Homology of semigroups with 7 elements

Description

Below, all semigroups of order 7 are sorted by their first several homology groups [H1, H2, H3, ...].

Under each heading is a comma-separated list of integers: the ids of the semigroup in the smallsemi GAP package. For example, the the data below indicates that SmallSemigroup(7, 835206) and SmallSemigroup(7, 835214) both have homology [0, Z x Z, 0, 0, 0] for dimensions 1-5.

Checking these with the repository here:

sage: for id_ in [835206, 835214]:
....:     op = op_from_id(7, id_)
....:     crs = find_best_gens_crs(op, 5)
....:     print(crs.homology_list(5))
....:
[0, Z x Z, 0, 0, 0]
[0, Z x Z, 0, 0, 0]

There are 836_021 semigroups of order 7, up to isomorphism and anti-isomorphism. One can also look up the ids by the row number in the text document here.

Data

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0] with zero element (811544 semigroups)

(Too many to list)

[C2, 0, C2, 0, C2, 0, C2, 0, C2, 0] (21982 semigroups)

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331,825332,825333,825334,825335,825336,825337,825338,825339,825340,825341,825342,825343,825344,825345,825346,825347,825348,825349,825350,825351,825352,825353,825354,825355,825356,825357,825358,825359,825360,825361,825362,825363,825364,825365,825366,825367,825368,825369,825370,825371,825372,825373,825374,825375,825376,825377,825378,825379,825380,825381,825382,825383,825384,825385,825386,825387,825388,825389,825390,825391,825392,825393,825394,825395,825396,825397,825398,825399,825400,825401,825402,825403,825404,825405,825406,825407,825408,825409,825410,825411,825412,825413,825414,825415,825416,825417,825418,825419,825420,827969,827970,827971,827972,827973,827974,827975,827976,827977,827978,827979,827980,827981,827982,827983,828227,828228,828229,828230,828231,828321,828380,828405,828423

[C3, 0, C3, 0, C3, 0, C3, 0, C3, 0] (1484 semigroups)

