ is injective and preserves the lexicographic ordering of $\\CCn$, we see that $f \\restriction_{U(\\ac{u}_{t})}$ is a bijection unto the set $U(\\ac{v}_{t})$ which preserves the lexicographic ordering of $\\CCn$. More specifically, since $f$ acts by a state of $T$ after a finite depth, we see that $f$ is a homeomorphism of $\\CCnr$ which is in fact an element of $\\TBnr$.\n\nThe proof of the other reading proceeds in an analogous fashion only here we do not have to worry about preserving the lexicographic ordering on $\\CCnr$.\n\\end{proof}\n\n\\begin{Remark}\nIt follows from results in \\cite{GriNekSus} that given an element $T \\in \\TOn [T \\in\\On]$ and $1 \\le r < n$, then it is possible to decide in finite time if $T \\in \\TOnr [T \\in\\Onr]$. Furthermore, by the above proposition, $T \\in \\TOns{1} [T \\in \\Ons{1}]$ if and only if $(T)\\sig \\equiv 1 \\mod{n-1}$.\n\\end{Remark}\n\nThe following result is a consequence of Proposition~\\ref{Proposition:sigdeterminesmembership}:\n\n\\begin{lemma}\\label{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}\nLet $n \\in \\N$ and suppose that $n\\ge 2$. Let $i, j, m \\in \\Z_{n}$ such that, $i$ and $j$ are non-zero and $m i \\equiv j \\mod n-1$ then $\\TOns{i} \\subseteq \\TOns{j}$ [$\\Ons{i} \\subseteq \\Ons{j}$].\n\\end{lemma}\n\\begin{proof}\nLet $i,j, m$ be as in the statement of the lemma. Let $T \\in \\TOns{i} [T \\in \\Ons{i}]$, then by Proposition~\\ref{Proposition:sigdeterminesmembership} $i (T)\\sig \\equiv i \\mod{n-1}$, therefore $mi (T)\\sig \\equiv mi \\mod{n-1}$ and so $j (T)\\sig \\equiv j \\mod{n-1}$ and $T \\in \\TOns{j} [T \\in \\Ons{j}]$ again by Proposition~\\ref{Proposition:sigdeterminesmembership}.\n\\end{proof}\n\n\\begin{comment}\n\\begin{proof}\nWe first consider elements $g \\in \\WTOns{i}$. Let $g \\in \\WTOns{i}$, and as usual let ${\\bf{\\dot{i}}} = \\{\\dot{1},\\ldots, \\dot{i} \\}$. Let $\\bar{m} \\in \\N$ be such that $\\bar{m}$ is non-zero and congruent to $m$ modulo $n-1$. Let $A_{q_0} \\in \\TBmr{n}{,i}$ such that $\\core(A_{q_0}) = g$. There is a complete antichain $\\ac{u}$ of $\\CCmr{n,j}$ with size precisely precisely $\\bar{m}i$. Suppose $$\\ac{u} = \\{u_{1,1}, \\ldots, u_{1,i}, u_{2,1},\\ldots, u_{2,i}, \\ldots, u_{m,1},\\ldots, u_{m,i}\\}.$$ For $1 \\le l \\le i$ Let $\\ac{u}_{l} := \\{u_{l,1}, u_{l,2}, \\ldots, u_{l,i} \\}$, observe that $\\ac{u}_{l}$ is lexicographically ordered since $\\ac{u}$ is lexicographically ordered, thus $u_{l,1}\\lelex u_{l,2}\\lelex \\ldots \\lelex u_{l,i}$. Let $U_{\\ac{u}_{l}}:= U_{u_{l,1}} \\sqcup \\ldots \\sqcup U_{u_{l,i}}$.\n\n Fix $1 \\le l \\le m$ and consider the set $U_{\\ac{u}_{l}}$. Define a map $f_{l}$ from $U_{\\ac{u}_{l}}$ to itself as follows: for $\\delta \\in \\CCn$ and $r, r' \\in \\{1, \\ldots, i\\}$, $(u_{l,r}\\delta)f = u_{l,r'}\\rho$ if and only if $(\\dot{r}\\delta)h_{A_{q_0}} = \\dot{r'}\\rho$. Notice that $f_{l}$ is a homeomorphism form $U_{\\ac{u}_{l}}$ to itself, since $h_{A_{q_0}}$ is a homeomorphism on $\\CCmr{n,i}$, and $\\ac{u}_{l}$ is an antichain. Moreover since $A_{q_0} \\in \\WTOns{i}$ and the antichain $\\ac{u}_{l}$ is lexicographically ordered, we have that $f_{l}$ preserves the lexicographic ordering on $U_{\\ac{u}_{l}}$. Now let $f: \\CCmr{n,j} \\to \\CCmr{n,j}$ be such that for $1 \\le l \\le m$ $f\\restriction_{U_{\\ac{u}_{l}}} = f_{l}$. Then $f$ is a homeomorphism from $\\CCmr{n,j}$ to itself, since for $1 \\le l \\le m$ $f_{l}$ is a homeomorphism from $U_{\\ac{u}_{l}}$ to itself, and $\\ac{u}$ is a complete antichain for $\\CCmr{n,j}$. Moreover $f$ preserve the lexicographic ordering of $\\CCmr{n,j}$ since each of th $f_{l}$'s ($1\\le l \\le m$) preserve the lexicographic ordering on $U_{\\ac{u}_{l}}$. Furthermore observe that if $B_{p_0}$ is a minimal, initial automaton such that $h_{B_{p_0}} = f$, then $B_{p_0}$ is finite, synchronizing, and $\\core(B_{p_0}) = g$. This is because each $f_{l}: U_{\\ac{u}_{l}} \\to U_{\\ac{u}_{l}}$ is defined such that after processing a finite prefix it acts as a state of $A_{q_0}$. Thus $f \\in \\TBmr{n}{,j}$ and so $g \\in \\WTOns{j}$.\n \n Since $g \\in \\WTOns{i}$ was arbitrary, we therefore conclude that $\\WTOns{i} \\subseteq \\WTOns{j}$. However since $\\pi_{R,n} \\in \\TOns{i} \\cap \\TOns{j}$, we therefore have that $\\TOns{i} \\subseteq \\TOns{j}$ as required.\n\\end{proof}\n\\end{comment}\n\n\\begin{Remark}\nObserve that the lemma above implies that whenever $j \\in \\Z_{n}$ is co-prime to $n-1$ then $\\TOns{j} = \\TOns{1}$ [$\\Ons{j} = \\Ons{1}$]. Thus it follows that for $n$ a natural number bigger than $2$ such that $n-1$ is prime, if $T \\in \\TOn$ [$T \\in \\On$], satisfies $(T)\\sig \\not\\equiv 1 \\mod{n-1}$ then $T \\in \\TOn \\backslash \\{\\TOns{1}\\}$[$T \\in \\On \\backslash \\{\\Ons{1}\\}$]. The following result generalises this observation.\n\\end{Remark}\n\n\\begin{comment}\n\\begin{proposition}\\label{Proposition:groupofunitscyclicimpliesonlyOn1andOnnminus1}\nLet $n \\in \\N_{2}$ and suppose that the group of units of $\\Z_{n-1}$ is cyclic. Then for any $1 \\le r \\le n-1$ either $\\TOnr = \\TOns{1}$ or $\\TOnr= \\TOns{n-1}$ [either $\\Onr = \\Ons{1}$ or $\\Onr= \\Ons{n-1}$]\n\\end{proposition}\n\\begin{proof}\nWe prove the first reading of the proposition, as the second reading follows by mechanical substitutions of $\\TOn$ with $\\On$ in the arguments below. \n\nLet $1 \\le r \\le n-1$ if there is no $T \\in \\TOnr$ such that $(T)\\rsig \\ne 1$, then $\\TOnr = \\TOns{1}$. Thus, assume that there is some $T \\in \\TOnr$ such that $(T)\\rsig \\ne 1$. Now observe that for any $m \\in \\N$, we have $r((T)\\rsig)^m \\equiv r \\mod{n-1}$ since $r(T)\\rsig \\equiv r \\mod{n-1}$ by Proposition~\\ref{Proposition:sigdeterminesmembership}. Now since $\\gen{(T)\\rsig}=\\{((T)\\rsig)^m \\mod{n-1} \\mid m \\in \\N \\}$ is precisely the group of units of $\\Z_{n-1}$, the result follows by Proposition~\\ref{Proposition:sigdeterminesmembership} once again.\n\\end{proof}\n\\end{comment}\n\nAn immediate corollary of the Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} is the following result, an analogous result appears in \\cite{BCMNO} for the groups $\\Onr$:\n\n\\begin{Theorem}\\label{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}\nLet $n \\in \\N$, $n \\ge 2$, then for all non-zero $i \\in \\Z_{n}$ we have $\\TOns{i} \\subseteq \\TOns{n-1}$. Hence $\\TOns{n-1} = \\TOn$ and $\\TOns{1} \\subseteq \\TOns{i}$ for all $1 \\le i \\le n-1$, hence $\\cap_{1 \\le r \\le n-1} \\TOns{r} = \\TOns{1}$.\n\\end{Theorem}\n\\begin{proof}\nLet $i$ be non-zero in $\\Z_{n}$, then observe that $(n-1)\\ast i \\equiv n-1 \\mod{n-1}$ and $i*1 \\equiv i \\mod{n-1}$. Therefore by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} above, we have $\\TOns{i} \\subseteq \\TOns{n-1}$ and $\\TOns{1} \\subseteq \\TOns{i}$. Thus $\\TOns{n-1} = \\TOn$ and $\\cap_{1 \\le r \\le n-1} \\TOns{r} = \\TOns{1}$.\n\\end{proof}\n\nThe following result is again a corollary of Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}, we observe that the result for $\\Onr$ was proved in \\cite{BCMNO}:\n\n\\begin{corollary}\\label{Corollary:TOniisequaltoTOnjforsomejdividingnminus1}\nLet $n$ be a natural number bigger than $2$, $j \\in \\Z_{n}\\backslash \\{0\\}$ and $d$ be the greatest common divisor of $n-1$ and $j$, then $\\TOns{j} = \\TOns{d}$ [$\\Ons{j} = \\Ons{d}$]. \n\\end{corollary}\n\\begin{proof}\nLet $n,j,d$ be as in the statement of the corollary. Since $d$ is the greatest common divisor of $n-1$ and $j$, there are co-prime numbers $a, b \\in \\Z_{n}\\backslash\\{0\\}$ such that $j = da$ and $n-1 = db$. Since $a$ and $b$ are co-prime, there are numbers $u, v \\in \\Z$ such that $ua = 1 + vb$. Multiplying both sides of the equation by $d$ it follows that $uad = d + vbd$ and so $uj = d + v(n-1)$. Therefore there is some $m_1 \\in \\Z_{n-1}$ such that $m_1j \\equiv d \\mod {n-1}$. Moreover, since $d$ divides $j$, there is some $m_2 \\in \\Z_{n-1}$ such that $m_2 d = j \\mod{n-1}$. It therefore follows by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} that $\\TOns{j} = \\TOns{d}$[$\\Ons{j} = \\Ons{d}$].\n\\end{proof}\n\nCorollary~\\ref{Corollary:TOniisequaltoTOnjforsomejdividingnminus1} should be compared with the result of Pardo \\cite{EPardo} showing that $G_{n,r} \\cong G_{m,s}$ if and only if $n=m$ and $\\gcd(n-1,r) = \\gcd(n-1,s)$. It is a question in \\cite{BCMNO} whether or not $\\Ons{r} \\cong \\Ons{s}$ if and only if $\\gcd(n-1,r) = \\gcd(n-1,s)$. Below (Remark~\\ref{Remark:negativesolutiontoquestionofCollinetal}) we show that this question has a negative answer. \n\nThe following Lemma can be thought of as a partial converse to Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}.\n\n\n\\begin{lemma}\\label{Lemma:partialconverselemma}\nLet $T \\in \\TOn$ [$T \\in \\On$] and let $r$ be minimal such that $T \\in \\TOnr$ [$T \\in \\Onr$], then $T \\in \\TOns{j}$ [$T \\in \\Ons{j}$] for some $1 \\le j \\le n-1$ if and only if there is some $m \\in \\Z_{n}$ such that $mr = j$.\n\\end{lemma}\n\\begin{proof}\nThe forward implication is a consequence of Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}. Therefore let $T \\in \\TOn$ and $r$ be minimal such that $T \\in \\TOnr$ Let $1 \\le j \\le n-1$ be such that $T \\in \\TOns{j}$. By assumption we must have that $r < j$. Since $T \\in \\TOnr$ then by Proposition~\\ref{Proposition:sigdeterminesmembership}, $r(T)\\sig \\equiv r \\mod{n-1}$ and since $T \\in \\TOns{j}$, then $j (T)\\sig \\equiv j \\mod{n-1}$. Thus we deduce that $(j-r)(T)\\sig \\equiv (j-r) \\mod{n-1}$. If $(j-r) > r$, then we may repeat the process otherwise $j-r = r$, by the minimality assumption on $r$ and Proposition~\\ref{Proposition:sigdeterminesmembership}, in which case $j = 2r$. Inductively there is some $k \\in \\N$ such that $j-kr = r$ and so $j = (k+1)r$ which concludes the proof. \n\nThe other reading of the lemma is proved analogously, simply replace $\\TOn$ with $\\On$ in the paragraph above.\n\\end{proof}\n\nWe now show that the map $\\rsig: \\On \\to \\Z_{n-1}$ is a homomorphism from $\\On$ to the group of units of $\\Z_{n-1}$ with kernel $\\Ons{1}$.\n\nWe begin with the following result:\n\n\\begin{proposition}\\label{Proposition:signatureisequaltosizeofimagecovermodnminus1}\nLet $T \\in \\On$, $q$ be any state of $T$ and $m_q$ be the size of the smallest subset $V$ of $\\Xns$ such that $U(V) = \\{ U_v \\mid v \\in V\\}$ is a clopen cover of $\\im(q)$ and $U_{v} \\subset \\im(q)$ for all $v \\in V$, then $m_{q} \\mod{n-1} = (T)\\rsig$.\n\\end{proposition}\n\\begin{proof}\n\nFor any state $p$ of $T$ let $V(p)= \\{v_{1,p}, v_{2,p}, \\ldots\n, v_{m_p,p}\\} \\subset \\Xns$ be such that $\\im(p) = \\cup_{1 \\le i \\le m_{p}} U_{v_{i,p}}$. Now fix a state $q$ of $T$, let $k$ be the minimal synchronizing level of $T$ and $j \\in \\N_{k}$ be such that for any word $\\Gamma \\in \\Xn^{j}$ and any state $p$ of $T$, $|\\lambda_{T}(\\Gamma, p)| \\ge \\max\\{ |v_{1,q}| \\mid v_{1,q} \\in V(q) \\}$. Order the set $X_{n}^{j}$ in the lexicographic ordering as follows $x_1 < x_2 < x_3 \\ldots < x_{n^{j}}$, for $1 \\le a \\le n^{j}$ let $q_{x_a}$ be the unique state of $T$ forced by $x_a$ and $\\rho_{x_a} = \\lambda_{A}(x_a, q)$. Observe that for all $1 \\le a \\le n^{j}$, $|\\rho_{x_a}| \\ge \\max\\{ |v_{1,q}| \\mid v_{1,q} \\in V(q) \\}$ and so $\\rho_{x_a}$ has a prefix in $V(q)$. Now since the state $q$ is induces a homeomorphism from $\\CCn$ unto its image, it follows that the set $\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}$ is a complete antichain for the subset of $\\Xns$ consisting of all elements with prefix in $V(q)$. From this we deduce that $|\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}| \\equiv m_q \\mod{n-1}$.\n\nOn the other hand \n\\[\n|\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}| = \\sum_{1 \\le a \\le n^{j}} m_{q_{x_a}}.\n\\]\nHowever, since $T$ is synchronizing at level $k$, the sequence $(q_{x_a})_{1 \\le a \\le n^j}$ is in fact equal to the sequence $(q_{x_a})_{(1 \\le a \\le n^{k})}$ repeated $n^{j-k}$ times. Therefore we have,\n\\[\n|\\cup_{1 \\le a \\le n^{j}} \\{\\rho_{x_a}v_{i,q_{x_a}} \\mid 1 \\le i \\le m_{q_{x_a}} \\}| = n^{j-k}\\sum_{1 \\le a \\le n^{k}} m_{q_{x_a}} \n\\]\n and so we conclude that $m_{q} \\mod {n-1} = n^{j-k}\\sum_{1 \\le a \\le n^{k}} m_{q_{x_a}} \\mod{n-1} = (T)\\rsig$. \n\\end{proof}\n\n\\begin{Theorem}\\label{Theorem:rsigisahomomorphism}\nThe map $\\rsig: \\On \\to \\Z_{n-1}$ is a homomorphism from $\\On$ into the group of units of $\\Z_{n-1}$ with kernel $\\Ons{1}$.\n\\end{Theorem}\n\\begin{proof}\nLet $T, U \\in \\On$, $j, k \\in \\N$ be the minimal synchronizing levels of $T$ and $U$. Consider the transducer product $T \\ast U$. Let $(p,q)$ be any state in the core of $T \\ast U$, and for $\\sharp \\in \\{p,q\\}$ let $V(\\sharp)= \\{v_{1,\\sharp}, v_{2,\\sharp}, \\ldots\n, v_{m_p,\\sharp}\\} \\subset \\Xns$ be such that $\\im(\\sharp) = \\cup_{1 \\le i \\le m_{\\sharp}} U_{v_{i,\\sharp}}$. We may assume that $V_{q}$ is the smallest subset of $\\Xns$ with this property. Let $m = \\max\\{ |v_{i,q} | \\mid v_{i,q} \\in V(q) \\}$ and $l \\in \\N$ be such that for any state $p'$ of $T$ and any word $\\Gamma \\in \\Xn^{l}$, $|\\lambda_{T}(\\Gamma, p')| \\ge m$. We may further assume that $V(p)$ is the smallest subset of $\\Xn^{l}$ satisfying $\\im(p) = \\cup_{1 \\le i \\le m_{p}} U_{v_{i,p}}$. For each $1 \\le i \\le m_{p}$ let $q_i = \\pi_{U}(v_{i,p}, q)$ and $\\rho_i = \\lambda_{U}(v_{i,p}, q)$. Since $\\im(p) = \\cup_{1 \\le i \\le m_{p}} U_{v_{i,p}}$, it follows that $\\im((p,q)) = \\sqcup_{1 \\le i \\le m_{p}} \\{ \\rho_i \\im(q_i)\\}$. For each $1 \\le i \\le m_{p}$, let $m_{q_i}$ be the size of the smallest subset $V(q_i) \\subset \\Xns$ such that $\\cup_{v \\in V(q_i)} U_{v} = \\im(q_i)$, it follows that $V:=\\cup_{1 \\le i \\le m_p} \\{ \\rho_i v \\mid v \\in V(q_i) \\}$ satisfies $\\cup_{v \\in V}(U_{v}) = \\im((p,q))$. Notice that since both $p$ and $q$ are injective, then $|V| = \\sum_{1 \\le i \\le m_{p}} m_{q_i}$.