A certified basis of 2 identities
This list of identities is a basis, proved in Lean, of 322 semigroups of order six, all of which generate the variety V[1229]. Removing the identities that follow from the others leaves 2 identities (Vampire); the variety page shows the shortest known basis.
Identifier sb-9cfb2fe381b25f59 in the data of the bases-min seal.
Certified for
[6, 1229] [6, 1241] [6, 1246] [6, 1251] [6, 1255] [6, 1258] [6, 1263] [6, 1275] [6, 1282] [6, 1288] [6, 1292] [6, 1296] [6, 1299] [6, 1301] [6, 1311] [6, 1326] [6, 1331] [6, 1337] [6, 1341] [6, 1344] [6, 1346] [6, 1355] [6, 1357] [6, 1365] [6, 1374] [6, 1381] [6, 1386] [6, 1390] [6, 1394] [6, 1397] [6, 1401] [6, 2939] [6, 2988] [6, 2998] [6, 3007] [6, 3023] [6, 3034] [6, 3042] [6, 3052] [6, 3054] [6, 3055] [6, 3080] [6, 3081] [6, 3086] [6, 3087] [6, 3090] [6, 3091] [6, 3093] [6, 3098] [6, 3109] [6, 3110] [6, 3115] [6, 3116] [6, 3119] [6, 3120] [6, 3124] [6, 3127] [6, 3142] [6, 3143] [6, 3148] [6, 3149] [6, 3152] [6, 3153] [6, 3155] [6, 3160] [6, 3168] [6, 4034] [6, 4037] [6, 4054] [6, 4063] [6, 4065] [6, 4074] [6, 4075] [6, 4084] [6, 4088] [6, 4092] [6, 4098] [6, 4101] [6, 4108] [6, 4111] [6, 4113] [6, 4140] [6, 4143] [6, 4158] [6, 4168] [6, 4170] [6, 4183] [6, 4184] [6, 4194] [6, 4197] [6, 4201] [6, 4207] [6, 4209] [6, 4215] [6, 4218] [6, 4220] [6, 4249] [6, 4252] [6, 4266] [6, 4275] [6, 4277] [6, 4284] [6, 4285] [6, 4292] [6, 4294] [6, 4298] [6, 4304] [6, 4307] [6, 4314] [6, 4317] [6, 4319] [6, 5799] [6, 5800] [6, 5808] [6, 5809] [6, 5814] [6, 5816] [6, 5818] [6, 5819] [6, 5824] [6, 5825] [6, 5830] [6, 5831] [6, 5839] [6, 5842] [6, 6066] [6, 6069] [6, 6206] [6, 6209] [6, 6256] [6, 6258] [6, 6299] [6, 6301] [6, 6360] [6, 6361] [6, 6453] [6, 6467] [6, 6490] [6, 6492] [6, 6533] [6, 6537] [6, 6549] [6, 6562] [6, 6564] [6, 6579] [6, 6587] [6, 6595] [6, 6597] [6, 6603] [6, 6606] [6, 6609] [6, 6627] [6, 6633] [6, 6637] [6, 6639] [6, 6743] [6, 6748] [6, 6751] [6, 6773] [6, 6795] [6, 6796] [6, 6797] [6, 6799] [6, 6800] [6, 6801] [6, 6811] [6, 6812] [6, 6814] [6, 6815] [6, 6817] [6, 6820] [6, 6821] [6, 6822] [6, 6823] [6, 6828] [6, 6829] [6, 6839] [6, 6840] [6, 6852] [6, 6859] [6, 6861] [6, 6870] [6, 6875] [6, 6888] [6, 6894] [6, 6895] [6, 6896] [6, 6897] [6, 6901] [6, 6904] [6, 6905] [6, 6919] [6, 6941] [6, 6942] [6, 6944] [6, 6945] [6, 6946] [6, 6954] [6, 6955] [6, 6957] [6, 6958] [6, 6960] [6, 6963] [6, 6964] [6, 6965] [6, 6966] [6, 6971] [6, 6972] [6, 6982] [6, 6983] [6, 6995] [6, 7002] [6, 7004] [6, 7013] [6, 7018] [6, 7031] [6, 7035] [6, 7036] [6, 7039] [6, 7040] [6, 7044] [6, 7047] [6, 7048] [6, 8838] [6, 8839] [6, 8843] [6, 8847] [6, 8849] [6, 8850] [6, 8868] [6, 8870] [6, 8871] [6, 8879] [6, 8881] [6, 8893] [6, 8894] [6, 8898] [6, 8910] [6, 8912] [6, 8922] [6, 8988] [6, 8989] [6, 8992] [6, 8995] [6, 8997] [6, 8998] [6, 9012] [6, 9014] [6, 9015] [6, 9021] [6, 9023] [6, 9035] [6, 9036] [6, 9039] [6, 9051] [6, 9053] [6, 9063] [6, 9828] [6, 9892] [6, 9894] [6, 9896] [6, 9900] [6, 9901] [6, 9902] [6, 9903] [6, 9904] [6, 9905] [6, 9939] [6, 9944] [6, 9945] [6, 9946] [6, 9947] [6, 9954] [6, 9955] [6, 10809] [6, 10810] [6, 10819] [6, 10820] [6, 10828] [6, 10829] [6, 10830] [6, 10833] [6, 10838] [6, 10843] [6, 11189] [6, 11191] [6, 11192] [6, 11359] [6, 11361] [6, 11366] [6, 11372] [6, 11378] [6, 11381] [6, 11509] [6, 11510] [6, 11513] [6, 11516] [6, 11519] [6, 11520] [6, 11590] [6, 11591] [6, 11593] [6, 11594] [6, 11596] [6, 11666] [6, 11667] [6, 11694] [6, 11696] [6, 11706] [6, 11707] [6, 11736] [6, 11738] [6, 11756] [6, 11772] [6, 11773] [6, 11783] [6, 11790] [6, 11791] [6, 11798] [6, 11799] [6, 11835] [6, 11836]
The identities
- x² ≈ x⁴
- xy ≈ yx