291027,291028,291029,291030,291031,291032,291033,291034,291035,291036,291037,291038,291039,291040,535382,535383,535384,535385,535386,535387,535388,535389,535390,535391,535392,535393,535394,535395,535396,535397,535398,535399,535400,535401,535402,535403,535404,535405,535406,535407,535408,535409,535410,535411,561968,561969,561970,561971,561972,561973,561974,561975,561976,561977,561978,561979,561980,561981,561982,561983,561984,561985,561986,561987,561988,561989,561990,561991,561992,561993,561994,561995,561996,561997,561998,561999,562000,562001,562002,562003,562004,562005,562006,562007,562008,562009,562011,562013,562015,562026,562037,562048,562051,562053,562054,562055,686651,686652,686653,686654,686655,686656,686657,686658,686659,686660,686661,686662,686663,686664,686665,686666,686667,686668,686669,686670,686671,686672,686673,686674,686675,686676,686677,686678,687274,687275,687276,687277,687278,687279,687280,687322,692740,692741,692742,692743,692744,692745,692746,692747,692748,692749,692750,692751,692752,692753,692754,692755,692756,692757,692759,692761,692763,692765,692768,692771,692772,692773,692774,692775,723747,723748,723749,723750,723751,723752,723753,723754,723755,723756,723757,723758,723759,723760,723761,723762,723763,723764,723765,723766,723767,723768,723769,723770,723771,723772,723775,723776,723778,723783,723784,723785,723789,723790,723791,748810,748811,748812,748813,748814,748815,748816,748817,748818,748819,748820,748821,748822,748823,748824,748825,748826,748827,748828,748829,748830,748831,748832,748833,748834,748835,748836,748837,748838,748839,748840,748841,748842,748843,748844,748845,748846,748847,748848,748849,748850,748851,748852,748853,748854,748855,748856,748857,748858,748859,748860,748861,748862,748863,748864,748865,748866,748867,748868,748869,748870,748871,748872,748873,748874,748875,748876,748877,748878,748879,748880,748881,748882,748883,748884,748885,748886,748887,748888,748889,748890,748891,748892,748893,748894,748895,748896,748897,748898,748899,748900,748901,748902,748914,748916,748918,748920,748921,748923,748925,748927,748929,748931,748933,748937,748941,748964,748965,748966,748968,748970,748992,748993,748994,748996,748998,749012,749013,749014,749015,749016,749017,749018,749019,749020,749021,749022,749023,749024,749025,749026,749027,749028,749029,749030,749031,749032,749033,749034,749035,749036,749037,749038,749039,749040,749041,749042,749043,749044,749045,749046,749048,749049,749050,749054,749055,749056,749060,749061,749062,749063,749064,749065,749066,749067,749068,749069,749070,749071,749072,749073,749076,749077,756253,756254,756255,756256,756257,756258,756259,756260,756261,756262,756263,756264,756265,757954,757955,757956,757957,757958,757959,757960,757961,757963,757965,757967,757969,757970,757971,757972,760242,760243,760244,760245,760246,760247,760250,760251,760253,761314,761315,761316,761317,761318,761319,761320,761324,761325,761327,761331,761335,761340,761341,761342,761343,761344,761345,761347,761348,761349,761350,761351,761352,761353,761354,761355,761356,761357,796866,796867,796868,796869,796870,796871,796872,796873,796874,796875,796876,796877,796878,796879,796880,796881,796882,796883,796884,796885,796886,796887,796888,796889,796890,796891,796892,796893,796894,796895,796896,796897,796898,796899,796900,796901,796902,796903,796904,796905,796906,796907,796908,796909,796910,796911,796912,796913,796914,796915,796916,796917,796918,796919,796920,796921,796922,796923,796924,796929,796931,796933,796935,796936,796938,796940,796952,796954,796956,796960,796965,796966,796967,796971,796972,796973,796974,796975,796976,796977,796978,796979,796980,796981,796982,796983,796984,796985,796986,796987,796988,796989,796990,827701,827702,827703,827704,827705,827706,827707,827708,827709,827710,827711,827712,827713,827714,827715,827716,827717,827718,827719,827720,827721,827722,827723,827724,827725,827726,827727,827728,827729,827730,827731,827732,827733,827734,827735,827736,827737,827738,827739,827740,827741,827742,827743,827744,827745,827746,827747,827748,827749,827750,827751,827752,827753,827754,827755,827756,827757,827758,827759,827760,827761,827762,827763,827764,827765,827766,827767,827768,827769,827770,827771,827772,827773,827774,827775,827776,827777,827778,827779,827780,827781,827782,827783,827784,827785,827786,827817,827819,827820,827822,827824,827826,827827,827828,827830,827832,827834,827835,827836,827837,827839,827841,827843,827845,827847,827848,827849,827850,827851,827852,827853,827855,827857,827881,827884,827887,827890,827891,827896,827899,827910,827914,827917,827921,827935,827936,827937,827948,827952,827953,827966,827967,827968,827987,827988,827989,827996,827997,828002,828009,828010,828011,828018,828020,828021,828027,828028,828029,828030,828077,828078,828079,828080,828081,828082,828083,828084,828085,828086,828087,828088,828089,828090,828091,828092,828093,828094,828095,828096,828097,828098,828099,828100,828101,828102,828103,828104,828105,828106,828107,828108,828109,828110,828111,828112,828113,828114,828115,828116,828117,828118,828119,828120,828121,828122,828123,828124,828125,828126,828127,828128,828129,828130,828131,828132,828133,828134,828135,828136,828137,828138,828139,828140,828141,828142,