\n\nObserve for any other set $V' \\subset \\Xns$ such that $\\cup_{v \\in V'} U_{v} = \\im((p,q))$ it must be the case that $|V'| \\equiv |V| \\mod{n-1}$. Thus if $V'$ is the smallest subset of $\\Xns$ with this property, we have $|V'| \\equiv |V| \\mod{n-1}$. Furthermore we observe that removing incomplete response from the states of $T\\ast U$ simply removes the greatest common prefix of $\\im(p,q)$. Therefore if the state $s$ of $TU$ is equal to the state $(p,q)$ after removing the incomplete response, then for $V'' \\subset \\Xns$ minimal such that $\\cup_{v \\in V''} U_{v} = \\im(s)$, we have $|V''| \\equiv |V| \\mod{n-1}$. By Proposition~\\ref{Proposition:signatureisequaltosizeofimagecovermodnminus1} it follows that\n\\[\n (TU)\\rsig \\equiv |V''| \\equiv \\sum_{1 \\le i \\le m_{p}} m_{q_i} \\equiv (T)\\rsig(U)\\rsig\\mod{n-1}.\n\\]\n\nSince $(\\id)\\rsig = 1$, it follows that $\\rsig$ is a homomorphism from $\\On$ to the group of units of $\\Z_{n-1}$.\n\nTo see that $\\ker(\\rsig) = \\Ons{1}$, observe that by Proposition~\\ref{Proposition:sigdeterminesmembership} $(T)\\rsig = 1$ if and only if $T \\in \\Ons{1}$.\n\n\\end{proof}\n\nGiven $T \\in \\On$ the following result enables us to compute the reduced signature of $T^{-1}$ directly from $T$ i.e. without computing the inverse.\n\n\\begin{proposition}\nLet $T \\in \\On$ and $q$ be any state of $T$. Let $\\nu \\in \\CCn$ be such that $U_{\\nu} \\subset \\im(q)$ and $j \\in \\N$ be such that for any word $\\Gamma \\in \\Xn^{j}$, $|\\lambda_{T}(\\Gamma, q)| \\ge |\\nu|$. Let $W \\subset \\Xn^{j}$ be maximal such that for any word $\\Delta \\in W$, $\\nu$ is a prefix $\\lambda_{T}(\\Delta, \\Gamma)$. Let $1 \\le w \\le n-1$ be such that $w \\equiv |W| \\mod{n-1}$, then $w$ depends only on $T$, in particular $w = (T^{-1})\\rsig$.\n\\end{proposition}\n\\begin{proof}\nBy Proposition~\\ref{Proposition:signatureisequaltosizeofimagecovermodnminus1} it suffices to show that there is a state $p'$ of $T^{-1}$ and a subset $V \\subset \\Xns$ such that $\\cup_{\\mu \\in V} U_{\\mu} = \\im(p)$ and $|V| = |W|$.\n\nLet $A_{q_0}$ a bi-synchronizing transducer with $\\core(A_{q_0}) = T$ and $k \\in \\N_{1}$ be the minimal bi-synchronizing level of $A_{q_0}$. Let $\\varphi = (\\nu)L_{q}$ i.e. $\\varphi$ is the greatest common prefix of the set $h_{q}^{-1}(U_{\\nu})$. Let $\\nu_1 = \\lambda_{T}(\\varphi, q)$, $p = \\pi_{T}(\\varphi, q)$ and $\\nu_2 = \\nu - \\nu_1$. We claim that $(\\nu_2,p)$ is a state of $A_{(\\epsilon, q_0)}$ and is $\\omega$-equivalent to a state of $T^{-1}$.\n\n For, let $l \\in \\N$ be such that for any word $\\Gamma \\in \\Xn^{l}$, $|\\lambda_{A}(\\Gamma, q_0)| \\ge k$.\n Let $\\Gamma \\in \\Xn^{l}$ be such that $\\pi_{A}(\\Gamma, q_0) = q \\in \\core(A_{q_0})$. Such a word $\\Gamma$ exists because of the bi-synchronizing condition. Let $\\Delta = \\lambda_{A}(\\Gamma, q_0)$. Then observe that $(\\Delta\\nu)L_{q_0} = \\Gamma (\\nu)L_{q} = \\Gamma \\varphi$ since $U_{\\nu} \\subset \\im(q)$. Thus, in the transducer $A_{(\\epsilon, q_0)}$, $\\pi_{A}(\\Delta\\nu, q_0) = (\\Delta\\nu - \\lambda_{A}(\\Gamma \\varphi, q_0), \\pi_{A}(\\Gamma\\varphi, q_0))= (\\nu_2,p)$. Now since $A_{(\\epsilon, q_0)}$ has no states of incomplete response, is $\\omega$-equivalent to the minimal transducer $B_{p_0}$ representing $h_{q_0}^{-1}$, and by the choice of $l$, we therefore have that $(\\nu_2,p)$ is $\\omega$-equivalent to a state in $\\core(B_{p_0}) = T^{-1}$.\n \n Now, suppose $W = \\{ \\rho_1', \\rho_2', \\ldots, \\rho_{m}'\\}$. We observe that by definition of $W$, $\\varphi$ is the greatest common prefix of $W$, thus for $1\\le i \\le m$, let $\\rho_i \\in \\Xns$ be such that $\\varphi\\rho_i = \\rho_i'$. Observe that for any $\\delta \\in \\CCn$, there is a unique $1 \\le i \\le m$, and $\\xi \\in \\CCn$ such that $\\lambda_{A}(\\rho_i'\\xi, q) = \\nu \\delta$, therefore $\\lambda_{A}(\\rho_i \\xi, p) = \\nu_2\\delta$. Moreover, for any $1 \\le i \\le m$ and $\\xi \\in \\CCn$, $\\lambda_{A}(\\rho_i\\xi, p) = \\nu_2 \\delta$ for some $\\delta \\in \\CCn$. Since $\\im((\\nu_2,p)) = \\{(\\nu_2\\delta)L_{p} \\mid \\delta \\in \\CCn \\} = (U_{\\nu_2})h_{p}^{-1}$, it follows that $\\im((\\nu_2,p)) = \\cup_{1 \\le i \\le m} U_{\\rho_i'}$. Therefore it follows, from Proposition~\\ref{Proposition:signatureisequaltosizeofimagecovermodnminus1}, that $m = |W| \\equiv (T^{-1})\\rsig \\mod{n-1}$.\n \n \n\\end{proof}\n\nThe corollary below follows straight-forwardly from Theorem~\\ref{Theorem:rsigisahomomorphism}\n\n\\begin{corollary}\nLet $1< r< n$, then the following hold:\n\\begin{enumerate}[label = (\\alph*)]\n\\item $\\Onr$ is a normal subgroup of $\\Ons{n-1} = \\On$, in particular for non-zero $i, j \\in \\Z_{n}$ such that $i$ divides $j$ in the additive group $\\Z_{n-1}$, $\\Ons{i} \\unlhd \\Ons{j}$,\n\\item $[\\On, \\On] \\le \\Onr$,\n\\item $\\Onr\/\\Ons{1}$ is isomorphic to a subgroup of the group of units of $\\Z_{n-1}$ and $\\On\/\\Onr$ is isomorphic to a quotient of the group of units of $\\Z_{n-1}$. \n\\end{enumerate}\n \n\\end{corollary}\n\\begin{proof}\nSince the group of units of $\\Z_{n-1}$ is abelian, it follows that $[\\On, \\On] \\le \\Ons{1}$ and by Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1} $[\\On, \\On] \\le \\Ons{r}$.\n\nTo see that $\\Onr$ is normal in $\\On$, we observe that given $T \\in \\Onr$, and $U \\in \\On$, then $(U^{-1}TU)\\rsig \\equiv (U^{-1})\\rsig(T)\\rsig(U)\\rsig \\mod{n-1}$ and so $(U^{-1}TU)\\rsig = (T)\\rsig$ and by Proposition~\\ref{Proposition:sigdeterminesmembership}, we have $U^{-1}TU \\in \\Onr$.\n\nThat $\\Onr\/\\Ons{1}$ is isomorphic to a subgroup of the group of units of $\\Z_{n-1}$, follows by the correspondence theorem since $\\Ons{1} \\le \\Onr$. \n\nThat $\\On\/\\Onr$ is isomorphic to a quotient of the group of units of $\\Z_{n-1}$, follows from the third isomorphism theorem: $\\On\/\\Onr \\cong (\\On\/\\Ons{1})\/(\\Onr\/\\Ons{1})$.\n\\end{proof}\n\n\\begin{Question}\\label{Question:issigsurjective}\nIs it the case that $[\\On, \\On] = \\Ons{1}$? Is the map $\\rsig$ from $\\On$ to the group of units of $\\Z_{n-1}$ surjective?\n\\end{Question}\n\n\\begin{Remark}\nSince $\\TOn \\le \\On$ the results above and questions can be restated with $\\TOn$ in place of $\\On$.\n\\end{Remark}\n\n\\section{ Nesting properties of the groups $\\TOnr$}\\label{Section:nestingproperties2}\n\nIn this section we focus on the group $\\TOn$ however, {\\bfseries{all the results below are equally valid when all occurrences of $\\TOn$ are replaced with $\\On$}}.\n\nThe following result is a direct consequence corollary of Proposition~\\ref{Proposition:sigdeterminesmembership} and generalises Corollary~\\ref{Corollary:TOniisequaltoTOnjforsomejdividingnminus1}:\n\n\\begin{corollary}\\label{Corollary:OnrisequaltoOnjifandonlyif}\nLet $1\\le r,s2$, and $1 \\le r, s \\le n-1$ such that, for $\\T{X}= \\T{TO}, \\T{O}$, $\\XOnr = \\XOns{s}$ but $\\gcd(n-1, r) \\ne \\gcd(n-1, s)$.\n\\end{Theorem}\n\nPartition the set $\\Z_{n} \\backslash \\{0\\}$ as follows. For $i \\in \\Z_{n} \\backslash \\{0\\}$, set $[i] := \\{j \\in \\Z_{n} \\backslash \\{0\\} \\mid \\TOns{j} = \\TOns{i}\\} \\subset \\Z_{n}$, $[\\Z_{n}^{0}] \\seteq \\{ [i] \\mid i \\in \\Z_{n} \\backslash \\{0\\} \\}$ and, for $[i] \\in [\\Z_{n}^{0}]$, $\\TOns{[i]} \\seteq \\TOns{i}$. Observe that the set $[\\Z_{n}^{0}]$ inherits an ordering from $\\Z_{n}$ where $[i] < [j]$ if the smallest element of $[i]$ is less than the smallest element of $[j]$. Further observe that if $n-1$ is prime, then $[\\Z_{n}^{0}]$ has size at most $2$ by Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}.\n\n\\begin{Definition}[Atoms]\nLet $[r] \\in [\\Z_{n}^{0}]$, then we say $[r]$ is \\emph{an atom (of $[\\Z_{n}^0]$)} if there is an element $T \\in \\TOns{[r]}$ which is not an element of $\\TOns{[s]}$ for any $[s] < [r]$. Let $[i] \\in [\\Z_{n}^{0}]$, an \\emph{atom of $[i]$} is an atom $[r]$ of $[\\Z_{n}^0]$ such that $\\TOns{[r]} \\le \\TOns{[i]}$. \n\\end{Definition}\n\n\\begin{Remark}\\label{Remark:atomsdividenminus1}\nLet $[r] \\in [\\Z_{n}^0]$ be an atom with $r$ the minimal element of $r$, then as a consequence of Lemma~\\ref{Lemma:partialconverselemma} and Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}, we have that $r|n-1$. Observe that by Remark~\\ref{Remark:negativesolutiontoquestionofCollinetal} it is not always the case that for every element $r$ dividing $n-1$ that $[r]$ is an atom of $[\\Z_{n}^{0}]$. \n\\end{Remark}\n\n\nObserve that if the map $\\rsig$ is surjective, then for each $i \\in \\Z_{n} \\backslash \\{0\\}$, we may completely determine the elements of $[i]$. \n\n\\begin{Question}\nIs it the case that all elements $[i] \\in [\\Z_{n}^{0}]$ are atoms?\n\\end{Question}\n\nIn the interim we make the following definitions.\n\n\\begin{Definition}\nLet $[i],[j] \\in [\\Z_{n}^{0}]$ where $i,j \\in \\Z_{n}$ are the minimal elements of $[i]$ and $[j]$ respectively. We say that $[i]$ \\emph{divides} $[j]$ if $r|j$ for any atom $[r]$ of $[i]$ with $r$ the minimal element of $[r]$.\n\\end{Definition}\n\n\n\\begin{Definition}\nGiven two elements $[i],[j] \\in [\\Z_{n}^0]$, the \\emph{lowest common multiple of $[i]$ and $[j]$} is the smallest element $[l]$ of $[\\Z_{n}^0]$ such that $[i]$ and $[j]$ divide $[l]$. Notice that by Remark~\\ref{Remark:atomsdividenminus1} the lowest common multiple of any pair of numbers $[i], [j] \\in [\\Z_{n}^0]$ always exists. We extend the definition of lowest common multiple to tuples of elements of $[\\Z_{n}^0]$ in the usual way.\n\\end{Definition}\n\n\n\n\\begin{Definition}\n Let $i, j \\in \\Z_{n} \\backslash \\{0\\}$ be the minimal elements of $[i]$ and $[j]$ respectively, then we define the \\emph{greatest common divisor of $[i]$ and $[j]$} to be $[r] \\in [\\Z_{n}^0]$ such that $r$, the minimal element of $[r]$, is the greatest common divisor of $i$ and $j$.\n\\end{Definition}\n\n\\begin{Remark}\nNotice that for $[i], [j], [r] \\in [\\Z_{n}^{0}]$ where $i,j,r$ are the minimal elements of $[i]$, $[j]$ and $[r]$ respectively, if $[r]$ is an atom of $[i]$ then, by Lemma~\\ref{Lemma:partialconverselemma}, $r|i$ and so if $i|j$ then $r|j$ and $[i]|[j]$. Further observe that for $[i] \\in [\\Z_{n}^{0}]$ either $[i]$ is an atom or $\\TOns{[i]}$ is a union of groups $\\TOns{[r]}$ for atoms $[r]$ of $[i]$. This is because for any element $T \\in \\TOns{[i]}$, either $T$ is not an element of $\\TOns{[r]}$ for any $[r]<[i]$ and so $[i]$ is an atom, or there is a minimal $[j] \\in [\\Z_{n}^0]$ such that $T \\in \\TOns{[j]}$ and so $[j]$ is an atom of $[i]$ by Lemma~\\ref{Lemma:partialconverselemma} and Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1}.\n\\end{Remark}\n\n As a consequence of Lemma~\\ref{Lemma:partialconverselemma} we have the following result.\n\n\\begin{proposition}\\label{Proposition:partiallatticestructure}\nLet $1 \\le i, \\le j \\le n-1$ be integers such $i$ is the smallest element of $[i]$ and $j$ is the smallest element of $[j]$, then the following things hold:\n\\begin{enumerate}[label=(\\alph*)]\n\\item $\\TOns{[i]} \\le \\TOns{[j]}$ if and only if $[i]$ divides $[j]$ \\label{Proposition:partiallatticestructure order},\n\\item if $[r]$ is the lowest common multiple of $[i],[j]$, then $[r]$ is minimal such that $\\gen{\\TOns{[i]}, \\TOns{[j]} } \\le \\TOns{[r]}$ \\label{Proposition:partiallatticestructure join},\n\\item if $[r]$, where $r \\in \\Z_{n}$ is the smallest element of $[r]$, is the greatest common divisor of $[i]$ and $[j]$ then $\\TOns{[i]} \\cap \\TOns{[j]} = \\TOns{[r]}$. \\label{Proposition:partiallatticestructure meet}\n\\end{enumerate}\n \n\\end{proposition}\n\\begin{proof}\n For part~\\ref{Proposition:partiallatticestructure order}, let $i, j$ be elements of $\\Z_{n} \\backslash \\{0\\}$ which are the minimal elements of sets $[i]$ and $[j]$ respectively. Suppose $[i]$ divides $[j]$. If $[i]$ is an atom then $i|j$, and so by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} we have that $\\TOns{i} \\le \\TOns{j}$, in particular $\\TOns{[i]} \\le \\TOns{[j]}$. If $[i]$ is not an atom, then $\\TOns{[i]}$ is the union of the groups $\\TOns{[r]}$ over all atoms $[r]$ of $[i]$. Therefore let $[r]$ be any atom of $[i]$, where $r$ is the smallest element of $[r]$, since $[i]$ divides $[j]$ then $r|j$ and so $\\TOns{r} \\le \\TOns{j}$ by Lemma~\\ref{Lemma:TOni is a subset of TOnj if mi is congruent to j mod n-1} once more. Since $[r]$ was an arbitrary atom of $[i]$, we conclude that $\\TOns{[i]} \\le \\TOns{[j]}$. On the other hand suppose that $\\TOns{i} \\le \\TOns{j}$.If $[i]$ is an atom then $i|j$ and so $[i]$ divides $[j]$. If $[i]$ is not an atom, then let $[r]$ be any atom of $[i]$ where $r$ is the minimal element of $[r]$. Since $\\TOns{[i]} \\le \\TOns{[j]}$, we also have $\\TOns{[r]} \\le \\TOns{[j]}$ by definition of an atom of $[i]$. Now making use of the definition of an atom together with Lemma~\\ref{Lemma:partialconverselemma}, we have that $r|j$. Therefore, since $[r]$ was an arbitrarily chose atom of $[i]$, we conclude that $[i]$ divides $[j]$.