A certified basis of 3 identities

This list of identities is a basis, proved in Lean, of 340 semigroups of order six, all of which generate the variety V[3303]. Removing the identities that follow from the others leaves 2 identities (Vampire); the variety page shows the shortest known basis.

Identifier sb-5f1a7142782b096e in the data of the bases-min seal.

Certified for

[6, 3303] [6, 3325] [6, 3443] [6, 3506] [6, 3508] [6, 3510] [6, 3562] [6, 3597] [6, 3601] [6, 3655] [6, 3705] [6, 3707] [6, 3709] [6, 3711] [6, 3719] [6, 3768] [6, 3770] [6, 3772] [6, 3774] [6, 3776] [6, 3778] [6, 3837] [6, 3869] [6, 3881] [6, 3907] [6, 3909] [6, 3933] [6, 3934] [6, 3936] [6, 3952] [6, 6044] [6, 6120] [6, 6137] [6, 6138] [6, 6149] [6, 6150] [6, 6202] [6, 6249] [6, 6329] [6, 6341] [6, 6342] [6, 6350] [6, 6351] [6, 6402] [6, 6405] [6, 6408] [6, 6416] [6, 6417] [6, 6420] [6, 6421] [6, 6424] [6, 6425] [6, 6429] [6, 6430] [6, 6431] [6, 6450] [6, 6486] [6, 6516] [6, 6520] [6, 6527] [6, 6555] [6, 6589] [6, 6592] [6, 6734] [6, 6753] [6, 7312] [6, 7337] [6, 7346] [6, 7360] [6, 7508] [6, 7524] [6, 7549] [6, 7565] [6, 7639] [6, 7640] [6, 7693] [6, 7696] [6, 7699] [6, 7745] [6, 7761] [6, 7767] [6, 7775] [6, 7859] [6, 7872] [6, 7894] [6, 7896] [6, 7898] [6, 7912] [6, 7914] [6, 7916] [6, 7919] [6, 7924] [6, 8064] [6, 8066] [6, 8068] [6, 8080] [6, 8084] [6, 8088] [6, 8129] [6, 8142] [6, 8144] [6, 8190] [6, 8198] [6, 8200] [6, 8238] [6, 8258] [6, 8297] [6, 8306] [6, 8342] [6, 8343] [6, 8387] [6, 8389] [6, 8400] [6, 8402] [6, 8404] [6, 8406] [6, 8442] [6, 8443] [6, 8455] [6, 8456] [6, 8460] [6, 8468] [6, 8469] [6, 8484] [6, 8511] [6, 8514] [6, 8517] [6, 8537] [6, 8548] [6, 8552] [6, 8553] [6, 8556] [6, 8558] [6, 8560] [6, 8619] [6, 8629] [6, 8650] [6, 8667] [6, 8669] [6, 8671] [6, 8684] [6, 8691] [6, 8693] [6, 8718] [6, 8732] [6, 8741] [6, 8743] [6, 8746] [6, 9888] [6, 9931] [6, 9933] [6, 9942] [6, 10041] [6, 10104] [6, 10123] [6, 10499] [6, 10544] [6, 10550] [6, 10584] [6, 10631] [6, 10632] [6, 10662] [6, 10694] [6, 10704] [6, 10721] [6, 10738] [6, 10740] [6, 10742] [6, 10746] [6, 10748] [6, 10756] [6, 10760] [6, 10772] [6, 10774] [6, 10776] [6, 10995] [6, 10996] [6, 11000] [6, 11001] [6, 11005] [6, 11006] [6, 11053] [6, 11055] [6, 11064] [6, 11088] [6, 11089] [6, 11093] [6, 11116] [6, 11140] [6, 11143] [6, 11172] [6, 11174] [6, 11180] [6, 11188] [6, 11258] [6, 11259] [6, 11263] [6, 11264] [6, 11266] [6, 11325] [6, 11327] [6, 11329] [6, 11330] [6, 11335] [6, 11339] [6, 11430] [6, 11465] [6, 11468] [6, 11472] [6, 11475] [6, 11477] [6, 11482] [6, 11485] [6, 11542] [6, 11557] [6, 11559] [6, 11567] [6, 11568] [6, 11574] [6, 11576] [6, 11578] [6, 11581] [6, 11584] [6, 11587] [6, 11657] [6, 11680] [6, 11681] [6, 11690] [6, 11703] [6, 11733] [6, 12536] [6, 12545] [6, 12548] [6, 12551] [6, 12587] [6, 12597] [6, 12609] [6, 12612] [6, 12617] [6, 12649] [6, 12657] [6, 12674] [6, 12682] [6, 12696] [6, 12717] [6, 12721] [6, 12731] [6, 12743] [6, 12785] [6, 12787] [6, 12789] [6, 12799] [6, 12801] [6, 12805] [6, 12810] [6, 12813] [6, 12819] [6, 13067] [6, 13079] [6, 13103] [6, 13108] [6, 13150] [6, 13202] [6, 13205] [6, 13223] [6, 13224] [6, 13245] [6, 13251] [6, 13267] [6, 13268] [6, 13275] [6, 13276] [6, 13277] [6, 13281] [6, 13284] [6, 13287] [6, 13288] [6, 13289] [6, 13291] [6, 13451] [6, 13452] [6, 13517] [6, 13518] [6, 13592] [6, 13594] [6, 13596] [6, 13599] [6, 13602] [6, 13604] [6, 13606] [6, 13608] [6, 13699] [6, 13700] [6, 13702] [6, 13703] [6, 13707] [6, 13781] [6, 13783] [6, 13785] [6, 13786] [6, 13788] [6, 13790] [6, 13791] [6, 13795] [6, 13797] [6, 13799] [6, 13804] [6, 13896] [6, 13905] [6, 13908] [6, 13911] [6, 13947] [6, 13957] [6, 13969] [6, 13972] [6, 13990] [6, 13991] [6, 14008] [6, 14010] [6, 14012] [6, 14096] [6, 14105] [6, 14135] [6, 14136] [6, 14137] [6, 14151] [6, 14171] [6, 14174] [6, 14175] [6, 14178] [6, 14254] [6, 14275] [6, 14357] [6, 14359] [6, 14361] [6, 14362] [6, 14365] [6, 14368] [6, 14427] [6, 14430]

The identities

x² ≈ x³
x²y ≈ xyx
xyz ≈ xzy