828143,828144,828145,828146,828147,828148,828149,828150,828151,828152,828153,828154,828155,828156,828157,828158,828159,828160,828161,828162,828163,828164,828165,828166,828167,828168,828174,828175,828176,828178,828179,828182,828183,828184,828187,828188,828189,828190,828191,828192,828193,828196,828197,828198,828200,828201,828202,828203,828204,828205,828206,828210,828211,828212,828213,828214,828215,828220,828221,828222,828224,828225,828226,828233,828234,828235,828236,828237,828238,828240,828241,828242,828243,828244,828245,828248,828249,828250,828252,828253,828254,828258,828259,828260,828261,828262,828263,828264,828265,828266,828267,828268,828269,828270,828271,828272,828273,828274,828275,828276,828277,828278,828279,828280,828281,828282,828283,828284,828285,828286,828287,828288,828289,828290,828291,828292,828293,828294,828295,828296,828297,828298,828299,828300,828301,828302,828303,828304,828307,828308,828309,828310,828311,828313,828314,828315,828316,828317,828318,828319,828320,828322,828323,828324,828325,828326,828327,828328,828329,828330,828331,828332,828333,828334,828338,828339,828340,828341,828342,828343,828344,828345,828346,828347,828348,828349,828350,828351,828352,828353,828354,828355,828356,828357,828358,828359,828360,828361,828362,828363,828364,828365,828366,828367,828368,828369,828373,828374,828375,828376,828377,828378,828379,828382,828383,828384,828385,828386,828387,828388,828393,828394,828395,828396,828397,828398,828399,828400,828403,828404,828406,828407,828408,828409,828410,828411,828412,828413,828414,828425,828426,828427,828428,828429,828430,828431,828432,828433,828434,828435,828436,828437,828438,828439,828440,828441,828442,828443,828446,828447,828448,828449,828450,828451,828452,828453,828454,828455,828456,828457,828458,828459,828460,828461,828462,828463,828464,835415,835416,835417,835418,835419,835420,835421,835422,835423,835424,835425,835426,835427,835428,835429,835430,835431,835432,835433,835434,835435,835436,835437,835438,835439,835440,835441,835442,835443,835444,835445,835446,835447,835448,835449,835450,835451,835452,835453,835454,835455,835456,835457,835458,835459,835460,835461,835521,835522,835523,835524,835525,835526,835527,835528,835529,835530,835531,835532,835533,835534,835535,835536,835537,835538,835539,835540,835541,835542,835543,835544,835545,835546,835547,835548,835549,835550,835551,835552,835553,835554,835555,835556,835557,835558,835559,835560,835561,835562,835563,835564,835565,835566,835567,835568,835569,835570,835571,835572,835573,835574,835575,835576,835577,835578,835579,835580,835581,835582,835583,835584,835585,835586,835587,835588,835589,835590,835591,835592,835593,835594,835595,835596,835597,835598,835613,835614,835615,835616,835617,835618,835619,835620,835621,835622,835623,835624,835625,835626,835627,835628,835629,835630,835631,835632,835633,835634,835635,835636,835637,835638,835639,835640,835641,835642,835643,835644,835645,835646,835647,835648,835649,835650,835651,835652,835653,835654,835655,835656,835657,835658,835659,835660,835661,835662,835663,835664,835665,835666,835667,835668,835669,835670,835671,835672,835673,835674,835675,835676,835677,835678,835679,835680,835681,835682,835683,835684,835685,835686,835687,835688,835689,835690,835691,835692,835693,835694,835695,835707,835708,835709,835710,835711,835712,835713,835714,835715,835716,835717,835718,835719,835720,835721,835722,835723,835724,835725,835726,835727,835728,835729,835730,835731,835732,835733,835734,835735,835736,835737,835738,835739,835740,835741,835742,835743,835744,835745,835746,835747,835748,835749,835750,835751,835752,835753,835754,835755,835756,835757,835758,835759,835760,835761,835768,835769,835770,835771,835772,835773,835774,835779,835780,835781,835782,835783,835784,835785,835795,835796,835797,835798,835799,835800,835801,835802,835803,835804,835805,835806,835807,835808,835809,835810,835811,835812,835813,835814,835815,835816,835817,835818,835819,835820,835821,835822,835823,835824,835825,835829,835830,835831,835832,835833,835834,835835,835836,835837,835838,835839,835840,835841,835842,835843,835844,835845,835846,835847,835848,835849,835850,835851,835852,835853,835854,835855,835856,835857,835858,835859,835860,835861,835862,835863,835864,835865,835866,835867,835868,835869,835870,835871,835873,835874,835875,835876,835877,835878,835879,835880,835884,835885,835886,835887,835888,835889,835890,835891,835892,835893,835894,835895,835896,835897,835901,835902,835903,835904,835905,835906,835907,835908,835909,835910,835911,835912,835913,835914,835915,835920,835921,835922,835923,835924,835925,835926,835927,835928,835929,835930,835931,835932,835933,835934,835935,835936,835937,835938,835939,835940,835941,835942,835943,835944,835945,835946,835947,835948,835949,835950,835951,835952,835953,835954,835955,835956,835957,835958,835959,835960,835961,835962,835963,835964,835965,835966,835967,835968,835969,835970,835971,835972,835973,835974,835975,835976,835977,835978,835979,835980,835981,835982,835983,835984,835985,835986,835987,835988,835989,835990,835991,835992,835993,835994,835995,835996,835997,835998,835999,836000,836001,836002,836003,836004,836005,836006,836007,836008,836009,836010,836011,836012,836013,836014,836015,836016