\n \n Part~\\ref{Proposition:partiallatticestructure join} follows from Part~\\ref{Proposition:partiallatticestructure order} since for any $s \\in \\Z_{n}\\backslash\\{0\\}$ such that $\\gen{\\TOns{[i]}, \\TOns{[j]}}\\le \\TOns{[s]}$, then $[i]$ and $[j]$ divide $[s]$. \n \n For Part~\\ref{Proposition:partiallatticestructure meet} let $[r]$ be the greatest common divisor of $[i]$ and $[j]$. First observe that if $[l]$ is an atom of $[r]$, where $l$ is the minimal element of $[l]$, then $l|i$ and $l|j$. Therefore $[r]$ divides $[i]$ and $[j]$ and by Part~\\ref{Proposition:partiallatticestructure order}, $\\TOns{[r]} \\le \\TOns{[i]} \\cap \\TOns{[j]}$. Now let $[s]$, with $s$ the minimal element of $[s]$ be an atom of $[i]$ and $[j]$. This means that $s|i$ and $s|j$ and so $s|r$, since $r$ is the greatest common divisor of $i$ and $j$. By Part~\\ref{Proposition:partiallatticestructure order} once more, we conclude that $s$ is an atom of $[r]$ and so $\\TOns{[s]} \\le \\TOns{[r]}$. Now since every element $T \\in \\TOns{[i]} \\cap \\TOns{[j]}$ lies in some atom of both $[i]$ and $[j]$, we conclude that $\\TOns{[i]} \\cap \\TOns{[j]} = \\TOns{[r]}$.\n\\end{proof}\n\nThe proposition above prompts the following question:\n\n\\begin{Question}\\label{Question:canjoinbereplacedwithubgroupgenerated}\nLet $n$ be a natural number and let $[i], [j] \\in [\\Z_{n}^0]$ is it true that $\\gen{\\TOns{[i]}, \\TOns{[j]}} = \\TOns{[r]}$ where $[r]$ is the lowest common multiple of $[i]$ and $[j]$? \n\\end{Question}\n\nThe proposition below addresses this question by showing that in the case where the map $\\rsig$ is unto the group of units of $\\Z_{n-1}$, then Question~\\ref{Question:canjoinbereplacedwithubgroupgenerated} has an affirmative answer.\n\n\\begin{proposition}\nLet $\\T{X}= \\T{TO}, \\T{O}$ and suppose the map $\\rsig: \\XOn \\to \\Z_{n-1}$ is unto the group of units of $\\Z_{n-1}$. Then for $[i],[j] \\in [Z_{n}^0]$, $\\gen{\\XOns{[i]},\\XOns{[j]}} = \\XOns{[r]}$ for $[r]$ the lowest common multiple of $[i]$ and $[j]$.\n\\end{proposition}\n\\begin{proof}\nThe result is essentially a consequence of the following claim, as we have not been able to find a reference for it, we also provide a proof.\n\n\\begin{claim}\\label{Claim:groupgeneratedbypairequallcmgroup}\nLet $n \\in \\N_{2}$ and consider the integer ring $\\Z_{n}$. Let $\\Z_{n}^{*}$ denote the group of units, and for each $i \\in \\Z_{n}$ let $(\\Z_{n}^{*})_{i} := \\{ a \\in \\Z_{n}^{*} \\mid ai \\equiv i \\mod{n}\\} \\le \\Z_{n}^{*}$. Let $i,j \\in \\Z_{n}$ let $i_1 = \\gcd(i,n)$, $j_1 = \\gcd(j,n)$ and $r = lcm(i_1,j_1)$ then $ \\gen{ (\\Z_{n}^{*})_{i_1}, (\\Z_{n}^{*})_{j_1}} = \\gen{ (\\Z_{n}^{*})_{i}, (\\Z_{n}^{*})_{j}} = (\\Z_{n}^{*})_{r}$.\n\\end{claim}\n\\begin{proof}\nWe first observe that for $i \\in \\Z_{n}$ the group $(\\Z_{n}^*)_{i} = (\\Z_{n}^*)_{d}$ where $d = \\gcd(i, n)$. This is because for any $a \\in \\Z_{n}^{*}$ such that $ad \\equiv d \\mod{n-1}$ then $ai \\equiv i \\mod{n-1}$ as well. If, on the other hand $ai \\equiv i \\mod{n-1}$, then observe that, as, by Bezout's lemma, there is a $u \\in \\Z_{n}$ such that $iu \\equiv d \\mod{n-1}$, then $ad \\equiv d \\mod{n-1}$ also.\n\nThus let $i, j \\in \\Z_{n}$ be divisors of $n$ with $r = lcm(i,j)$ (a divisor of $n$). First assume that $n = p^{\\alpha}$ for a prime $p$. Then $i = p^{\\beta}$ and $j = p^{\\gamma}$ for $\\beta, \\gamma \\le \\alpha$. Without loss of generality we may assume that $i \\le j$, and so, in particular, that $i$ divides $j$. However, it then follows that $(\\Z_{n}^{*})_{i} \\le (\\Z_{n}^{*})_{j}$, from which we conclude that $\\gen{ (\\Z_{n}^{*})_{i}, (\\Z_{n}^{*})_{j}} = (\\Z_{n})^{*}_{j}$ noting that $lcm(i,j) = j$.\n\nNow suppose that $n= p_1^{\\alpha_{1}}p_2^{\\alpha_{2}}\\ldots p_m^{\\alpha_{m}}$ where the $p_{l}$'s, $1 \\le l \\le m$ are distinct primes. We may further assume that $\\gcd(i,j) = 1$. This is because if $\\gcd(i,j) = d$, then setting $i_1 = i\/d$ and $j_1 = j\/d$ we observe that $(\\Z_{n}^{\\ast})_{i_1} \\le (\\Z_{n}^{\\ast})_{i}$, $(\\Z_{n}^{\\ast})_{j_1} \\le (\\Z_{n}^{\\ast})_{j}$ and $lcm(i,j) = lcm(i_1, j_1)$. Thus, noting that $\\gen{ (\\Z_{n}^{\\ast})_{i}, (\\Z_{n}^{\\ast})_{j}} \\le \\gen{(\\Z_{n}^{\\ast})_{lcm(i,j)}}$, the result holds for $i_1$ and $j_1$ precisely if it holds for $i$ and $j$. \n\nMaking use of the Chinese remainder theorem, there is a ring isomorphism from $\\Z_{n} \\to \\Z_{p_1^{\\alpha_{1}}} \\times \\Z_{p_2^{\\alpha_{2}}} \\times \\ldots \\times \\Z_{p_m^{\\alpha_{m}}}$ defined by $k \\mapsto (k\\mod{p_1^{\\alpha_1}},k\\mod{p_2^{\\alpha_1}}, \\ldots, k\\mod{p_m^{\\alpha_m}})$. For $1 \\le l \\le m$ let $i_l = i \\mod{p_l^{\\alpha_{m}}}$ likewise define the sequence $j_{l}$ and $r_l$, where $r = ij$. It follows that $(\\Z_{n}^{\\ast})_{i} \\cong (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{i_1} \\times (\\Z_{p_1^{\\alpha_{2}}}^{\\ast})_{i_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{i_m}$, likewise $(\\Z_{n}^{\\ast})_{j} \\cong (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_1} \\times (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{j_m}$. Observe that $$\\gen{\\left((\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{i_1} \\times (\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{i_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{i_m}\\right),\\left((\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_1} \\times (\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{j_2} \\times \\ldots \\times (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{j_m}\\right) }$$ is precisely the group $$\\gen{(\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{i_1}, (\\Z_{p_1^{\\alpha_{1}}}^{\\ast})_{j_1}} \\times \\gen{(\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{i_2}, (\\Z_{p_2^{\\alpha_{2}}}^{\\ast})_{j_2}} \\times \\ldots \\times \\gen{(\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{i_m}, (\\Z_{p_m^{\\alpha_{m}}}^{\\ast})_{j_m}}.$$ Since $i$ and $j$ are divisors of $n$ then $i = p_1^{\\beta_1}p_2^{\\beta_2} \\ldots p_{m}^{\\beta_m}$, $j= p_1^{\\gamma_1}p_2^{\\gamma_2} \\ldots p_{m}^{\\gamma_m}$ where for $1 \\le l \\le m$, $0 \\le \\beta_l, \\gamma_{l} \\le \\alpha_{l}$ and $\\gamma_{l}+ \\beta_{l} = \\max\\{\\gamma_{l}, \\beta_{l}\\}$ (since $i,j$ are assumed co-prime). It therefore follows that $i \\mod p_l^{\\alpha_{l}}$ is either a unit of $\\Z_{p_l^{\\alpha_{l}}}$ otherwise it is equal to a unit of $\\Z_{p_l^{\\alpha_{l}}}$ times a non-trivial (appropriate) power of $p_l$ and likewise for $j$. We note that if one of $i_l$ or $j_l$ is equal to a unit of $\\Z_{p_l^{\\alpha_{l}}}$ times a non-trivial (appropriate) power of $p_l$, then the other must be equal to a unit of $\\Z_{p_l^{\\alpha_{l}}}$. Therefore for $1 \\le l \\le m$, $\\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$ is either equal to the trivial group or it is equal to $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{p_l^{\\delta}}$ where $p_l^{\\delta} \\ne 1$ is the maximum power of $p_l$ dividing $ij$. In the case that $\\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$ is the trivial group, then $i_l$ and $j_l$ are both units of $\\Z_{p_l^{\\alpha_{l}}}$ and so $r_l = i_lj_l \\mod p_{l}^{\\alpha_{l}}$ is also a unit of $\\Z_{p_l^{\\alpha_{l}}}$. In the case that $\\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$ is equal to $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{p_l^{\\delta}}$ where $p_l^{\\delta} \\ne 1$ is the maximum power of $p_l$ dividing $ij$, then it follows that one of $i_l$ or $j_l$ is a unit and the other is equal to a unit times $p_{l}^{\\delta}$. In this case we have that $r_l$ is equal to a unit time $p_{l}{\\delta}$ and so $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{p_l^{\\delta}} = (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{r_l}$. In either case we see that $(\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{r_l} = \\gen{ (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{i_l}, (\\Z_{p_l^{\\alpha_{l}}}^{\\ast})_{j_l} }$. Thus we conclude that that $\\gen{(\\Z_{n}^{\\ast})_{i}, (\\Z_{n}^{\\ast})_{j}} = (\\Z_{n}^{\\ast})_{r}$ as required.\n\\end{proof}\n\nWe can now prove the proposition. We observe that for $1 \\le r < n$ since $\\rsig$ is surjective, then $(\\XOnr)\\rsig = (\\Z_{n-1}^*)_{r}$. Let $1 \\le i,j \\le n-1$ such that $i$ the the minimal element of $[i]$ and $j$ is the minimal element of $[j]$ so that both $i$ and $j$ are divisors of $n$. Let $r = lcm(i,j)$. By Claim~\\ref{Claim:groupgeneratedbypairequallcmgroup} we have that $\\gen{(\\Z_{n-1}^*)_{i}, (\\Z_{n-1}^*)_{j}} = (\\Z_{n-1}^*)_{r}$. Thus, let $T \\in \\XOnr$ be any element. By Proposition~\\ref{Proposition:sigdeterminesmembership} $(T)\\rsig \\in (\\Z_{n-1}^*)_{r}$ and so there are elements $u \\in (\\Z_{n-1}^*)_{i}$ and $v \\in (\\Z_{n-1}^*)_{j}$ such that $(T)\\rsig = uv \\mod{n-1}$. Let $U \\i \\XOns{i}$ and $V \\in \\XOns{j}$ be such that $u= (U)\\rsig$ and $v = (V)\\rsig$, then $(TU^{-1}V^{-1})\\rsig = 1$ and so $TU^{-1}V^{-1} \\in \\XOns{1} \\le \\gen{\\XOns{i}, \\XOns{j}}$. It therefore follows that $\\XOns{r}\\le \\gen{\\XOns{i}, \\XOns{j}}$. Proposition~\\ref{Proposition:sigdeterminesmembership} guarantees that $\\gen{\\XOns{i}, \\XOns{j}} \\le \\XOns{r}$ Thus by Proposition~\\ref{Proposition:partiallatticestructure} $[r]$ is the lowest common multiple of $[i]$ and $[j]$.\n\\end{proof}\n\n \n \nWe conclude our investigation of the nesting properties of the groups $\\TOns{r}$ for $1 \\le r < n$ by making use of Proposition~\\ref{Proposition:sigdeterminesmembership} to construct an element of $\\T{TO}_{4}$ which is not an element of $\\T{TO}_{4,r}$ for any $1 \\le r \\le 3$. This indicates that in general the group $\\TOns{r}$ depends on $r$.\n\nLet $g$ be the transducer below:\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,node distance=3cm,on grid,auto] \n \\node[state] (q_0) {$a$};\n \\node[state, xshift=0.5cm] (q_1) [right=of q_0] {$b$}; \n \\path[->] \n (q_0) edge[in=170, out=10] node {$1|0$, $3|1$} (q_1)\n edge[in=180, out=150,loop] node[swap] {$0|0$} ()\n edge[in=180, out=210, loop] node {$2|1$} () \n (q_1) edge[in=350, out=190] node {$0|2$, $2|3$} (q_0)\n edge[in=0, out=30,loop] node {$3|3$}()\n edge[in=0, out=330,loop] node[swap] {$1|2$}();\n\\end{tikzpicture}\n\\caption{An element $g \\in \\T{TO}_{4}$ which is not in $\\T{TO}_{4,r}$ for any $0 \\le r <3$}.\n\\label{Figure:elementinTo4butnotinanyotherTOnr}\n\\end{figure}\n\n\nWe make the following observations about $g$ which the reader may verify:\n\\begin{enumerate}[label=(\\roman*)]\n\\item For all states $\\alpha$ of $g$ the map $g_{\\alpha}: \\CCn \\to \\CCn$ preserves the lexicographic ordering on $\\CCn$. Thus, since $g^2 = \\id$ under the product defined for $\\TOn$, by Theorem~\\ref{Thm: Equivalent conditions for an element of $On$ to belong to TOnr} there is some non-zero $r \\in \\Z_{4}\\backslash$ such that $g \\in \\TOnr$. \\label{Observation:gisanelementofTOn}\n\\item We have $\\im(a) \\cap \\im(b) = \\emptyset$ and $\\im(a)\\sqcup\\im(b) = \\CCmr{4}$. In particular we have $\\im(a) = U_{0} \\cup U_{1}$ and $\\im_{b} = U_{2}\\cup U_{3}$.\\label{Observation:gneedstostatestofillacone}\n\n\\item The following fact is not essential to our discussion here but is nevertheless worth mentioning. The paper forthcoming article \\cite{BleakCameronOlukoya} shows that there is subgroup $\\T{L}_{n} \\le \\On$ which is isomorphic to the quotient $\\aut{\\Xn^{\\Z}, \\sigma_{n}}\/\\gen{\\sigma_{n}}$ of the automorphisms of the two-sided full shift on $n$ letters by the group generated by the shift ($\\sigma_{n}$ denotes the shift map on $n$ letters). The transducer $g$ is in fact an element of $\\T{L}_{4}$. \n\\end{enumerate}\n\n\\begin{lemma}\nLet $1 \\le r \\le 3$ and let $A_{q_0} \\in \\TBmr{4}{r}$ be such that $\\core(A_{q_0})$ is equal to the transducer $g$ of Figure~\\ref{Figure:elementinTo4butnotinanyotherTOnr}, then $r = 3$.\n\\end{lemma}\n\\begin{proof}\n\nWe compute $(g)\\sig$. First observe that $g$ is synchronizing at level 1 and moreover the states of $g$ forced by $0$ and $2$, $q_{0}$ and $q_{2}$ respectively are both equal to $a$ and the states of $g$ forced by $1$ and $3$, $q_{1}$ and $q_3$ respectively are both equal to $b$. Further observe that $m_{a} = 2$ and $m_{b} = 2$ (where $m_{a}$ and $m_{b}$ are as in Definition~\\ref{Definition:signature}). Therefore $(g)\\sig = 2.4 = 8 \\equiv 2 \\mod 3$. The only number $r \\in \\Z_{4} \\backslash \\{0\\}$ such that $ 2r \\equiv r \\mod{n-1}$ is $3$. Therefore by Proposition~\\ref{Proposition:sigdeterminesmembership} we conclude that if $g \\in \\TBmr{4}{r}$ then $r$ must be equal to 3.