A certified basis of 4 identities
This list of identities is a basis, proved in Lean, of 160 semigroups of order six, all of which generate the variety V[1231]. Removing the identities that follow from the others leaves 3 identities (Vampire); the variety page shows the shortest known basis.
Identifier sb-c5ea964ebed33d39 in the data of the bases-min seal.
Certified for
[6, 1231] [6, 1233] [6, 1235] [6, 1240] [6, 1242] [6, 1244] [6, 1245] [6, 1249] [6, 1252] [6, 1253] [6, 1254] [6, 1257] [6, 1277] [6, 1279] [6, 1281] [6, 1283] [6, 1286] [6, 1287] [6, 1291] [6, 1293] [6, 1294] [6, 1295] [6, 1298] [6, 1300] [6, 1313] [6, 1315] [6, 1317] [6, 1321] [6, 1325] [6, 1327] [6, 1329] [6, 1330] [6, 1335] [6, 1338] [6, 1339] [6, 1340] [6, 1343] [6, 1345] [6, 1376] [6, 1378] [6, 1380] [6, 1382] [6, 1384] [6, 1385] [6, 1389] [6, 1391] [6, 1392] [6, 1393] [6, 1396] [6, 2943] [6, 2947] [6, 2952] [6, 2953] [6, 2973] [6, 2990] [6, 2994] [6, 2996] [6, 3005] [6, 3009] [6, 3025] [6, 3027] [6, 3029] [6, 3030] [6, 3039] [6, 3084] [6, 3085] [6, 3088] [6, 3089] [6, 3113] [6, 3114] [6, 3117] [6, 3118] [6, 3146] [6, 3147] [6, 3150] [6, 3151] [6, 4036] [6, 4039] [6, 4047] [6, 4049] [6, 4056] [6, 4060] [6, 4064] [6, 4066] [6, 4094] [6, 4095] [6, 4110] [6, 4112] [6, 4142] [6, 4145] [6, 4151] [6, 4153] [6, 4160] [6, 4165] [6, 4169] [6, 4171] [6, 4172] [6, 4173] [6, 4203] [6, 4204] [6, 4217] [6, 4219] [6, 4251] [6, 4254] [6, 4260] [6, 4262] [6, 4268] [6, 4272] [6, 4276] [6, 4278] [6, 4300] [6, 4301] [6, 4316] [6, 4318] [6, 5805] [6, 5806] [6, 5811] [6, 5812] [6, 6068] [6, 6071] [6, 6105] [6, 6107] [6, 6207] [6, 6233] [6, 6235] [6, 6237] [6, 6257] [6, 6281] [6, 6283] [6, 6285] [6, 6300] [6, 6302] [6, 6501] [6, 6503] [6, 6563] [6, 6573] [6, 6574] [6, 6575] [6, 6631] [6, 6635] [6, 6745] [6, 6749] [6, 6775] [6, 6787] [6, 6798] [6, 6818] [6, 6854] [6, 6860] [6, 6889] [6, 6921] [6, 6933] [6, 6943] [6, 6961] [6, 6997] [6, 7003] [6, 7032] [6, 9832] [6, 9866] [6, 9898] [6, 9941]
The identities
- x² ≈ x⁴
- xyx ≈ yx²
- x²y² ≈ xy²x
- xyz ≈ yxz