[0, 0, 0, 0, 0] (273 semigroups)

744256,744341,744379,744407,744418,744577,744589,744593,744768,744775,744779,744915,744920,746273,747444,747447,748558,748561,760918,760948,760975,761243,793546,793630,793666,793730,793768,793836,793838,793841,793850,793855,793860,793869,793873,793876,793884,793891,794041,794050,794053,794055,794119,794150,794179,794186,794772,794773,794776,795031,795718,795722,795726,795727,795729,795814,817675,817704,817710,817732,817734,817744,817745,817820,817821,817824,817827,817831,817833,817835,817838,817877,817878,818011,818012,818026,818027,818049,818051,818062,818081,818084,818086,818172,818173,818192,818193,818209,818219,818221,818258,818262,818263,818266,818330,818331,818334,818337,818338,818405,818406,818409,818565,818566,818582,818583,818589,818590,818611,818612,818618,818619,818636,818637,818639,818721,818735,818755,818756,818765,818766,818871,818872,818876,818879,818880,818882,818884,818887,818888,818892,818894,818896,818899,818900,819192,819199,819201,819204,819205,819208,819211,819220,819222,819225,819226,819229,820326,820692,820758,820760,820762,820763,820764,820765,820771,821212,821231,821233,821234,821239,823490,823491,823516,823532,823559,823561,823580,823591,823599,823616,823629,823630,823655,824580,824581,824602,824612,824613,824839,824840,825159,826048,826049,826052,826078,826081,826626,826627,826648,826658,826659,826664,826670,826671,826924,826925,827366,835082,835084,835086,835087,835089,835090,835091,835093,835094,835096,835097,835099,835100,835102,835104,835106,835107,835108,835110,835111,835113,835114,835116,835117,835118,835120,835121,835124,835126,835127,835129,835133,835134,835136,835137,835139,835141,835142,835144,835145,835147,835148,835150,835151,835153,835154,835155,835157,835158,835160,835161,835163,835165,835166,835167,835169,835171,835173,835174,835175,835177,835178,835180,835182,835184,835186,835187,835197,835199,835202,835204,835205,835207,835213,835463

[C4, 0, C4, 0, C4, 0, C4, 0, C4, 0] (222 semigroups)

203363,203364,203365,203366,203367,203368,203369,203370,203371,203372,383539,383540,383541,521165,521166,521167,521168,521169,521170,521171,521172,521173,521174,521175,521176,526790,526791,526792,526793,526794,526795,526796,526797,526798,526799,526801,526802,526803,526804,526805,526806,526807,526808,526809,526810,526812,526813,526814,526815,526816,526817,526818,526819,526820,526821,597868,597869,597870,597871,597872,597873,597874,597875,597876,597877,597878,597879,643763,643764,643765,643766,643767,643768,643769,646766,646767,646768,646769,646770,646771,646772,646773,646774,646775,646777,646778,646779,646780,646781,646782,646783,646784,646785,646786,681327,681328,681329,681330,681331,681332,681333,681334,681335,681336,681337,682581,682582,682584,682585,682587,682588,684782,684783,684784,684785,684786,684787,684788,684794,684795,684796,684797,684798,684799,684800,686606,686607,686608,686609,686610,686611,686612,686613,686614,686615,686616,686617,686618,686619,686620,686629,686630,686631,686632,686633,686634,686635,686636,686637,686638,686639,686640,686641,686642,695769,695813,695815,695859,695861,754237,754238,755003,755004,755005,755006,755007,755008,755009,755010,755011,755012,755014,755015,755016,755017,755018,755019,755020,755674,755675,755676,755677,755678,755679,755680,756231,756232,756233,756234,756235,756236,756237,756238,756239,756240,756241,756242,756243,756244,756358,757192,757193,757194,757195,757196,757197,757198,757199,757200,757203,757204,757903,757904,757905,757906,757907,757908,757909,757910,757911,757912,757968