\n\n\\begin{comment}\nLet $A_{q_0}$ be as in the statement of the lemma and $k \\in \\N$ be the synchronizing level of $A_{q_0}$. Let $Q_{g} \\subset Q_{A}$ be the states of $g$. Since $k$ is the synchronizing level of $A$, for any state $q \\in Q_{A}$ and any word $\\Gamma \\in \\Xnr^{k}$, $\\pi_{A}(\\Gamma, q) \\in Q_{g}$. Let $P_{IN}:= \\Xnr^{k+1}$ and let $P_{OUT} = \\{\\lambda_{A}(\\Delta, q_0) \\mid \\Delta \\in \\Xnr^{k+1} \\}$. Since, for $(i,j) \\in \\{(0,1), (2,3)\\}$ and $\\alpha \\in \\{a, b\\}$, we have $\\pi_{g}(i, \\alpha) = a$, $\\pi_{g}(j, \\alpha) = b$ and $\\lambda_{A}(i, \\alpha) = \\lambda_{A}(j, \\alpha)$, then, for every element $\\Delta$ of $P_{OUT}$, there are two elements $\\Gamma i, \\Gamma j$, $(i,j) \\in \\{(0,1), (2,3)\\}$, such that $ \\lambda_{A}(\\Gamma i, q_0) = \\lambda_{A}(\\Gamma i, q_0) = \\Delta$ and $(\\pi_{A}(\\Gamma i, q_0),\\pi_{A}(\\Gamma i, q_0)) = (a,b)$. Let $\\Delta \\in P_{OUT}$ and $\\Gamma i, \\Gamma j \\in P_{IN}$, $(i,j) \\in \\{(0,1), (2,3)\\}$, be such that $ \\lambda_{A}(\\Gamma i, q_0) = \\lambda_{A}(\\Gamma i, q_0) = \\Delta$ and $(\\pi_{A}(\\Gamma i, q_0),\\pi_{A}(\\Gamma i, q_0)) = (a,b)$. By point~\\ref{Observation:gneedstostatestofillacone} in the list of observations about $g$ and since $A_{q_0}$ represents a homeomorphism of $\\CCnr$, it must be the case that, for $\\Theta \\in P_{IN}$ such that $\\Theta \\ne \\{\\Gamma i, \\Gamma j\\}$, $\\lambda_{A}(\\Theta, q_0)$ is incomparable to $\\Delta$. Now since $A_{q_0}$ is a homeomorphism and $g$ is synchronous, then for any element of $\\rho \\in \\CCnr$ there is an element $\\nu \\in P_{OUT}$ such that $\\nu \\le \\rho$. Therefore, we deduce that $P_{OUT}$ is in fact a complete antichain for $\\Xnrs$ moreover, $2|P_{OUT}| = |P_{IN}|$. Since any complete antichain $P$ of $\\Xnrs$ satisfies $|P| \\equiv r \\mod{3}$, and since $P_{IN}$ is a complete antichain of $\\Xnrs$, we have $2r \\equiv r \\mod{3}$. Hence $r \\equiv 0 \\mod{3}$ and, since $1 \\le r \\le 3$, we conclude that $r =3$.\n\\end{comment}\n\\end{proof}\n\nIn the next section we address the question of the surjectivity of the homomorphism $\\rsig$.\n\n\\section{On the surjectivity of \\texorpdfstring{$\\rsig$}{Lg}}\\label{Section:onsurjectivityofrsig}\n\nIn this section we show that there are infinitely many numbers $n \\in \\N$ such that the map $\\rsig$ from $\\On$ to the group of units of $\\Z_{n-1}$ is surjective. More specifically, for $n \\in \\N$ we show that for every divisor $d$ of $n$, $1 \\le d |A_{l+1}|$ or $A_{l} = A_{l+1}$. Thus let $k$ be minimal in $\\N$ such that $A_{k} = A_{k+1}$. The automaton $A$ is synchronizing if and only if $A_{k}$ is the single state automaton over $\\Xn$ (\\cite{BCMNO}). If $T = \\gen{\\Xn, Q_{T}, \\pi_{T}, \\lambda_{T}}$ is a transducer, then $T$ is synchronizing if and only if the automaton $\\T{A}(T) = \\gen{\\Xn, Q_T, \\pi_T}$ is synchronizing. Typically we when applying the collapsing procedure, we shall not distinguish between $T$ and $\\T{A}(T)$.\n\nWe have the following general construction, which is in some sense an extension of a construction given in the paper \\cite{Olukoya1} for embedding direct sums of the group of automorphisms of one-sided shift over alphabets sizes summing to $n$ into the group of automorphisms of the one-sided shift on $n$ letters.\n\nLet $n \\in \\N_{2}$ and $d$ be a divisor of $n$ not equal to $n$. Let $m \\in \\N$ be such that $md =n$ and fix an element $T \\in \\hn{d}$. For $0\\le i \\le m-1$, partition the set $\\Xn$ into sets $X_{n,i}:= \\{di, di+1, \\ldots, d(i+1)-1 \\}$, likewise form sets $Q_{T,i} = \\{ q(i) \\mid q \\in Q_{T}\\}$. Form a transducer $\\oplus_{d}T = \\gen{ \\Xn, \\cup_{0 \\le i \\le m-1} Q_{T,i}, \\pi_{\\oplus_{d}T}, \\lambda_{\\oplus_{d}T}}$ with transition and output defined such that:\n\n\\begin{enumerate}[label=(\\arabic*.)]\n\\item the restriction $\\oplus_{d}T\\restriction_{Q_{T,i}} = \\gen{X_{n,i}, Q_{T,i},\\pi_{\\oplus_{d}T}\\restriction_{Q_{T,i}}, \\lambda_{\\oplus_{d}T}\\restriction_{Q_{T,i}}}$ of $\\oplus_{d}T$ to the set of states $Q_{T,i}$ is a transducer equal to $T$ up to relabelling the alphabet and the set of states,\n\n\\item for $0 \\le i,j \\le m-1$ such that $i \\ne j$, for any $b \\in X_{d}$, and for any $q \\in Q_{T}$, $\\pi_{\\oplus_{d}T}(dj +b, q(i)) = (q_{b})(j)$, where $\\pi_{T}(b, q_{b})= b$, and $\\lambda_{\\oplus_{d}T}(dj+b, q(i)) = di+b$.\n\n\\end{enumerate}\n\n\nWe have the following result:\n\n\\begin{proposition}\\label{Propositon:forncompositeforeverydivisorofnthereisanelementwithsigequaltodivisor}\nLet $n \\in \\N_{2}$, $d$ be a divisor of $n$ such that $n= md$ for some $m \\in \\N_{2}$, and $T \\in \\hn{d}$. Then, the transducer $\\oplus_{d}T$ is bi-synchronizing, and is fact an element of $\\Ln{n}$. Moreover, for $0 \\le i \\le m-1$, and any state $q(i) \\in Q_{T,i}$, $\\bigcup_{0 \\le b \\le d-1} U_{di+b}=\\im(q_i)$. In particular, $(\\oplus_{d}T)\\rsig = d$.\n\\end{proposition}\n\\begin{proof}\nWe begin by arguing that the transducer $\\oplus_{d}T$ is synchronizing. However, this follows straight-forwardly from the observation that for $0 \\le i,j \\le m-1$ such that $i \\ne j$, any pair of states $q(i), p(i) \\in Q_{T,i}$, and any $0 \\le b < d$, $\\pi_{\\oplus_{d}T}(dj+b, q(i)) = \\pi_{\\oplus_{d}T}(dj+b, p(i)) = (q_b)(i)$. Thus since, each transducer $\\oplus_{d}(T)\\restriction_{Q_{T,i}}$ for $0 \\le i \\le m-1$ at the same level $k \\in \\N$, then after applying the collapsing procedure at most $k+1$ times we have reduced $\\oplus_{d}T$ to the single state automaton.\n\nFurther observe that as $T$ is in fact a synchronous transducer, then if $T \\in \\On$ it is in fact an element of $\\Ln{n}$.\n\nTherefore in order to show that $\\oplus_{d}T \\in \\Ln{n}$ it suffices to show that each state is injective and has clopen image and that it has a synchronizing inverse.\n\n We begin by showing that each state of $\\oplus_{d}T$ is injective and has clopen image. Let $0 \\le i \\le m-1$ and $q(i)$ be a state of $T$. Let $\\mu, \\nu \\in \\Xnp$ such that $|\\mu| = |\\nu| \\ge 2$ and $\\mu \\ne \\nu$. We now show that either $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$ or for any $x \\in \\Xn$, $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$. If $\\mu, \\nu \\in X_{n,i}^{+}$, then this follows from the fact that $T \\in \\hn{n}$ and so each state of $T$ induces a permutation of $X_{d}$. Thus, we may assume that there are $1 \\le j_1, j_2 \\le m-1$ with $j_1$ and $j_2$ not both equal to $i$, $a_1 \\in X_{n, j_1}$, $a_2 \\in X_{n, j_2}$, and $\\mu_1, \\nu_1 \\in \\Xn^{+}$ such that $\\mu = a_1 \\mu_1$ and $\\nu= a_2 \\nu_1$. If $j_2 \\ne j_2$, then as $\\pi_{\\oplus_{d}T}(a_1, q(i)) \\in Q_{T, j_1}$ and $\\pi_{\\oplus_{d}T}( a_2, q(i)) \\in Q_{T, j_2}$, it follows that the first letter of $\\lambda_{\\oplus_{d}T}(\\mu_1, \\pi_{\\oplus_{d}T}(a_1, q(i)))$ is an element of $X_{n, j_1}$ and the first letter of $\\lambda_{\\oplus_{d}T}(\\nu_1, \\pi_{\\oplus_{d}T}(a_2, q(i)))$ is an element of $X_{n, j_2}$. Thus $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$. Therefore we may assume that $j_1 = j_2 = j$. In this case, since $\\mu \\perp \\nu$, there are words $\\mu_2, \\nu_2, \\phi \\in \\Xn^{\\ast}$ and $a \\ne b \\in \\Xn$ such that $\\mu= \\phi a \\mu_2$ and $\\nu = \\phi b \\nu_2$. If $a, b \\in X_{n,l}$ for some $0 \\le l \\le m-1$, then since for any state $t \\in \\cup_{0\\le i \\le m-1} Q_{T,i}$, $\\lambda_{\\oplus_{d}T}(a,t) \\ne \\lambda_{\\oplus_{d}T}(b, t)$, we have $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$. Thus, we may assume that $a \\in X_{n, l_1}$, $b \\in X_{n,l_2}$, by adding a letter to $\\mu_2$ and $\\nu_2$ if necessary, we may assume they are non-empty. In this case, the fact that $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) \\ne \\lambda_{\\oplus_{d}T}(\\nu, q(i))$ follows from the observation above that for any state $t \\in \\cup_{0\\le i \\le m-1} Q_{T,i}$, the first letter of $\\lambda_{\\oplus_{d}T}(\\mu_2, \\pi_{\\oplus_{d}T}(a,t))$ is an element of $X_{n,l_1}$ and the first letter of $\\lambda_{\\oplus_{d}T}(\\nu_2, \\pi_{\\oplus_{d}T}(b,t))$ is an element of $X_{n,l_2}$. Therefore, since $T$ is synchronous, each state must induce an injective map from $\\CCn$ to itself. \n \n Let $q(i) \\in Q_{T,i}$ be an arbitrary state of $T$, $b \\in \\Z_{d}$ be arbitrary. We now show, by induction, that $U_{di+b} \\subset \\im(q_i)$. \n \n Let $j \\in \\Z_{m}$ and $a \\in \\Z_{d}$ be arbitrary so that $jd +a \\in X_{n,j} \\subset \\Xn$ is arbitrary. There is a letter $x \\in X_{n,j}$ such that $\\lambda_{\\oplus_{d}T}(x, q(i)) = di+b$ and $\\pi_{\\oplus_{d}T}(x, q(i)) \\in Q_{T, j}$. Since $\\pi_{\\oplus_{d}T}(x, q(i)) \\in Q_{T, j}$, by construction, for any $l \\in \\Z_{m}$, there is a letter $y \\in X_{n,l}$ such that $\\lambda_{A}(y, \\pi_{\\oplus_{d}T}(x, q(i))) = jd +a$ and $\\pi_{\\oplus_{d}T}(y, \\pi_{\\oplus_{d}T}(x, q(i))) \\in Q_{T,l}$. Thus, for any $j \\in \\Z_{m}$, $a \\in \\Z_{d}$ and any $l \\in \\Z_{m}$, there is a word $xy \\in \\Xn^{2}$ such that $\\pi_{\\oplus_{d}T}(xy, q(i)) \\in Q_{T,l}$ and $\\lambda_{\\oplus_{d}T}(xy, q(i)) = (di+b) (jd+a)$.\n \n Assume by induction that there for any $\\nu \\in \\Xn^{M}$ and any $l \\in \\Z_{m}$, there is a word $\\mu \\in \\Xn^{M+1}$, such that $\\lambda_{\\oplus_{d}T}(\\mu, q(i)) = (di+b) \\nu$ and $\\pi_{\\oplus_{d}T}(\\mu, q(i)) \\in Q_{T,l}$. Let $\\xi \\in \\Xn^{M+1}$ be arbitrary. Let $j \\in \\Z_{m}$, $a \\in \\Z_{d}$ be such that $ \\xi = \\chi(jd + a)$ for some $\\chi \\in \\Xn^{M}$. By the inductive assumption, there is a word $\\zeta \\in \\Xn^{M+1}$ such $\\lambda_{\\oplus_{d}T}(\\zeta, q(i)) = (di+b) \\chi$ and $\\pi_{\\oplus_{d}T}(\\zeta, q(i)) = p(j) \\in Q_{T,j}$. Now by construction, for any $l \\in \\Z_{m}$, there is a word $y \\in X_{n,l}$ such that $\\lambda_{\\oplus_{d}T}(y, p(j)) = dj + a$ and $\\pi_{\\oplus_{d}T}(y, p(j)) \\in Q_{T,l}$. Therefore, we have that for any $l \\in \\Z_{m}$, there is a $y \\in X_{n,l}$ such that $\\lambda_{\\oplus_{d}T}(\\zeta y, q(i)) = (di+b) \\chi (dj+a)$ and $\\pi_{\\oplus_{d}T}(\\zeta y, q(i)) \\in Q_{T,l}$. Therefore we see, by transfinite induction, that $U_{di+b} \\subset \\im(q_i)$. Therefore every state of $T$ is injective and has clopen image.\n \n We now show that $\\oplus_{d}T$ has an inverse in $\\On$. We do this by showing that for any transducer $A_{q_0}$, with $\\core(A_{q_0}) = \\oplus_{d}T$, then set of states of $A_{(\\epsilon, q_0)}$ which are reached by arbitrarily long words, form a synchronizing transducer. To this end, fix a transducer $A_{q_0}$ with $\\core(A_{q_0}) = \\oplus_{d}T$. \n \n Let $0 \\le i \\le m-1$ and $q(i) \\in Q_{T,i}$, we observe that for $a \\in \\Z_{d}$, $(di + a)L_{q(i)} = \\epsilon$. Set $Q_{U} = \\{ (di + a, q(i)) \\mid i \\in \\Z_{m}, a \\in \\Z_{d} \\}$. Define $\\lambda_{U}: \\Xn \\times Q_{U} \\to \\Xn$ by $\\lambda_{U}(dj+ b, (di+a, q(i))) = ((di+a)(dj+b))L_{q(i)}$ for all $i, j \\in \\Z_{m}$ and $a,b \\in \\Z_{d}$. Further define $\\pi_{U}: \\Xn \\times Q_{U} \\to Q_{U}$ by $\\pi_{U}(dj+ b, (di+a, q(i))) = ((di+a)(dj+b))L_{q(i)} = ( (di+a)(dj+b) - \\lambda_{A}(((di+a)(dj+b))L_{q(i)}, q(i)), \\pi_{A}(((di+a)(dj+b))L_{q(i)}, q(i)))$. Now, observe that $((di+a)(dj+b))L_{q(i)}$ is precisely the element $x \\in X_{n,j}$ such that $\\lambda_{\\oplus_{d}T}(x, q(i)) = di +a$, thus if $p(j) = \\pi_{\\oplus_{d}T}(x, (q(i)))$, then $\\pi_{U}(dj+ b, (di+a, q(i))) = (dj+b, p(j))$. Let $U = \\gen{ \\Xn, Q_{U}, \\pi_{U}, \\lambda_{U} }$. We now show that $U$ is synchronizing, then we argue that the set of states of $A_{(\\epsilon, q_0)}$ reached by arbitrarily long words forms a transducer precisely equal to $U$.\n \n We partition the sets of $U$ as follows: for $0 \\le i \\le m-1$ and $a \\in \\Z_{d}$, set $Q_{U,i,a}:= \\{ (di+ a, q(i)) \\mid q(i) \\in Q_{T,i} \\}$. Fix $0 \\le i \\le m-1$ an $a \\in \\Z_{d}$. \n \n Let $0 \\le j \\le n-1$ such that $i \\ne j$ and $(di+ a, q(i)), (di+ a, p(i)) \\in Q_{U,i,a}$ and $dj +b$, $b \\in \\Z_{d}$, be arbitrary. We observe that, by construction, if $x, y \\in X_{n,j}$ satisfy $\\lambda_{\\oplus_{d}T}(x, q(i)) = \\lambda_{\\oplus_{d}T}(y, p(i)) = di+a$, then $x = y = dj +a$. Observe, moreover, that $\\pi_{\\oplus_{d}T}(dj + a, q(i)) = \\pi_{\\oplus_{d}T}(dj + a, p(i)) = (q_{a})(j)$. Therefore it follows that $\\pi_{U}(dj+ b, (di+a, q(i))) = (dj+b, (q_a)(j)) = \\pi_{U}(dj+ b, (di+a, p(i)))$. Thus we conclude that for any $0 \\le i \\le m-1$ and any $a \\in \\Z_{d}$, the set of elements $Q_{U,i,a} := \\{ (di+a, q(i)) \\mid q(i) \\in Q_{T,i}\\}$ transition identically on all elements of $X_{n,j}$ for $j \\ne i$. \n \n \n \n Now let, $T^{-1} = \\gen{ \\Xn, Q_{T^{-1}}, \\pi_{T^{-1}}, \\lambda_{T^{-1}}}$, a synchronous and synchronizing transducer, denote the inverse of $T$ and denote by $(a)q^{-1}$, for $a \\in \\Z_{d}$, the element of $X_{d}$ such that $\\lambda_{T}((a)q^{-1}, q) = a$. Consider a letter $di +c$, we observe that $\\pi_{U}(di + c, (di+a,q(i))) = (di + c, \\pi_{\\oplus_{d}T}(di+(a)q^{-1}, q(i)))$. Note moreover that $\\pi_{T^{-1}}(a, q^{-1}) = (\\pi_{T}((a)q^{-1}, q))^{-1}$. Thus, $$(di + c, \\pi_{\\oplus_{d}T}(di+(a)q^{-1}, q(i))) = (di+c, (\\pi_{T}((a)q^{-1}, q))^{-1}(i) ) = (di+c, (\\pi_{T^{-1}}(a, q^{-1}))^{-1}(i)). $$ A simple induction argument now shows that given a word $\\nu = v_1 v_2 \\ldots v_{l} \\in X_{d}^{+}$, with corresponding word $\\nu(i) = (di + v+1) (di+v_2) \\ldots (di+ v_{l}) \\in X_{n,i}^{+}$, if $\\pi_{T^{-1}}(\\nu, q^{-1}) = p^{-1}$, for a state $q \\in Q_{T}$, then, setting $\\nu(i)_{1} = (di+v_2) \\ldots (di+ v_{l})$, $\\pi_{U}(\\nu(i)_{1}(di+b), (di+v_1,q(i))) = (di+b, p(i))$. Let $k$ be the synchronizing level of $T^{-1}$, it therefore follows that, all elements of $Q_{U,i,a}$ transition identically on all words in $X_{n,i}^{k}$.