A certified basis of 8 identities
This list of identities is a basis, proved in Lean, of 131 semigroups of order six, all of which generate the variety V[3308]. Removing the identities that follow from the others leaves 4 identities (Vampire); the variety page shows the shortest known basis.
Identifier sb-ef7ad74e1491e24e in the data of the bases-min seal.
Certified for
[6, 3308] [6, 3312] [6, 3316] [6, 3392] [6, 3446] [6, 3452] [6, 3454] [6, 3515] [6, 3517] [6, 3519] [6, 3658] [6, 3660] [6, 3662] [6, 3714] [6, 3716] [6, 3718] [6, 6036] [6, 6039] [6, 6042] [6, 6122] [6, 6129] [6, 6130] [6, 6143] [6, 6146] [6, 6147] [6, 6185] [6, 6232] [6, 6331] [6, 6333] [6, 6334] [6, 6347] [6, 6348] [6, 6404] [6, 6407] [6, 6410] [6, 6500] [6, 7317] [6, 7342] [6, 7365] [6, 7380] [6, 7386] [6, 7448] [6, 7453] [6, 7467] [6, 7473] [6, 7479] [6, 7513] [6, 7518] [6, 7533] [6, 7534] [6, 7535] [6, 7629] [6, 7636] [6, 7643] [6, 7647] [6, 7654] [6, 7658] [6, 7663] [6, 7665] [6, 7666] [6, 7668] [6, 7683] [6, 7701] [6, 7716] [6, 7727] [6, 7748] [6, 7764] [6, 7778] [6, 7787] [6, 7792] [6, 7862] [6, 7866] [6, 7877] [6, 7878] [6, 7879] [6, 8333] [6, 8339] [6, 8346] [6, 8350] [6, 8355] [6, 8357] [6, 8358] [6, 8360] [6, 8519] [6, 8530] [6, 8624] [6, 8653] [6, 8728] [6, 8738] [6, 8769] [6, 8774] [6, 9871] [6, 10046] [6, 10110] [6, 10156] [6, 10262] [6, 10296] [6, 10328] [6, 10497] [6, 10505] [6, 10513] [6, 10522] [6, 10527] [6, 10530] [6, 10532] [6, 10542] [6, 10548] [6, 10578] [6, 10579] [6, 10580] [6, 10596] [6, 10603] [6, 10608] [6, 10615] [6, 10619] [6, 10622] [6, 10623] [6, 10625] [6, 10626] [6, 10660] [6, 10724] [6, 10759] [6, 10801] [6, 11132] [6, 11426] [6, 11442] [6, 11471] [6, 11490] [6, 11533] [6, 11566] [6, 11655]
The identities
- x² ≈ x³
- x²yx ≈ xyx
- xyx ≈ xyx²
- x²y² ≈ xyxy
- x²y² ≈ xy²x
- x²yz ≈ xyxz
- x²yzy ≈ xy²zx
- x²yz² ≈ xyz²x