[0, Z, 0, 0, 0] (163 semigroups)

744300,744342,744380,744384,744408,744586,744772,744914,744919,746329,760938,760974,761299,793617,793731,793739,793839,793842,793851,793858,794040,794044,794049,794052,794149,794185,795716,795717,795719,817703,817708,817733,817823,817829,817834,817879,817999,818003,818013,818028,818050,818061,818080,818085,818101,818174,818194,818210,818220,818329,818336,818404,818530,818557,818567,818581,818585,818588,818610,818614,818617,818635,818640,818757,818767,818870,818875,818878,818886,818891,818898,819191,819200,819203,819207,819210,819221,819224,819228,820525,820541,820554,820693,820715,820759,821211,821229,821230,821232,821272,823515,823530,823560,823628,823995,824037,824101,824151,824601,824950,825158,825160,826047,826077,826625,826647,826657,826672,826923,827367,835081,835083,835085,835088,835092,835095,835098,835103,835105,835109,835112,835115,835119,835122,835125,835128,835130,835131,835132,835135,835138,835140,835143,835146,835149,835152,835156,835159,835164,835168,835170,835172,835176,835179,835181,835183,835190,835191,835192,835193,835194,835195,835196,835198,835200,835201,835203,835208,835209,835210,835211,835212,835462

[C2 x C2, C2, C2 x C2 x C2, C2 x C2, C2 x C2 x C2 x C2, C2 x C2 x C2, C2^5, C2 x C2 x C2 x C2, C2^6, C2^5] (147 semigroups)

11464,11465,11466,11592,11760,11761,11762,11763,11764,11765,11767,11768,11769,11770,11771,82687,82688,82689,82690,82691,83146,83147,83148,83149,83150,83151,86240,86241,86242,86243,86244,86245,86246,86247,86248,86249,86583,86584,86585,86586,86587,86588,86589,86590,86591,86592,86692,86693,86694,86695,86696,86697,86698,86699,86700,86701,86703,86704,86705,86706,86707,86708,86709,203355,203356,203357,203358,203359,203360,203361,203362,279904,279905,279907,279908,281853,281854,281855,281856,281857,281858,281859,281974,281975,281976,281977,281978,281979,281980,281981,290459,290460,290461,290462,290463,290464,290465,290466,290467,290468,290469,290470,290471,290472,290473,291004,291005,291006,291007,291008,291009,291010,291011,291012,291013,291014,291015,291016,291017,536266,536267,536268,536269,536270,536516,544626,544627,544628,544629,544630,544631,544632,544633,544634,544637,544638,561879,561880,561881,561882,561883,561884,561885,561886,561887,561888,562049

[0, Z, Z, Z, Z, Z] (72 semigroups)

692400,744308,744346,744576,744579,744590,744592,744767,744770,744776,744778,747443,748557,760917,760947,760977,761070,761242,793629,793675,793844,793847,793854,793861,793870,793872,793875,793890,794047,794118,794141,794178,794774,794775,795721,795724,795730,795732,795815,795950,817674,817743,817819,817826,817837,818257,818261,818265,818268,818333,818408,818883,818895,819215,819217,820325,820757,820761,820770,821221,821224,822797,822798,822799,823489,823579,823590,823654,824579,824611,824616,824838

[0, Z, Z x Z, Z x Z x Z x Z, Z^8, Z^16] (18 semigroups)

692313,744255,744417,744917,745667,746272,757786,759491,760209,793545,793835,793883,794056,794771,795728,795731,795813,795949

[C5, 0, C5, 0, C5, 0, C5, 0, C5, 0] (18 semigroups)

749078,761358,828418,828419,828420,828422,828424,828465,835791,835792,835793,835917,835918,835919,836018,836019,836020,836021

[0, 0, Z, 0, 0, 0] (16 semigroups)

817709,817830,818063,818082,818586,818615,818641,818874,818890,820555,823531,835101,835123,835162,835185,835188