\n \n Finally, since by an observation above we have that for any state $(dj+b, q(j)) \\in Q_{U}$, and any letter $dl +c \\in X_{n}$ $\\pi_{U}(dl+c,(dj+b, q(j))) \\in Q_{U,l,c}$, it therefore follows that $U$ is synchronizing at level $k+1$. \n \n\n\nTo conclude the proof we have to demonstrate that $(\\oplus_{d}T)^{-1} = U$. We do this by showing that $A_{(\\epsilon, q_0)}$ is synchronizing and has core equal to $U$. Let $q(i)$ be an arbitrary state of $\\oplus_{d}T$. Since $A_{q_0}$ is synchronizing, there is a word $\\Gamma \\in \\Xn^{+}$ such that $\\Delta = \\lambda_{A}(\\Gamma, q_0) \\ne \\epsilon$ and $\\pi_{A}(\\Gamma, q_0) = q(i)$. It therefore follows that, for $0 \\le a \\le d-1$, $(\\Gamma (di + a))L_{q_0}= \\Delta (di+a)L_{q}$ and so $\\pi'_{A}(\\Gamma (di+a), (\\epsilon,q_0)) = (di+a, q(i))$. Thus we see that $A_{(\\epsilon, q_0)}$ is synchronizing and has $\\core(A_{(\\epsilon, q_0)}) = U$. Therefore $T^{-1} = U$. This concludes the proof. \n\\end{proof}\n\n\\begin{corollary}\nLet $n \\in \\N_{2}$, then the image of the map $\\rsig$ from $\\On$ to $\\Z_{n-1}^{*}$, the group of units of $Z_{n-1}$, contains the subgroup of $\\Z_{n-1}^{*}$ generated by the divisors of $n$. \\qed \n\\end{corollary} \n\\begin{corollary}\nLet $n \\in \\N_{2}$ be such that the group of units of $\\Z_{n-1}$ is generated by the divisors of $n$, then the map $\\rsig: \\On \\to \\Z_{n-1}^{*}$ is surjective. \\qed\n\\end{corollary} \n\n\\begin{corollary}\nThere are infinitely many number $n \\in \\N$ such that the map $\\rsig$ from $\\On$ to the group of units of $Z_{n-1}$ is surjective.\n\\end{corollary}\n\\begin{proof}\nIt is an elementary result in number theory that for an $m \\ge 1$, $2$ generates the group of units of $3^{m}$, thus we may take $n = 3^{m} + 1$.\n\\end{proof}\n\nThus, we are left with the question:\n\n\\begin{Question}\nfor $n \\in \\N_{2}$ and $p$ in the group $\\Z_{n-1}^{\\ast}$ of units of $\\Z_{n-1}$ such that $p$ is not an element of the subgroup of $\\Z_{n-1}$ generated by the divisors of $n$, is there an element $T \\in \\On$ such that $(T)\\rsig = p$? \n\\end{Question}\n\nWe observe that in order to answer this question, it suffices to show that for any element $p \\in \\Z_{n-1}$ co-prime both to $n$ and $n-1$, there is an element $T \\in \\On$ such that $(T)\\rsig = p$.\n \n\n\\begin{comment}\nWe address this question below. \n\nLet $n \\in \\N_{2}$, let $p \\in \\Z_{n-1}$ be co-prime both to $n$ and $n-1$. Since $p$ is co-prime to $n$ and $n-1$, we may find an integer $q \\in \\N$ such that $n + q(n-1) \\equiv 0 \\mod{p}$. Let $d \\in \\N$ be such that $n + q(n-1) = dp$. Set $\\ac{u}:= \\{(n-1)^q a \\mid a \\in \\Xn\\} \\cup \\{ (n-1)^i a \\mid 0 \\le i < q, a \\in \\Xn\\backslash\\{n-2\\} \\}$ a complete antichain for $\\Xnp$. We recall that antichains are ordered lexicographically, thus for $u \\in \\ac{u}$, set $(u)\\iota = j \\in X_{dp}$ if and only if $(u)\\iota$ is the $j+1$'st element of $\\ac{u}$. As $ \\iota$ is defined on elements of $\\ac{u}$ we extend it to a bijection from $\\ac{u}^{\\ast} \\to X_{dp}^{\\ast}$, and from $\\ac{u}^{\\omega} \\to X_{dp}^{\\omega}$. We write $\\iota^{-1}$ for the inverse of $\\iota$. Let $\\shift{n}: \\Xnz \\to \\Xnz$ be defined by $(x_{i})_{i \\in \\Z} \\mapsto (y_i)_{i \\in \\Z}$ where $y_{i} = x_{i+1}$ i.e. $\\shift{n}$ is the usual shift map on $\\Xnz$. Let $\\shift{dp}: X_{dp}^{\\Z} \\to X_{dp}^{\\Z}$ be defined analogously. \n\nLet $\\psi \\in \\Ln{dp}$ be a fixed element such that $\\psi$ is synchronous and let $k \\in \\N$ be the synchronizing level of $\\psi$. Then $\\psi$ defines a shift commuting homeomorphism map from $X_{dp}^{\\Z}$ to itself by a sequence $x \\in X_{dp}^{\\Z}$ maps to the sequence $y$ defined by $y_i = \\lambda_{\\psi}(x_i, q_{x_{i+k}x_{i+k-1}\\ldots x_{i-1}})$ where $q_{x_{i+k}x_{i+k-1}\\ldots x_{i-1}}$ is the unique state in the image of the map $\\pi(x_{i+k}x_{i+k-1}\\ldots x_{i-1}, \\centerdot): Q_{\\psi}\\to Q_{\\psi}$. \n\nAs we now need also to read words in the opposite orientation we set-up some notation for this. For a word $w = w_1w_2\\ldots w_{|w|}$ over an finite alphabet let $\\rev{w} =w_{|w|}w_{|w|-1}\\ldots w_2w_1$. \n \nNow fix a right infinite sequence $x:=x_0x_1 \\ldots \\in \\Xn^{\\Z}$ and we assume that for some $l > k(q+1)$ $x_l \\ne n-1$. We further assume that the sequence $x_{l-1}x_{l-2}\\ldots x_0$ can be decomposed into a sequence $u_{l'}u_{l'-1}\\ldots u_{0}$ such that $x_0 x_1 \\ldots x_{l-1} = \\rev{u_{0}}\\rev{u_{1}}\\ldots \\rev{u_{l'}}$. We observe that as we have assumed this decomposition exists, it is unique. Also observe that $l'$ must be at least $k$ since $|u_{i}| \\le q+1$ for all $0 \\le i \\le l'$. Let $v = \\phi(u_{l'}u_{l'-1}\\ldots u_{0}) \\in X_{dp}^{l'}$.\n\n Define a map $h_{\\psi,x}: \\CCn \\to \\CCn$ as follows. Let $\\delta \\in \\CCn$ and write $\\delta = \\delta_{-1}\\delta_{-2} \\ldots$. There is a unique sequence $u_{-1}, u_{-2} \\ldots$ of elements of $\\ac{u}$ such that $\\delta_{-1} \\delta_{-2} \\ldots = u_{-1} u_{-2} \\ldots$. Thus $ \\ldots\\delta_{-3}\\delta_{-2}\\delta_{-1}x_0x_1 \\ldots x_{l-1} = \\ldots\\rev{u_{-2}}\\rev{u_{-1}}\\rev{u_{0}}\\rev{u_{1}}\\ldots \\rev{u_{l'}}$. Let $v_{i} = (u_i)\\iota$ for $i \\in \\{-1,-2,\\ldots\\}$. Then $\\ldots \\rev{v_2}\\rev{v_1}\\rev{v}$ is a left infinite sequence in $X_{dp}$. Let $\\rho_1\\rho_2\\ldots=\\rho = \\lambda_{\\psi}(v_1v_2\\ldots, q_{v})$. Set $(\\delta)h_{\\psi,x} = (\\rho)\\iota^{-1}$.\n \n\nWe have the following claim:\n\n\\begin{claim}\nThe map $h_{\\psi, x}$ is injective, has finitely many local actions and is synchronizing. Furthermore, for any local action $g$ of $h_{\\psi,x}$, $\\im(g)$ is clopen, and if $W \\subset \\Xnp$ is minimal such that $\\cup_{\\eta \\in W}U_{\\eta} = \\im(g)$, then $|W| \\equiv (\\psi)\\rsig \\mod{n-1}$.\n\\end{claim}\n\\begin{proof}\nLet $\\nu \\in \\Xnp$ be a word of length $> k(q+1)$ and let $\\delta \\in \\CCn $ be arbitrary. We observe that as $\\nu$ has length at least $k(q+1) +1$ then $\\nu = u_{1}u_{2} \\ldots u_{m} \\phi$ for some $u_{i} \\in \\ac{u}$, $m \\ge k$, and $\\phi \\in \\Xns$ such that $\\phi$ is a prefix of some element of $\\ac{u}$. Let $\\phi\\delta = u_{m+1}u_{m+2}\\ldots$. Then, $\\nu\\phi\\delta = u_{1}u_{2}\\ldots$ and this decomposition is unique. Let $v_i = (u_i)\\iota$ for $i \\in \\N_{1}$ and let $\\rho = v_1v_2 \\ldots\n \\in X_{dp}^{\\omega}$. By definition we have $(\\nu\\phi\\delta)h_{\\psi,x} = (\\lambda_{\\psi}(\\rho, q_{v}))\\iota^{-1}$. Let $v_{m-k+1}v_{m-k+1}\\ldots v_{m}$ be the length $k$ suffix of $v_{1}v_{2}\\ldots v_{m}$ and let $\\rho'$ be such that $v_{1}\\ldots v_{m} \\rho' = \\rho$, then $\\lambda_{\\psi}(\\rho, q_{v}) = \\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}) \\lambda_{\\psi}(\\rho', q_{v_{m-k+1}v_{m-k+1}\\ldots v_{m}})$. Thus we have that $$(\\nu\\phi\\delta)h_{\\psi,x} = (\\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}))\\iota^{-1}(\\lambda_{\\psi}(\\rho', q_{v_{m-k+1}v_{m-k+2}\\ldots v_{m}}))\\iota^{-1}.$$\n \n We now observe that the greatest common prefix of the set $(U_{\\nu}) h_{\\psi,x}$ is precisely $(\\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}))\\iota^{-1}$ since $\\psi$ has no states of incomplete response, $\\iota$ is a bijection and the greatest common prefix of $\\ac{u}$ is $\\epsilon$. Now by a similar argument we also have that, setting $\\nu' = u_{m-k+1}u_{m-k+2}\\ldots u_{m}$,: \n \n $$(\\nu'\\phi\\delta)h_{\\psi,x} = (\\lambda_{\\psi}(v_{m-k+1}v_{m-k+2}\\ldots v_{m}, q_{v}))\\iota^{-1}(\\lambda_{\\psi}(\\rho', q_{v_{m-k+1}v_{m-k+1}\\ldots v_{m}}))\\iota^{-1}.$$\n \n Thus, we see that $(\\nu\\phi\\delta)h_{\\psi,x} = (\\lambda_{\\psi}(v_1\\ldots, v_{m}, q_{v}))\\iota^{-1} (\\phi\\delta)(h_{\\psi,x})_{\\nu'}$. In particular, the local actions $h_{\\nu}$ and $h_{\\nu'}$ are equal and so the local action $h_{\\nu \\phi}$ is equal to $h_{\\nu'\\phi}$. Therefore we conclude that $h_{\\phi, x}$ has finitely many local actions. This argument also demonstrates that $h_{\\psi,x}$ is synchronizing, since the local action of a long enough word is determined by some uniformly bounded suffix.\n \n That $h_{\\psi,x}$ is injective, follows from the facts that $\\iota$ is a bijection and all states of $\\psi$ induce injective function.\n \n\\end{proof}\n\nThe above claim means that $h_{\\psi,x}$ can be represented by a synchronizing transducer $A_{q_0}$, set $\\Psi:= \\core(A_{q_0})$, we now need to argue that $\\Psi$ is bi-synchronizing. We begin with the lemma below.\n\\end{comment}\n\\begin{comment}\nWe now define a map $\\varPsi: \\Xn^{\\Z} \\to X_{dp}^{\\Z}$ in the following way. Fix $x:=(x_i)_{i \\in \\Z} \\in \\Xn^{\\Z}$. We define a sequence $y:=(y_i)_{i \\in \\Z} \\in X_{dp}^{\\Z}$ as follows.\nFirst suppose that $(x_i)$ has no occurrence of a symbol in the set $\\{0,1,\\ldots, n-2\\}$. In this case $x$ is the sequence such that $x_i = n-1$ for all $i \\in \\Z$. In this case $y$ is the sequence such that $y_i = dp-1$ for all $i \\in \\Z$. We set $(x)\\varPsi = y$.\n\nThus we may assume that $(x_i)$ has an occurrence of a symbol in $\\{0,1,\\ldots, n-2\\}$. Let $l \\in \\Z$ be minimal in absolute value such that $x_{l}$ is an element of $\\{0,1,\\ldots, n-2\\}$. Let $x_{2$ and $1\\le r < n$ contain an isomorphic copy of Thompson's group $F$}\\label{Section:OutTncontainscopyofF}\n\nWe now show that for $n >2$, $n \\in N$, and for all valid $r$, the group $\\TOns{r}$ is infinite. This extends the result of Brin and Guzm{\\'a}n stating that the group $\\out{\\mathcal{T}_{n,n-1}}$ is infinite whenever $n>3$ (\\cite{MBrinFGuzman}) to the groups $\\out{\\Tnr}$. We observe that the group $\\out{T_{2}}$ is the cyclic group of order $2$ and so finite. By Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1} it suffices to demonstrate that $\\TOns{1}$ is infinite. First we have the following straight-forward lemma.\n\n\\begin{lemma}\\label{Lemma:possessingahomeostatemeanslementsofTOn1}\nLet $T \\in \\TOn$ be an element with a homeomorphism state, then $T \\in \\TOns{1}$.\n\\end{lemma}\n\\begin{proof}\nLet $q \\in Q_{T}$ be an homeomorphism state of $T$, then the initial transducer $T_{q}$ is an element of $\\TBmr{n}{1}$.\n\\end{proof}\n\nTherefore to show that $\\TOns{1}$ is infinite, for $n>2$, it suffices to find an element of infinite order of $\\TOn$ which has a homeomorphism state.\n\nLet $T$ be the element of $\\TOn$ depicted in Figure~\\ref{Figure:elementinTon1ofinfiniteorder}. In this figure we use the symbol $x$ to represent an element of $\\Xn$ strictly greater than zero and less than or equal to $n-2$.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,node distance=3cm,on grid,auto] \n \\node[state] (q_0) {$a$};\n \\node[state, xshift=4cm] (q_1) {$b$}; \n \\node[state, xshift=2cm, yshift=-2cm] (q_2) {$c$};\n \\path[->] \n (q_0) edge[in=170, out=10] node {$n-1|n-1n-1$} (q_1)\n edge[in=105, out=75,loop] node[swap] {$x|n-1x$} ()\n edge[in=180, out=265] node[swap] {$0|\\epsilon$} (q_2) \n (q_1) edge[in=350, out=190] node {$x|x$}(q_0)\n edge[in=90, out=120,loop] node {$0|0$}()\n edge[in=90, out=60,loop] node[swap] {$n-1|n-1$}()\n (q_2) edge[in=275, out=170] node[swap] {$x|x$} (q_0)\n edge[in=270, out=0] node[swap, yshift=0.2cm] {$n-1|n-10$} node {$0|0$} (q_1);\n\\end{tikzpicture}\n\\caption{An element $T \\in \\T{TO}_{n,1}$ of infinite order.}\n\\label{Figure:elementinTon1ofinfiniteorder}\n\\end{figure}\n\nThe states $a$ and $b$ of $T$ are homeomorphism states, and so $T$ is in fact an element of $\\TOns{1}$.\n\nIn order to show that $T$ has infinite order we make use of the action of the group $\\On$, as introduced in the paper \\cite{BCMNO}, on the space $\\Xnz\/ \\gen{\\sigma_{n}}$ where $\\sigma_{n}$ is the shift map on $\\Xnz$. Let $U \\in \\On$ and suppose that $k \\in \\N$ is the minimal synchronizing level of $U$. The action of $U$ on $\\Xnz\/\\gen{\\sigma_{n}}$ is given as follows: let $y= \\ldots y_{-1}y_{-1}y_0y_1y_2\\dots \\in \\Xnz$ represent an equivalence class of $\\Xnz\/\\gen{\\sigma_{n}}$, the image of this class under $U$ is the equivalence class of the bi-infinite word $w$ defined as follows: for $i \\in \\Z$, let $y(i+1,k) = y_{i+1}y_{i+1}\\ldots y_{i+k}$ and $q_{y(i+1,k)}$ be the state of $U$ forced by $y(i+1,k)$, then set $w_i = \\lambda_{U}(y_i, q_{y(i+1,k)}$ and let $w = \\ldots w_{-2}w_{-1}w_{-1}w_0w_1w_2\\ldots$. Consider the bi-infinite word $z:=\\ldots(x n-1)(x n-1)\\ldots$ where $ x \\in \\Xn \\backslash \\{0,n-1\\}$, then $(z)T = \\ldots(xn-1n-1)(xn-1n-1)\\ldots$, $(z)T^2 = \\ldots x(n-1)^3x(n-1)^3\\ldots$, and $(z)T^{i} = \\ldots x(n-1)^{i+1}x(n-1)^{i+1}\\ldots$. Therefore we see that $z$ is on an infinite orbit under the action of $T$, demonstrating that $T$ is an element of infinite order. We have shown the following:\n\n\\begin{Theorem}\\label{Theorem:Tnrinfinitewhenevernbiggerthan2}\nFor $n>3$ and $1 \\le r \\le n-1$, the group $T_{n,r}$ is infinite.\n\\end{Theorem}\n\nLet $U$ be the following transducer where, once more, $n >2$ and $x \\in \\Xn$ is any element strictly bigger than $0$ and strictly less than $n-1$:\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,on grid,auto] \n \\node[state] (q_0) {$p$};\n \\node[state, xshift=4cm] (q_1) {$q$}; \n \\node[state, xshift=2cm, yshift=-4.3cm] (q_2) {$s$};\n \\node[state, xshift=2cm, yshift=-2cm] (q_3) {$t$};\n \\path[->] \n (q_0) edge[in=170, out=10] node {$0|0$} (q_1)\n edge[in=105, out=75,loop] node[swap] {$x|x$} ()\n edge[in=195, out=185] node[swap] {$n-1|n-1$} (q_2) \n (q_1) edge[in=350, out=190] node {$x|n-1x$}(q_0)\n edge[in=10, out=260] node[swap] {$0|\\epsilon$}(q_3)\n edge[in=0, out=0] node{$n-1|(n-1)^2$}(q_2)\n (q_2) edge[in=200, out=180] node[swap] {$x|x$}(q_0)\n edge[in=270, out=240,loop] node[swap, yshift=0.1cm] {$0|0$}()\n edge[in=270, out=300,loop] node[yshift=0.1cm] {$n-1|n-1$}()\n (q_3) edge[in=275, out=170] node[swap] {$x|x$} (q_0)\n edge node[swap] {$n-1|n-10$} node {$0|0$} (q_2);\n\\end{tikzpicture}\n\\caption{An element $U \\in \\T{TO}_{n,1}$ of infinite order. }\n\\label{Figure:ThetransducerUinTOn1}\n\\end{figure}\n\nThe state $p$ of $U$ is a homeomorphism state and moreover $U$ is an element of $\\TOns{1}$ of infinite order.\n\n\n\n\nIn order to state our next result we require the following notion and result from \\cite{BCMNO}.\n\n\\begin{Definition}\nGiven an element $g = \\gen{\\Xn, Q_{g}, \\pi_g, \\lambda_g} \\in \\On$ a state $q$ of $g$ is called a \\emph{loop state} if there is some $i \\in \\Xn$ such that $\\pi_{g}(i, q) = q$.\n\\end{Definition}\n\nThe following lemma is proven in \\cite{BCMNO}:\n\n\\begin{lemma}\\label{Lemma: reading loops give bijection from set of Xns to Xns}\nLet $g = \\gen{\\Xn, Q_g, \\pi_g, \\lambda_g} \\in \\On$ and let $w \\in \\Xnp$ then there is a unique state $q_{w} \\in Q_{g}$ such that $\\pi_{g}(w, q_w) = q_w$. Moreover for any periodic equivalence class of the tail equivalence $\\sim_{t}$ with minimal period $w$ there exists a unique $j \\in \\N_{1}$ and a unique circuit, decorated by some prime word $v$, in $A$ with output $w$. \n\\end{lemma}\n\n\n\\begin{lemma}\\label{Lemma:TandUgenerateF}\nThe elements $T$ and $U$ of $\\TOns{1}$ depicted in Figures~\\ref{Figure:elementinTon1ofinfiniteorder} and \\ref{Figure:ThetransducerUinTOn1} generate a subgroup of $\\TOns{1}$ isomorphic to R. Thompson's group $F$.\n\\end{lemma}\n\\begin{proof}\nFirst we make some observations about $T$ and $U$. Let $q$ be a state of $T$ or of $U$,let $x \\in \\Xn \\backslash \\{0, n-1\\}$, and let $(D,d) \\in \\{(T,a),(U,p)\\}$, then all transitions $\\pi_{D}(x,q) = d$ and $\\lambda_{D}(x,q)$ end in the symbol $x$. The output of all other transitions in $T$ or $U$ do not involve the symbol $x$. Therefore the transducers $A$ and $B$ below are in fact sub-transducers of $T$ and $U$ respectively.\n\n\\begin{figure}[H]\n\\centering\n\\begin{tikzpicture}[shorten >= .5pt,node distance=3cm,on grid,auto] \n \\node[state] (q_0) {$a$};\n \\node[state, xshift=4cm] (q_1) {$b$}; \n \\node[state, xshift=2cm, yshift=-2cm] (q_2) {$c$};\n \\node[state, xshift = 8cm] (p_0) {$p$};\n \\node[state, xshift=12cm] (p_1) {$q$}; \n \\node[state, xshift=8cm, yshift=-2cm] (p_2) {$s$};\n \\node[state, xshift=12cm, yshift=-2cm] (p_3) {$t$};\n \\path[->] \n (q_0) edge[in=170, out=10] node {$n-1|n-1n-1$} (q_1)\n edge[in=180, out=265] node[swap] {$0|\\epsilon$} (q_2) \n (q_1) edge[in=90, out=120,loop] node {$0|0$}()\n edge[in=90, out=60,loop] node[swap] {$n-1|n-1$}()\n (q_2) edge[in=270, out=0] node[swap, yshift=0.2cm] {$n-1|n-10$} node {$0|0$} (q_1)\n (p_0) edge[in=170, out=10] node {$0|0$} (p_1)\n edge node[swap] {$n-1|n-1$} (p_2) \n (p_1) edge node {$0|\\epsilon$}(p_3)\n edge node[swap, xshift=0.5cm]{$n-1|(n-1)^2$}(p_2)\n (p_2) edge[in=270, out=240,loop] node[swap, yshift=0.1cm] {$0|0$}()\n edge[in=270, out=300,loop] node[yshift=0.1cm] {$n-1|n-1$}()\n (p_3) edge[out=190, in=350] node[swap] {$n-1|n-10$} node {$0|0$} (p_2);\n\\end{tikzpicture}\n\\caption{The subtransducers $A$ and $B$ of $T$ and $U$ respectively.}\n\\label{Figure:subtransducersAandB}\n\\end{figure}\n\nNotice that $A_{a}$ and $B_{p}$ induce self-homeomorphisms of the Cantor space $\\{0,n-1\\}^{\\omega}$ equal, respectively, to the restrictions of the homeomorphisms $T_{a}$ and $U_{p}$ to the space $\\{0,n-1\\}^{\\omega}$. It is not hard to see that $F \\cong \\gen{A_{a}, B_{p}} \\le H(\\{0,n-1\\}^{\\omega})$. In fact $A_{a}^{-1}$ and $A_{a}^{-2}B_{p}A_{a}$ are the standard generators for $F$ acting on the space $\\{0,n-1\\}^{\\omega}$. Moreover $A_{a}$ and $B_{p}$ satisfy the defining relations for $F$, and so we see that $\\gen{A_{a}, B_{p}} = \\gen{A_{a}, B_{p}\\mid [B_{p}^{-1}A_{a}, A_{a}B_{p}A_{a}^{-1}], [B_{p}^{-1}A_{a}, A_{a}^{2}B_{p}A_{a}^{-2}]}$. Moreover $T_{a}^{-1} \\restriction_{\\{0, n-1\\}^{\\omega}} = A_{a}^{-1}$ and $U_{p}^{-1}\\restriction_{\\{0, n-1\\}^{\\omega}} = B_{p}^{-1}$. We also observe (one can verify this by direct computation) that the transducers $T^{-1}$ and $U^{-1}$ representing the inverses of $T$ and $U$ in $\\TOn$ respectively, have states $a^{-1}, b^{-1}$ of $T^{-1}$ and $p^{-1}, s^{-1}$ of $U^{-1}$ such that $T_{a^{-1}}^{-1} = T_{a}^{-1}$, $T_{s^{-1}}^{-1} = T_{s}^{-1}$, $U_{p^{-1}}^{-1} = U_{p}^{-1}$ and $U_{s^{-1}}^{-1} = U_{s}^{-1}$. \n\nWe now define a homomorphism from $\\gen{T, U}$ to $\\gen{A_{a}, B_{p}}$. To do this we make the following observation. Let $W(T,U) = w_1 w_2 \\ldots w_l$ be a word in $\\{T,U,T^{-1},U^{-1}\\}^{+}$. For each letter $w_i$ of $W(T,U)$ let $\\alpha_{i}$ be the state of $w_i$ so that $\\alpha_{i}$ is the unique loop state of $x$ in $w_i$. Observe that, for all $1 \\le i \\le l$, $\\alpha_i \\in \\{a,a^{-1}, p, p^{-1}\\}$. Let $S_{W(T,U)}$ be the element of $\\TOns{1}$ representing the word $W(T,U) \\in \\gen{T,U}$. Under the product defined on $\\TOns{1}$, we have that $S_{W(T,U)}$ is $\\omega$-equivalent to the core of the minimal transducer representing the product of the initial transducers $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$. However, since $\\alpha_i$ is the unique loop state state of $x$ in $w_i$, $\\lambda_{w_i}(x, \\alpha_i)$ ends in $x$ and for any state $q$ of $T$, $U$, $T^{-1}$ or $U^{-1}$, the resulting state when $x$ is read from $q$ is the appropriate element of the set $\\{a, a^{-1}, p, p^{-1}\\}$, we must have that the state $(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})$ is the unique loop state of $x$ in the product $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$. Therefore, since the core is strongly connected, $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$ is equal to its core. Therefore if $s$ is the unique loop state of $x$ in $S_{W(T,U)}$, we must have that $(S_{W(T,U)})_{s}$ is $\\omega$-equivalent to $(w_{1}\\ast w_{2} \\ast \\ldots \\ast w_{l})_{(\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{l})}$. Now let $D(A,B) = d_1d_2\\ldots d_l$ be the word in $\\{A_{a}, B_{p}, A_{a}^{-1}, B_{p}^{-1} \\}^{+}$ defined as follows, for all $1 \\le i \\le l$, if $w_i = T^{\\pm 1}$ then $d_i = A_{a}^{\\pm 1}$ and if $w_i = U^{\\pm 1}$ then $d_i = B_{p}^{\\pm 1}$. Let $C_{D(A,B)}$ be the element of $\\gen{A_{a}, B_{p}}$ representing the word $D_{(A,B)}$, then it follows that $C_{D(A,B)} = (S_{W(T,U)})_{s}\\restriction_{\\{0,n-1\\}^{\\omega}}$ since for all $i$, $(w_{i})_{\\alpha_{i}}\\restriction_{\\{0,n-1\\}^{\\omega}} = d_{i}$. Therefore the map $\\phi: \\gen{T,U} \\to \\gen{A_{a}, B_{p}}$ by $S \\mapsto S_{s}\\restriction_{\\{0,1\\}^{\\omega}}$, where $s$ is the unique loops state of $x$ in $S$, is a surjective homomorphism. \n\nIn order to conclude that $\\gen{T,U} \\cong \\gen{A_{a}, B_{p}}$ it suffices to show that the map $\\varphi$ which maps $A_{a}$ to $T$ and $B_{p}$ to $U$ extends to a group homomorphism $\\overline{\\varphi}: \\gen{T,U} \\to \\gen{A_{a}, B_{p}}$. To see this, it suffices to show that $[U^{-1}T, TUT^{-1}] =1$ and $[U^{-1}T, T^{2}UT^{-2}] =1$. This is easily verified by direct computation.\n\n\\begin{comment}\nWe now argue that $\\phi$ is injective. To do this we introduce a tree pair representation for the elements $T$, $U$ and $A$, $B$ that correspond to the products in $\\gen{T,U}$ and $\\gen{A_{a}, B_{p}}$. The pictures below deal with the case $n=3$ however for the case for $n>3$ is dealt with analogously, and in this case all leaves corresponding to a words ending in $x$, for $x \\in \\Xn \\backslash \\{0,n-1\\}$, have label $p$ attached in the range tree. \n\n\\begin{tikzpicture}\n\\Tree [.{} [ {} {} {} ] {} {} ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{T_{a}}{\\longrightarrow}$};\n\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{} ${}_{b}$ ${}_{p}$ [ ${}_{b}$ ${}_{p}$ ${}_{b}$ ] ]\n\\end{tikzpicture} \\qquad\n\\begin{tikzpicture}\n\\Tree [.{} [ [ {} {} {} ] {} {} ] {} {} ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{U_{p}}{\\longrightarrow}$};\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{} [ ${}_{s}$ ${}_{p}$ [ ${}_{s}$ ${}_{p}$ ${}_{s}$ ] ] ${}_{p}$ [ ${}_{b}$ ${}_{p}$ ${}_{b}$ ] ]\n\\end{tikzpicture}\n\n Notice that all leaves in the range trees corresponding to words in $\\Xn^{+}$ with final letter $0$ or $n-1$ have the state $b$ attached, while all leaves corresponding to words with final letter $x \\in \\Xn \\backslash \\{0,1\\}$ have the label $p$ attached. Deleting all leaves corresponding to words ending in $x$, $x \\in \\Xn \\backslash \\{0,1\\}$, in the domain and range trees and deleting the labels $b$ from the resulting range tree. The resulting tree pairs represent the elements $A_{a}$ and $B_{p}$ of $H(\\{0,1\\}^{\\omega})$. Notice that as the states $b$ and $p$ of $T$ and the states $p$ and $s$ of $U$ are homeomorphism states, it follows that the tree pair representing the maps $T_{a}^{-1}$ and $U_{p}^{-1}$ are as follows:\n\\begin{tikzpicture}\n\\Tree [.{} ${}$ ${}$ [ ${}$ ${}$ ${}$ ] ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{T_{a}^{-1}}{\\longrightarrow}$};\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{} [ ${}_{b^{-1}}$ ${}_{p^{-1}}$ ${}_{b^{-1}}$ ] ${}_{p^{-1}}$ ${}_{b^{-1}}$ ]\n\\end{tikzpicture}\\qquad\n\\begin{tikzpicture}\n\\Tree [.{} [ ${}$ ${}$ [ ${}$ ${}$ ${}$ ] ] ${}$ [ ${}$ ${}$ ${}$ ] ]\n\\node[xshift=1cm, yshift=-0.5cm] {$\\stackrel{U_{p}^{-1}}{\\longrightarrow}$};\n\\end{tikzpicture\n\\begin{tikzpicture}\n\\Tree [.{} [ [ ${}_{s^{-1}}$ ${}_{p^{-1}}$ ${}_{s^{-1}}$ ] ${}_{p^{-1}}$ ${}_{s^{-1}}$ ] ${}_{p^{-1}}$ ${}_{s^{-1}}$ ]\n\\end{tikzpicture}\n\nHence we see that after performing the operation of deleting the ignoring all leaves corresponding to words in $\\Xn^{+}$ ending in a letter $x \\in \\Xn \\backslash \\{0,1\\}$ we see that the resulting tree pairs actually represent the elements $A^{-1}$ and $B^{-1}$. Now observe that since for $i \\in \\{0, n-1\\}$, $x \\in \\Xn \\backslash \\{0,1\\}$, and for $(E,e,t) \\in \\{ (T,b,a), (T^{-1}, b^{-1}, a^{-1}),(U,s, p), (U^{-1}, s^{-1},p^{-1}) \\}$, we have $\\pi_{S}(i,s) = s$ and, $\\pi_{S}(x,s) = t$. Therefore adding a caret to a leaf corresponding to a word ending in the symbol $0$ or $n-1$ in the domain tree, results in adding a caret with leaves labelled $e,t,e$ for $(e,t) \\in \\{ (b,a), (b^{-1}, a^{-1}), (s,p), (s^{-1},p^{-1})\\}$ to the corresponding leaf in the range tree. Let $D_{(A,B)} = d_1 d_2 \\ldots d_l$ be a word over the alphabet $(A_{a}, B_{p}, A_{a}^{-1}, B_{p^{-1}})$, and let $\\delta_i \\in \\{a, a^{-1}, p, p^{-1}\\}$ be such that $d_i = D_{\\delta_i}$ where $D = T^{\\pm 1}$ if $\\delta_i = a^{\\pm 1}$ and $D = U^{\\pm 1}$ if $\\delta_i = p^{\\pm 1}$. Let $\\varepsilon_i$ be defined by $\\varepsilon_i = b^{\\pm 1}$ if $\\delta_i = a^{\\pm 1}$ and $\\varepsilon_i = s^{\\pm 1}$ if $\\delta_i = p^{\\pm 1}$. As noted above $D(A,B)$ corresponds to a word $W(T,U)$ over the alphabet $(T_{a}, U_{p}, T_{a}^{-1}, U_{p}^{-1})$. Now a given a tree pair $(F_1, F_2)$ representing the product of $D_{(A,B)}$ in $\\gen{A_{a}, B_{p}}$, we obtain a tree pair for the product $W(T,U)$ initialised at the loop state for $x$ as follows. Form finite $n$-ary tree $\\overline{F}_i$, $i \\in \\{1,2\\}$, by adding at each level the `missing leaves'. That is $F_{i}$ is the resulting tree obtained by deleting the leaves of $\\overline{F}_{i}$ corresponding to words in $X_{n}^{+}$ which do not end in the letters $0$ or $n-1$. Decorate the leaves of $\\overline{F}_{2}$ as follows, all leaves corresponding to words in $\\Xn^{+}$ ending in $0$ or $n-1$ are decorated with the word $\\epsilon_1\\epsilon2\\ldots\\epsilon_l$ otherwise leaves are decorated with the symbol $\\delta_1\\delta_2\\ldots\\delta_i$. \n\nSince $\\epsilon_1\\epsilon2\\ldots\\epsilon_l$ is the unique loop state of $0$ and $n-1$ and acts as the identity on $\\Xn$, and since $\\delta_1\\delta_2\\ldots\\delta_i$ is the loop state for $x$, we see that if the product $D(A,B)$ is trivial in $\\gen{A_{a}, B_{p}}$, then $W(T,U)$ initialised at the state $\\delta_1\\delta_2\\ldots\\delta_i$ is also trivial. Now since the core is strongly connected, and $\\delta_1\\delta_2\\ldots\\delta_i$ is a state of the core, it follows that $W(T,U)$ is the identity transducer. \n\n\\end{comment}\n\\end{proof}\n\n\\begin{Remark}\\label{Remark:TaandUpgenerateF}\nNotice that, for $T$ and $U$ as in Figures~\\ref{Figure:elementinTon1ofinfiniteorder} and \\ref{Figure:ThetransducerUinTOn1}, the proof above shows that the subgroup $\\gen{T_{a}, U_{p}} \\le \\TBmr{n}{1}$, for $n \\ge 3$ is isomorphic to $F$. \n\\end{Remark}\n\nAs a corollary we have the following theorem generalising the result of Brin and Guzman (\\cite{MBrinFGuzman}) stating that the groups $\\out{T_{n,n-1}}$ and $\\aut{T_{n,n-1}}$ contain an isomorphic copy of $F$.\n\n\\begin{Theorem}\nFor $n \\ge 3$ and $1 \\le r \\le n-1$, the groups $\\TOnr \\cong \\out{T_{n,r}}$ and $\\TBnr \\cong \\aut{T_{n,r}}$ contain an isomorphic copy of Thompson's group $F$.\n\\end{Theorem}\n\\begin{proof}\nThat $\\TOnr$, for $n\\ge 3$ and $1 \\le r \\le n-1$, contains a group isomorphic to $F$ is a consequence of Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}. To deduce implications for $\\TBnr$, let $T$ and $U$ be the transducers in Figures~\\ref{Figure:elementinTon1ofinfiniteorder} and \\ref{Figure:ThetransducerUinTOn1}. Observe that the map $f$ defined by, for all $\\dot{d} \\in \\dotr$ and $\\xi \\in \\CCn$, $\\dot{d}\\xi \\mapsto \\dot{d}(\\xi)T_{a}$, and the map $g$ given by, for all $\\dot{d} \\in \\dotr$ and $\\xi \\in \\CCn$, $\\dot{d}\\xi \\mapsto \\dot{d}(\\xi)U_{p}$, are elements of $\\TBnr$ since $\\core(f) = T$ and $\\core(g) = U$. Moreover, they generate a subgroup isomorphic to $F$ by Remark~\\ref{Remark:TaandUpgenerateF}. \n\\end{proof}\n\\begin{comment}\nLet $\\rot$ be the equivalence relation on $\\Xnp$ such that $u \\simrot v$ if and only if $u$ is equal to a rotation of $v$. That is if $v = v_1 v_2 \\ldots v_{|v|}$, then, setting $v_0 = \\epsilon$, there is some $1 \\le i \\le |v|$ such that $u = v_i v-{i+1}\\ldots v_{|v|} v_{1}\\ldots v_{|i-1|}$. Let $\\RXnp := \\Xnp \/ \\simrot$. For a word $w \\in\\Xnp$, we denote by $\\rotclass{w}$ the equivalence class of $w$ under $\\simrot$. Let $\\sym(\\RXnp)$ be the symmetric group on $\\RXnp$. For each element $T$ of $\\TOn$,it is a result in \\cite{BCMNO} that for every word $w \\in \\Xnp$ there is a unique state $q_{w}$ of $T$ such that $\\pi_{T}(w, q_{w}) = q_{w}$. Let $\\overline{T}$ be the map from $\\RXnp \\to \\RXnp$ given by $u \\mapsto v$, where $v$ is the output of the unique state $q_u$ such that $\\pi_{T}(u,q_{u} )= q_u$. The following result is straight-forward.