[0, 0, Z, Z, Z, Z] (14 semigroups)

744309,744347,744587,744773,793676,793845,793848,793857,794046,795720,817870,818269,819213,824100

[0, Z, C2, 0, C2, 0] (12 semigroups)

747521,747527,748636,748642,793662,793769,795030,826201,826207,827071,827192,827365

[0, Z, 0, Z, 0, Z] (9 semigroups)

744581,744584,793863,793867,795723,795816,818088,818179,819231

[0, Z, Z x Z, Z x Z x Z, Z x Z x Z x Z, Z^5] (9 semigroups)

558510,558574,692321,692350,716616,757770,757778,759501,766522

[0, 0, Z, 0, Z, 0] (7 semigroups)

744582,793864,793866,795725,818089,818178,819232

[0, C2, 0, C2, 0, C2] (6 semigroups)

747525,748640,793667,794110,826205,827156

[C6, 0, C6, 0, C6, 0, C6, 0, C6, 0] (5 semigroups)

749074,749075,828381,828444,828445

[0, 0, Z x Z, 0, 0, 0] (4 semigroups)

744385,794043,818102,835189

[0, Z, Z x Z x Z, Z^9, Z^27, Z^81] (4 semigroups)

558464,716602,719964,766521

[C2, 0, C6, 0, C2, 0, C6, 0] (4 semigroups)

291041,291042,562050,562056

[0, Z x Z, 0, 0, 0, 0] (2 semigroups)

835206,835214

[0, 0, C2, 0, C2, 0] (1 semigroup)

794109

[0, 0, Z, C2, 0, C2] (1 semigroup)

793740

[0, Z x Z, Z x Z, Z x Z, Z x Z, Z x Z] (1 semigroup)

820778

[0, Z, C3, 0, C3, 0] (1 semigroup)

835464

[0, Z, Z x C2, Z x C2 x C2, Z x C2^5, Z x C2^10] (1 semigroup)

794172

[C7, 0, C7, 0, C7, 0, C7, 0, C7, 0] (1 semigroup)

836017

In Lexicographic Order

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0] with zero element : x811544
[0, 0, 0, 0, 0] : x273
[0, 0, C2, 0, C2, 0] : x1
[0, 0, Z x Z, 0, 0, 0] : x4
[0, 0, Z, 0, 0, 0] : x16
[0, 0, Z, 0, Z, 0] : x7
[0, 0, Z, C2, 0, C2] : x1
[0, 0, Z, Z, Z, Z] : x14
[0, C2, 0, C2, 0, C2] : x6
[0, Z x Z, 0, 0, 0, 0] : x2
[0, Z x Z, Z x Z, Z x Z, Z x Z, Z x Z] : x1
[0, Z, 0, 0, 0] : x163
[0, Z, 0, Z, 0, Z] : x9
[0, Z, C2, 0, C2, 0] : x12
[0, Z, C3, 0, C3, 0] : x1
[0, Z, Z x C2, Z x C2 x C2, Z x C2^5, Z x C2^10] : x1
[0, Z, Z x Z x Z, Z^9, Z^27, Z^81] : x4
[0, Z, Z x Z, Z x Z x Z x Z, Z^8, Z^16] : x18
[0, Z, Z x Z, Z x Z x Z, Z x Z x Z x Z, Z^5] : x9
[0, Z, Z, Z, Z, Z] : x72
[C2 x C2, C2, C2 x C2 x C2, C2 x C2, C2 x C2 x C2 x C2, C2 x C2 x C2, C2^5, C2 x C2 x C2 x C2, C2^6, C2^5] : x147
[C2, 0, C2, 0, C2, 0, C2, 0, C2, 0] : x21982
[C2, 0, C6, 0, C2, 0, C6, 0] : x4
[C3, 0, C3, 0, C3, 0, C3, 0, C3, 0] : x1484
[C4, 0, C4, 0, C4, 0, C4, 0, C4, 0] : x222
[C5, 0, C5, 0, C5, 0, C5, 0, C5, 0] : x18
[C6, 0, C6, 0, C6, 0, C6, 0, C6, 0] : x5
[C7, 0, C7, 0, C7, 0, C7, 0, C7, 0] : x1