\n\n\\begin{proposition}\\label{Proposition:grouphomomorphismfromOntoSym}\nThe map $\\phi: \\TOn \\to \\sym(\\RXnp)$ is a group homomorphism.\n\\end{proposition}\n\\begin{proof}\nLet $w \\in \\Xnp$, and $T, U \\in \\TOn$. Suppose that $\\rotclass{v} = (\\rotclass{w})\\overline{T}$ for $v \\in \\Xnp$. This means $\\lambda_{T}(w, q_{w})$ is a rotation of $v$, for $q_{w} \\in Q_{T}$ satisfying $\\pi_{T}(w, q_w) = w$. Let $p_{v}$ be the state of $B$ such that $\\pi_{U}(v, p_{v}) = p_{v}$. Observe that $(\\rotclass{v})\\overline{T}$ is a rotation of the word $\\lambda_{U}(v, p_{v})$. Therefore $(\\rotclass{w})\\overline{T}\\overline{U} = \\rotclass{\\lambda_{U}(v, p_v)}$. Now observe that the unique state of $TU$ \n\\end{proof}\n\nWe now show that $\\Ons{r}$ contains a subgroup isomorphic to the Thompson's group $F_{2}$ for all $1 \\le r \\le n-1$.\n\n\\begin{Theorem}\nThe groups $\\TOns{r}$ for $1 \\le r < n-1$ and $n >2$ contain a subgroup isomorphic to Thompson's group $F_{2}$.\n\\end{Theorem}\n\\begin{proof}\nWe show that for $n>2$ the group $\\TOns{1}$ contains a subgroup isomorphic to $F_{2}$ the result then follows by Theorem~\\ref{Theorem:TOnisubsetofTOnjifidividesjinZnminus1}.\n\nWe construct an isomorphism $\\phi: F_2 \\to \\TOns{1}$ which maps $x_0$ and $x_1$ the standard generators for Thompson's group $F$ (see e.g. \\cite{CannFlydPar}) to transducers $X$ and $Y$ in $\\TOns{n-1}$ such that the $n$ loop state is a homeomorphism state, from this it follows that $X, Y \\in \\TOns{1}$. We then show that $X$ and $Y$ satisfy the relations of $F_2$, and that the map $\\phi$ is injective.\n\nNote that $x_1$ and $x_2$ are homeomorphisms of $\\CCn$ induced by the transducers below. We identify the transducers and the homeomorphism of $\\CCn$ they induce.\n\n\\begin{figure}[H]\n\\begin{center}\n \\begin{tikzpicture}[shorten >=0.5pt,node distance=3cm,on grid,auto] \n \\node[state, initial,initial text=] (q_0) {$q_0$}; \n \\node[state] (q_1) [xshift=-1cm, yshift=-2cm] {$q_1$}; \n \\node[state] (q_2) [xshift=1cm, yshift=-2cm] {$q_2$};\n \\node[state, initial,initial text=] (p_0)[xshift=4cm, yshift=0cm] {$p_0$};\n \\node[state] (p_1)[xshift=6cm, yshift=0cm] {$p_1$};\n \\node[state] (p_2) [xshift=8cm, yshift=0cm] {$p_2$};\n \\node[state] (p_3) [xshift=4cm, yshift=-2cm] {$p_3$};\n \\node [xshift=0cm, yshift=-3cm] {$x_0$};\n \\node [xshift=6cm, yshift=-3cm] {$x_1$};\n \\path[->] \n (q_0) edge node [swap] {$1|\\epsilon$} (q_1)\n edge node {$0|00$} (q_2)\n (q_1) edge node {$0|01$} node[swap] {$1|1$}(q_2) \n (q_2) edge[in=15, out=345, loop] node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n (p_0) edge node {$1|1$} (p_1)\n edge node[swap] {$0|0$} (p_3)\n (p_1) edge node[xshift=0cm] [swap] {$0|00$} (p_3)\n edge node {$1|\\epsilon$} (p_2)\n (p_2) edge node {$0|00$} node[xshift=0.1cm, yshift=-0.3cm] {$1|1$}(p_3) \n (p_3) edge[in=15, out=345, loop] node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ();\n \\end{tikzpicture}\n \\end{center}\n \\caption{Transducers representing generators of Thompson's group $F$.}\n \\label{Figure:transducersrepresentinggeneratorsofF}\n\\end{figure}\n\nWe now construct the element $X_0, X_1 \\in \\TOns{1}$ such that $(x_0)\\phi = X_0$ and $(x_1)\\phi = X_1$. Below transitions labelled by $a$ indicate how states of the transducer act on any element of $\\Xn \\backslash \\{ 0,1 \\}$.\n\n\\begin{figure}[H]\n\\begin{center}\n \\begin{tikzpicture}[shorten >=0.5pt,node distance=3cm,on grid,auto] \n \\node[state] (q_0) {$q_0$}; \n \\node[state] (q_1) [xshift=-1cm, yshift=-2cm] {$q_1$}; \n \\node[state] (q_2) [xshift=1cm, yshift=-2cm] {$q_2$};\n \\node[state] (p_0)[xshift=4cm, yshift=0cm] {$p_0$};\n \\node[state] (p_1)[xshift=6cm, yshift=0cm] {$p_1$};\n \\node[state] (p_2) [xshift=8cm, yshift=0cm] {$p_2$};\n \\node[state] (p_3) [xshift=4cm, yshift=-2cm] {$p_3$};\n \\node [xshift=0cm, yshift=-3cm] {$X_0$};\n \\node [xshift=6cm, yshift=-3cm] {$X_1$};\n \\path[->] \n (q_0) edge[in=80, out=200] node[xshift=0cm, yshift=0cm] {$1|\\epsilon$} (q_1)\n edge[out=340, in=100] node[xshift=0cm, yshift=0cm,swap] {$0|00$} (q_2)\n edge[in=105, out=75, loop] node[swap] {$a|a$} () \n (q_1) edge node {$0|01$} node[swap] {$1|1$}(q_2)\n edge[in=190, out=90] node {$a|0a$} (q_0) \n (q_2) edge[in=15, out=345, loop] node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n edge[in=350, out=90] node[swap] {$a|a$} (q_0)\n (p_0) edge node[swap] {$1|1$} (p_1)\n edge[in=120, out=240] node[swap] {$0|0$} (p_3)\n edge[in=105, out=75, loop] node[swap] {$a|a$} ()\n (p_1) edge[in=30, out=250] node[xshift=0.2cm] [swap] {$0|00$} (p_3)\n edge node {$1|\\epsilon$} (p_2)\n edge[in=10, out=170] node[swap] {$a|a$} (p_0)\n (p_2) edge[in=20,out=230] node {$0|00$} node[xshift=0.1cm, yshift=-0.3cm] {$1|1$}(p_3)\n edge[in=45, out=135] node[swap]{$a|0a$} (p_0) \n (p_3) edge[in=15, out=345, loop] node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n edge[in=250, out=110] node[swap]{$a|a$} (p_0);\n \\end{tikzpicture}\n \\end{center}\n \\caption{Images of $x_0$ and $x_1$ under $\\phi$.}\n \\label{Figure:embeddingFintoTOn}\n\\end{figure}\n\nObserve that since $X_0$ and $X_1$ have homeomorphism states, and since each stateThe inverses of $X_0$ and $X_1$ are given below:\n\n\\begin{figure}[H]\n\\begin{center}\n \\begin{tikzpicture}[shorten >=0.5pt,node distance=3cm,on grid,auto] \n \\node[state] (q_0) {$q_0$}; \n \\node[state] (q_1) [xshift=-1cm, yshift=-2cm] {$q_1$}; \n \\node[state] (q_2) [xshift=1cm, yshift=-2cm] {$q_2$};\n \\node[state] (p_0)[xshift=4cm, yshift=0cm] {$p_0$};\n \\node[state] (p_1)[xshift=6cm, yshift=0cm] {$p_1$};\n \\node[state] (p_2) [xshift=8cm, yshift=0cm] {$p_2$};\n \\node[state] (p_3) [xshift=4cm, yshift=-2cm] {$p_3$};\n \\node [xshift=0cm, yshift=-3cm] {$X_0^{-1}$};\n \\node [xshift=6cm, yshift=-3cm] {$X_1^{-1}$};\n \\path[->] \n (q_0) edge[in=80, out=200] node[xshift=0cm, yshift=0cm] {$0|\\epsilon$} (q_1)\n edge[out=340, in=100] node[xshift=0cm, yshift=0cm,swap] {$1|11$} (q_2)\n edge[in=105, out=75, loop] node[swap] {$a|a$} () \n (q_1) edge node {$1|10$} node[swap] {$0|0$}(q_2)\n edge[in=190, out=90] node {$a|1a$} (q_0) \n (q_2) edge[in=15, out=345, loop] node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n edge[in=350, out=90] node[swap] {$a|a$} (q_0)\n (p_0) edge node[swap] {$1|1$} (p_1)\n edge[in=120, out=240] node[swap] {$0|0$} (p_3)\n edge[in=105, out=75, loop] node[swap] {$a|a$} ()\n (p_1) edge[in=30, out=250] node[xshift=0.2cm] [swap] {$1|11$} (p_3)\n edge node {$0|\\epsilon$} (p_2)\n edge[in=10, out=170] node[swap] {$a|a$} (p_0)\n (p_2) edge[in=20,out=230] node {$1|10$} node[xshift=0.1cm, yshift=-0.3cm] {$0|0$}(p_3)\n edge[in=45, out=135] node[swap]{$a|1a$} (p_0) \n (p_3) edge[in=15, out=345, loop] node[swap] {$0|0$} node[xshift=0.7cm,yshift=-0.3cm]{$1|1$} ()\n edge[in=250, out=110] node[swap]{$a|a$} (p_0);\n \\end{tikzpicture}\n \\end{center}\n \\caption{The inverses of $X_0$ and $X_1$ in $\\TOns{1}$}\n \\label{Figure:embeddingFintoTOn}\n\\end{figure}\n\\end{proof}\n\\end{comment}\n\\section{Further properties of \\texorpdfstring{$\\TOn$}{Lg} and the \\texorpdfstring{$R_{\\infty}$}{Lg} property for \\texorpdfstring{$\\Tn$}{Lg} }\\label{Section:RftyTn}\n\nIn this section we deduce some properties of elements of $\\TOn$ which will enable us to demonstrate that the groups $\\Tnr$ for $1< r0$. Suppose that $\\ac{u} = \\{ u_0, u_1, \\ldots, u_{l-1} \\}$ (recall that antichains are always assumed to be ordered lexicographically). We form a homeomorphism of $\\CCn$ as follows. Let $\\sigma: \\ac{u} \\to \\ac{ u}$ be given by $u_{i} \\mapsto u_{(i+1)\\mod{l}}$ for $1 \\le i \\le l-1$. Let $h_{l}: \\CCn \\to \\CCn$ be given by $(u_i\\delta) h_{l} = (u_{i})\\sigma (\\delta)h_{A_{q_0}}$. Then since $A_{q_0}$ is a homeomorphism of $\\CCn$, $h_{l}$ is also homeomorphism of $\\CCn$. Moreover, as $h_{q_0}$ is orientation preserving, and preserves the relation $\\simeqI$, by choice of the permutation $\\sigma$ it follows that $h_{l}$ is an orientation preserving element of $\\TBmr{n}{1}$. Moreover, if $B_{p_0}$ is the transducer such that $h_{p_0} = h_{l}$, then $\\core(B_{p_0}) = \\core(A_{q_0})$. Thus there is an element of $f \\in \\Tmr{n}{1}$ such that $h_{l}f = h_{q_0}$. Furthermore, observe that as $A_{q_0}$ fixes $00\\ldots$, we have that the point $00\\ldots$ is on an orbit of length precisely $l$ under $h_{l}$. Moreover, any point on a finite orbit under $h_{l}$ must have orbit length a multiple of $l$. Thus for every $l \\in \\mathbb{N}$ such that $l \\cong 1 \\mod{n-1}$, let $h_{l} \\in \\TBmr{n}{1}$ be the map constructed as above, and let $f_{l}$ be such that $h_{l} = f_{l}h_{q_0} $. Therefore we conclude that for any $A_{q_0} \\in \\widetilde{\\TBmr{n}{1}}$ there are infinitely may $h_{q_0}$-twisted conjugacy classes.\n\nNow fix $r>1$, and let $\\bar{r}$ be minimal such that $\\TOns{\\bar{r}} = \\TOns{r}$, by Lemma~\\ref{Lemma:partialconverselemma} $\\bar{r}$ divides $r$ and $\\bar{r}$ divides $n-1$. Let $A_{q_0} \\in \\TBmr{n}{\\bar{r}}$ be an element which fixes $\\dot{0}00\\ldots$ and $\\dot{r-1}n-1n-1\\ldots$. Let $\\ac{u}$ be a complete antichain of $X_{n,r}^{*}$ of length $l$. Observe that as $l \\cong r \\mod{n-1}$ hence there is an $m \\in \\N$ such that $m\\bar{r} = l$. For $1 \\le i \\le m$ let $\\ac{u}_{i} : = \\{ u_{i,0}, \\ldots, u_{i,r-1}\\} \\subset \\ac{u}$ be such that for $1 \\le i_1 < i_2 \\le m$ all elements of $\\ac{u}_{i_1}$ are less than (in the lexicographic ordering) all elements of $\\ac{u}_{i_2}$. Let $\\sigma: \\{1,2 \\ldots, m\\} \\to \\{1,2 \\ldots, m\\}$ be given by $ (i)\\sigma = i+1 \\mod{m}$. We construct an element $h_{l} \\in \\TBnr$ as follows: for $\\Gamma \\in \\CCn$ and $1 \\le i \\le m$ and $0 \\le a \\le r-1$ let $(u_{i,a}\\Gamma) h_{l} = u_{(i)\\sigma,b}\\delta$ if and only if $(\\dot{a}\\Gamma)A_{q_0} = \\dot{b}\\delta$. Therefore since $A_{q_0}$ induces an orientation preserving homeomorphism on $\\CCmr{\\bar{r}}$, it follows that the ma $h_{l}$ induces a orientation preserving homeomorphism of $\\CCnr$. Moreover since $\\core(A_{q_0}) \\in \\TOnr$ it follows that $h_{l}$ is in fact and element of $\\TOnr$ and moreover (identifying $h_{l}$ with the initial transducer inducing inducing the homeomorphism $h_{l}$) $\\core(h_{l}) = \\core(A_{q_0})$. Lastly observe that the point $\\dot{0}00\\ldots$ is on an orbit of length $m$ under the action of $h_{l}$ and any other finite orbit of $h_{l}$ has length at least $l$.\n\nNow let $A_{q_0} \\in \\TBnr$ be arbitrary. Since $\\core(A_{q_0}) \\in \\TOns{\\bar{r}} = \\TOnr$, there is an element $B_{p_0} \\in \\TBmr{n}{\\bar{r}}$ such that $\\core(B_{p_0}) = \\core(A_{q_0})$. As before for each $l \\in \\N$ such that $l \\equiv r \\mod n-1$, there is an element $h_{l}$ with a minimal finite orbit of length $l$ and such that $\\core(h_{l}) = \\core(A_{q_0})$. Thus, there is a $f_{l} \\in \\Tnr$ such that $h_{q_0}f_{l} = h_{l}$. Therefore as in the case $r=1$, we conclude that for any there are infinitely many $h_{q_0}$-twisted conjugacy classes for any element $A_{q_0} \\in \\TBnr$.\n\n\\begin{comment}\nNow let $A_{q_0} \\in \\TBmr{n}{1}$ be orientation reversing such that $h_{q_0}$ fixes the point $00\\ldots$ and $n-1 n-1 \\ldots$ of $\\CCn$. Consider the antichains $\\ac{u} = \\{0, 10, 11, \\ldots 1n-1,2,3 \\ldots, n-1\\}$ and $\\ac{v} = \\{0, 1, \\ldots, n-2, n-10, n-11,\\ldots, n-1n-1\\}$. Let the elements of $\\ac{u}$ and $\\ac{v}$ be denoted in order by $u_1, u_2, \\ldots, u_{2n-1}$ and $v_1, v_2, \\ldots, v_{2n-1}$. Let $\\sigma: \\ac{u} \\to \\ac{v}$ be given by $u_i \\mapsto u_{n-i+1}$. Now let $h_{1}$ be defined by $u_i \\rho \\mapsto (u_i)\\sigma(\\rho)A_{q_0}$ for $\\rho \\in \\CCn$ and $1 \\le i \\le 2n-1$. Then $h_{1}^{2}$ has precisely 3 fixed points $00\\ldots$ $11\\ldots$ and $n-200\\ldots$.\n\\end{comment}\n\n{\\bfseries{Orientation reversing case:}} We now consider the orientation reversing case. Let $A_{s_0} \\in \\TBmr{n}{1}$ be orientation reversing. By Lemma~\\ref{Lemma: automorphisms can be modified to contain fixed point} we may assume that $(000\\ldots)A_{s_0} = n-1n-1n-1\\ldots$ and $(n-1n-1n-1\\ldots)A_{s_0} = n-1n-1n-1\\ldots$. Let $i$ be minimal such that $\\pi_{A}(0^{i}, s_0) = q_0$ and $\\pi_{A}((n-1)^{i}, s_0) = q_{n-1}$ (where $q_0$ and $q_{n-1}$ are the zero and one loop state of $A_{s_0}$ respectively). Let $(n-1)^{l_1} = (0^{i+1})A_{q_0}$ and $0^{l_2} = ((n-1)^{i+1})A_{q_0}$. Observe that $0<\\min\\{l_1, l_2\\}$ since $\\lambda_{A}(0, q_0) = n-1$ and $\\lambda_{A}(n-1, q_0) = n-1$ by Lemma~\\ref{Lemma: elements of TOn either fix 0 or swap 0 and n-1}. There is an element $h \\in \\Tn$ such that for any $\\delta \\in \\CCn$, $((n-1)^{l_1}\\delta)h = (n-1)^{i}\\delta$ and $(0^{l_2}\\delta h) = 0^{i}\\delta$. Let $C_{s'_0} \\in \\TBnr$ be such that $h_{s'_0} = h_{s_0} h$. Observe that $C_{s'_0}$ satisfies, $\\pi_{C}(0^{i}, s'_0) = q'_{0}$, $\\pi_{C}((n-1)^{i}, s'_0) = q'_{n-1}$, where $q'_{0}$ and $q'_{(n-1)}$ are the $0$ and $n-1$ loop states of $C_{s'_0}$ respectively, and, for $j \\ge i$, $\\lambda_{C}(0^{j}, s'_0) = (n-1)^{j}$ and $\\lambda_{C}((n-1)^{j}, s'_0) = 0^{j}$. As before we may replace $A_{s_0}$ with $C_{s'_0}$ without loss of generality, since there infinitely many $A_{s_0}$-twisted conjugacy classes if and only if there are infinitely many $C_{s'_0}$-twisted conjugacy classes. Therefore we may assume that there is an $i \\in \\N$, $i>0$ such that $A_{s_0}$ satisfies $\\pi_{A}(0^{i}, s_0) = q_{0}$, $\\pi_{A}((n-1)^{i}, s_0) = q_{n-1}$, where $q_{0}$ and $q_{(n-1)}$ are the $0$ and $n-1$ loop states of $A_{s_0}$ respectively, and, for $j \\ge i$, $\\lambda_{A}(0^{j}, s_0) = (n-1)^{j}$ and $\\lambda_{A}((n-1)^{j}, s_0) = 0^{j}$.\n\nWe now construct an element $g \\in \\Tn$ such that $g h_{s_0}$ such $(g h_{s_0})^2$ has the point $000\\ldots$ and $n-1n-1\\ldots$ as attracting fixed points.\n\n\nBy a branch of length $k$, for $k \\in \\N$, $k>0$, we shall mean the finite rooted $n$-ary tree with $1+ k(n-1)$ leaves and $k$ the length of the geodesic from the root to the left-most leaf. \n\nLet $\\bar{f} \\in \\Tn$ be defined as follows. Let $E$ and $F$ be the finite trees where the tree $E$ is obtained from the single caret, by attaching a branch of length $i+1$ to the first and penultimate leaves of the branch and the tree $F$ is obtained from the single caret by attaching a branch to the first and last leaves of the caret. Label the leaves of $E$ left to right by $1,2, \\ldots, 1 + (2k+1)(n-1)$ and the leaves of $F$ from right to left by $1,2, \\ldots, 1 + (2k+1)(n-1)$. The map $\\bar{f}$ is now defined by, for $\\Gamma \\in \\CCn$ and $\\mu_{j}$ and $\\nu_{j}$ the addresses of the $j$'th leaves of $E$ and $F$ respectively, $\\mu_{j}\\Gamma \\mapsto \\nu_{j}\\Gamma$.\n\nLet $f \\in \\TBmr{n}{1}$, be defined by, for $\\Gamma \\in \\CCn$, $\\mu_{j}\\Gamma \\mapsto \\nu_{j}(\\Gamma)A_{s_0}$ where $\\mu_{j}$ and $\\nu_{j}$ are the addresses of the $j$'th leaves of $E$ and $F$ respectively. Essentially $f$ is obtained from $\\bar{f}$ by attaching the state $s_0$ to all the leaves of $F$ the range tree of $\\bar{f}$. Moreover since $\\core(f) = \\core(A_{s_0})$ there is an element $g \\in \\Tn$ such that $f = gA_{s_0} $.\n\nBelow we shall consider the orbit of cones $U_{\\nu}$ for $\\nu \\in X_{n}^{+}$ under $f$, and to simplify notation, we identify the cone $U_{\\nu}$ with the finite word $\\nu$. Let us consider the orbit of the cone $(n-1)$ (a leaf of $E$) under $f$: \\[(n-1)\\mapsto 00^{i} \\mapsto n-1 (n-1)^i \\mapsto 00^i 0^i \\mapsto n-1 (n-1)^i (n-1)^i \\mapsto 00^i(0)^{2i} \\ldots \\]\nThis is because for $(\\alpha, \\bar{\\alpha}) \\in \\{ (0, n-1), (n-1, 0) \\}$ and $j \\in \\N$ $j\\ge i$, $\\lambda_{A}(\\alpha^{j}, s_0) = \\bar{\\alpha}^{j}$ and $\\pi_{A}(\\alpha^{j}, s_0) = q_{\\alpha}$. Hence we see that the point $n-1n-1\\ldots$ is an attracting fixed point of $f^2$ and in a similar way the point $00\\ldots$ is an attracting fixed point of $f$. In particular we have, for $j \\in \\N_{1}$, $(n-1)f^{2j} = (n-1)(n-1)^{ji}$ and $(00^{i})f^{2j} = 00^{i}0^{ji}$. We call the words $(n-1)^{i}$ and $0^{i}$ the attracting paths (in $f^2$) of $00\\ldots$ and $n-1n-1\\ldots\n$ respectively. \n\nLet $h \\in \\Tn$ be arbitrary and consider $h^{-1} f h$. Now as $h$ is a prefix exchange map, by an abuse of notation for a finite word $\\Gamma \\in \\Xn^{\\ast}$ such that $h$ acts as a prefix exchange on the cone $U_{\\Gamma} \\subset \\CCn$, we shall write, $(\\Gamma)h$ for the word $\\Delta \\in \\Xn^{*}$ such that $(U_{\\Gamma}) h = U_{\\Delta}$. There is an $l \\in \\N$ such that $((n-1)^{l}) h$ is a finite word in $X_n^{+}$ and we may assume that $l \\ge i+1$. Thus we have,\n\\[\n ((n-1)^{l})h h^{-1}f^2h = ((n-1)^{l})f^2h = ((n-1)^{l})h(n-1)^{i}\n\\]\n\nHence we deduce that $((n-1)^l)h h^{-1}f^{2j}h = ((n-1)^{l})h(n-1)^{ji}$ and we see that conjugation of $f$ by an element of $\\Tn$ maps the attracting fixed point $n-1n-1\\ldots$ to an attracting fixed point and preserves the path $(n-1)^{i}$. We also observe that since an orientation reversing homeomorphism of the circle has precisely two fixed points (on the circle), then $h$ must map the pair of fixed points of $f$ to the pair of fixed of $h^{-1}fh$. Notice that thinking of the fixed points as elements of $\\CCn$, since $\\Tn$ elements $\\Z[1\/n] \\to \\Z[1\/n]$, this means that either $h$ fixes the points $00\\ldots$ and $n-1 n-1 \\ldots$ otherwise, the other fixed point of $f$ is an element of $\\Z[1\/n]$ of the form $\\eta_1 \\simeqI \\eta_2$ (where $\\eta_1 > \\eta_2$ in the lexicographic ordering on $\\CCn$), $(00\\ldots)h =\\eta_1$ and $(n-1n-1\\ldots)h = \\eta_2$. We make use of these facts to show that there are infinitely many $h_{s_0}$-twisted conjugacy classes.\n\nLet $j$ be any integer greater than $i$, as before we may construct a map $f_{j} \\in \\TBmr{n}{1}$ such that $f_{j} = g_{j}h_{s_0}$ for some $g_{j} \\in \\Tn$, and $00\\ldots$ and $n-1n-1\\ldots$ are attracting fixed points of $f_{j}^2$ with attracting paths $0^{j}$ and $(n-1)^{j}$. We have the following claim:\n\n\\begin{claim}\nThere is an infinite subset $\\T{J} \\subset \\N_{i}$ for which the set $\\{f_{j} \\mid j \\in \\T{J}\\}$ are pairwise not conjugate by an element of $\\Tn$.\n\\end{claim}\n\\begin{proof}\nSuppose there is an $N \\in \\N$ such that every element $f_{j}$, $j \\in \\N_{i}$ is conjugate to some element of $\\{f_{i+1}, \\ldots\n f_{i+N}\\}$ by an element of $\\Tn$. Since $\\N_{i}$ is infinite, there is an infinite subset $\\T{I} \\subset \\N_{i}$ and $1 \\le M \\le N$ such that every element of $\\{f_{l} \\mid l \\in \\T{I}\\}$ is conjugate to $f_{i+M}$ by an element of $\\Tn$. Let $l > i+M$. Since conjugation by an element of $\\Tn$ preserves attracting paths and $l>M$ it must be the case that any conjugator $h \\in \\Tn$ such that $h^{-1}f_{l}h = f_{i+M}$ must satisfy $(n-1n-1\\ldots)h \\ne n-1 n-1 \\ldots$. Hence the other fixed point of $f_{i+m}$ must be a dyadic rational of the form $\\eta_1 \\simeqI \\eta_2 \\in \\CCn\/\\simeqI$ where $\\eta_1$ is greater than $\\eta_2$ in the lexicographic ordering of $\\CCn$. This means by arguments above that $(n-1n-1\\ldots)h = \\eta_2$ and $\\eta_2$ is an attracting fixed point of $f_{i+M}$ with attracting path $(n-1)^i$. However for $l' \\in \\T{I}$ such that $l' >l$, it must be the case that $f_{l'}$ is not conjugate to $f_{i+M}$ since the attracting path in $f_{l'}^2$ of $n-1n-1\\ldots$ is longer than the attracting path in $f_{i+M}^2$ of both attracting fixed points of $f_{i+M}$.\n\\end{proof}\n\nNow fix $1< r < n$. Let $A_{s_0} \\in \\TBnr$ be an orientation reversing element, $\\bar{r}$ be minimal such that $\\TOns{\\bar{r}} = \\TOns{r}$ and $B_{p_0}$ be an orientation reversing element of $\\TBmr{n}{\\bar{r}}$ with $\\core(B_{p_0}) = \\core(A_{s_0})$. As in the case $r=1$ we may assume that there is an $i \\in \\N$, $i>0$ such that $B_{p_0}$ satisfies $\\pi_{B}(\\dot{0}0^{i}, p_0) = q_{0}$, $\\pi_{B}(\\dot{r-1}(n-1)^{i}, p_0) = q_{n-1}$, where $q_{0}$ and $q_{(n-1)}$ are the $0$ and $n-1$ loop states of $\\core(A_{s_0}) = \\core(B_{p_0})$ respectively, and, for $j \\ge i$, $\\lambda_{B}(\\dot{0}0^{j}, p_0) = \\dot{r-1}(n-1)^{j}$ and $\\lambda_{B}(\\dot{r-1}(n-1)^{j}, p_0) = \\dot{0}0^{j}$. Now as in the orientation preserving case we construct elements $f_{j}$, for $j \\in \\N_{i}$ with desirable properties by simulating the element $B_{p_0}$ appropriately on cones.\n\nLet $\\ac{u} = \\{ \\dot{a} \\mid 1 \\le a \\le r \\}$, that is, $\\ac{u}$ corresponds to the roots of the disjoint union of $r$ copies of the $n$-ary tree. Since $\\bar{r}$ divides $r$ there is an $M \\in \\N$ such that $M\\bar{r} = r$. Let $\\ac{u}_{k}:= \\{ u_{k,1}, \\ldots, u_{k,r}\\}$, $1 \\le k \\le M$ be subsets of $\\ac{u}$, such that for $1 \\le k_1 < k_2 < M$, all elements of $\\ac{u}_{k_1}$ are less than (in the lexicographic ordering) all elements of $\\ac{u}_{k_2}$. Observe that $\\ac{u}_{1}$ corresponds to the roots of the disjoint union of the $\\bar{r}$ copies of the $n$-ary tree. Replace $\\ac{u}_1$, still retaining the symbol $\\ac{u}_{1}$ for the resulting antichain, with the antichain corresponding to attaching a branch of length $i+1$ to the root $u_{1,1}$. Likewise replace $\\ac{u}_M$ with the antichain corresponding to attaching a branch of length $i+1$ to the root $u_{M, r-1}$. In a similar way let $\\ac{v}_M$ be the antichain obtained from $\\ac{u}_{M}$ corresponding to attaching a branch of length $i+1$ to the root $\\ac{u}_1$, and let $\\ac{v}_{k}:= \\ac{u}_{k}$ for $1 \\le k < M$. Observe that $\\ac{u}_{1}$, $\\ac{u}_{M}$ and $\\ac{v}_{M}$ all have equal length $d$ congruent to $\\bar{r}$ modulo $n-1$. Since $\\bar{r}$ divides $n-1$ let $m \\in \\N$ be such that $m\\bar{r} = d$. Let $\\ac{u}_{1} = \\cup_{1 \\le a \\le m} \\ac{u}_{1,a}$ where $\\ac{u}_{1,a} = \\{u_{1,a,b} \\mid 1 \\le b \\le r \\}$, likewise let $\\ac{u}_{M} = \\cup_{1 \\le a \\le m} \\ac{u}_{M,a}$ where $\\ac{u}_{M,a} = \\{u_{M,a,b} \\mid 1 \\le b \\le r \\}$ and $\\ac{v}_{M} = \\cup_{1 \\le a \\le m} \\ac{v}_{M,a}$ where $\\ac{v}_{M,a} = \\{v_{M,a,b} \\mid 1 \\le b \\le r \\}$. \n\nWe construct a homeomorphism $f_{i}$ of $\\CCnr$ as follows. Let $\\sigma: \\{1,2,\\ldots,M\\} \\to \\{1,2, \\ldots M\\}$ by $k \\mapsto M-k +1$ and let $\\rho: \\{1,2,\\ldots, m\\} \\to \\{1,2,\\ldots m\\}$ by $a \\mapsto m-a+1$. For $\\Gamma \\in \\CCn$, $1< k < M$ and $1 \\le b \\le r$, $(u_{k,b}\\Gamma)f_{i} = v_{(k)\\sigma, b'}\\delta$ if and only if $\\lambda_{B}(\\dot{b}\\Gamma, p_0) \\in U_{\\dot{b'}}$ and $\\delta = \\lambda_{B}(\\Gamma, \\pi_{B}(\\dot{b}, p_0))$. For $\\Gamma \\in \\CCn$ $1 \\le a \\le m$ and $1 \\le b \\le r$, $(u_{1,a,b}\\Gamma)f_{i} = v_{M,((a)\\rho),b'}\\delta$ if and only if $\\lambda_{B}(\\dot{b}\\Gamma,p_0) \\in U_{\\dot{b'}}$ $\\delta = \\lambda_{B}(\\Gamma, \\pi_{B}(\\dot{b}, p_0))$. For $\\Gamma \\in \\CCn$ $1 \\le a \\le m$ and $1 \\le b \\le r$, $(u_{M,a,b}\\Gamma)f_{i} = v_{1,((a)\\rho),b'}\\delta$ if and only if $\\lambda_{B}(\\dot{b}\\Gamma,p_0) \\in U_{\\dot{b'}}$ $\\delta = \\lambda_{B}(\\Gamma, \\pi_{B}(\\dot{b}, p_0))$. Since $B_{p_0}$ induces an orientation reversing homeomorphism on $\\CCmr{\\dotr}$, we see that $f_{i}$ is in fact an element of $\\TBnr{r}$. Moreover, $\\core(f_i) = \\core(B_{p_0}) = \\core(A_{s_0})$ and so there is an element $g_i \\in \\Tnr$ such that $f_i = h_{s_0}g_i$.\n\nWe now argue that the points $\\dot{0}0\\ldots$ and $\\dot{r-1}n-1n-1\\ldots$ are attracting fixed points of $f_{i}$ with attracting paths $0^{i}$ and $(n-1)^{i}$. We consider the orbit of the cone $\\dot{r-1}(n-1)^{i}$:\n\\[\n\\dot{r-1}(n-1)^{i} \\mapsto \\dot{0}0^{i}0^{i} \\mapsto \\dot{r-1}(n-1)^{i}(n-1)^{i} \\mapsto \\dot{0}0^{i}0^{i}0^{i} \\mapsto \\dot{r-1}(n-1)^{i}(n-1)^{2i} \\ldots\n\\] \n\nthis follows by making use of the definition of $f_{i}$ and the facts: $\\pi_{B}(\\dot{0}0^{i}, p_0) = q_{0}$, $\\pi_{B}(\\dot{r-1}(n-1)^{i}, p_0) = q_{n-1}$, where $q_{0}$ and $q_{(n-1)}$ are the $0$ and $n-1$ loop states of $\\core(A_{s_0}) = \\core(B_{p_0})$ respectively, and, for $j \\ge i$, $\\lambda_{B}(\\dot{0}0^{j}, p_0) = \\dot{r-1}(n-1)^{j}$ and $\\lambda_{B}(\\dot{r-1}(n-1)^{j}, p_0) = \\dot{0}0^{j}$. In general we see that, for $l \\in \\N_{1}$, $(\\dot{r-1}(n-1)^{i})f_{i}^{2l} = \\dot{r-1}(n-1)^{i}(n-1)^{l}$ and $(\\dot{0}0^{i}0^{i})f_{i}^{2l} = \\dot{0}0^{i}0^{i}0^{l}$.\n\nFor each $j > i$ we may repeat the construction above to get elements $h_{j}$ such that $f_{j} = h_{s_0}g_{j}$ for some $g_{j} \\in \\Tnr$ and the points $\\dot{0}0\\ldots$ and $\\dot{r-1}n-1n-1\\ldots$ are attracting fixed points of $f_{j}$ with attracting paths $0^{j}$ and $(n-1)^{j}$. Since $f_{j}$ for $j \\in \\N_{i}$ induces a homeomorphism on circle of length $r$, $[0,r]$ with end points identified, we may repeat the arguments as in the case $r =1$ to conclude that there is an infinite subset $\\T{J} \\subset \\N_{i}$ for which the set $\\{f_{j} \\mid j \\in \\T{J}\\}$ are pairwise not conjugate by an element of $\\Tnr$.\n\n\nThus we have demonstrated the following:\n\n\\begin{Theorem}\\label{Theorem:TnrhastheRftyproperty}\nThe group $\\Tnr$ for $1 \\le r < n-1$ has the $R_{\\infty}$ property.\n\\end{Theorem}\n \n\n\\begin{comment}\nWe require a few result about conjugacy in homeomorphisms of the circle. Let $f$ be an orientation preserving homeomorphism of the circle, then, as usual we may identify $f$ with a map from $[0,1] \\to [0,1]$. Define a function $\\Delta_{f}$ as follows:\n\\[\n(x)\\Delta_f = \\begin{cases}\n1 \\ \\mbox{if } x < (x)f \\mbox{ in } [0,1], \\\\\n0 \\ \\mbox{if } x = (x)f, \\\\\n-1 \\ \\mbox{if } x > (x)f \\mbox{ in } [0,1]\n\n\\end{cases}.\n\\]\n\nThe following result is standard:\n\n\\begin{lemma}\nLet $f, h$ be orientation preserving homeomorphisms of the circle such that $f$ has a fixed point, then\n\\begin{enumerate}[label = (\\roman{*})]\n\\item $\\Delta_{h^{-1}fh} = h^{-1}\\Delta_{f}$\n\\item $\\Delta_{f}^{-1} = - \\Delta_{f}$.\n\\end{enumerate}\n\\end{lemma}\n\nThe following key result is standard in the literature concerning homeomorphisms of the circle.\n\n\\begin{Theorem}\nTwo orientation reversing homeomorphisms of the circle $f$ and $g$ are conjugate if and only if $f^2$ and $g^2$ are conjugate by a homeomorphism of the circle that maps the pair of fixed points of $f$ to the pair of fixed points of $g$.\n\\end{Theorem}\n\\end{comment}\n\n\\printbibliography \n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}