A certified basis of 3 identities

This list of identities is a basis, proved in Lean, of 345 semigroups of order six, all of which generate the variety V[3266]. Removing the identities that follow from the others leaves 3 identities (Vampire); the variety page shows the shortest known basis.

Identifier sb-1b71104a9509f281 in the data of the bases-min seal.

Certified for

[6, 3266] [6, 3267] [6, 3268] [6, 3269] [6, 3282] [6, 3283] [6, 3292] [6, 3293] [6, 3300] [6, 3386] [6, 3388] [6, 3416] [6, 3417] [6, 3418] [6, 3419] [6, 3424] [6, 3425] [6, 3436] [6, 3437] [6, 3440] [6, 3470] [6, 3472] [6, 3476] [6, 3478] [6, 3483] [6, 3484] [6, 3496] [6, 3500] [6, 3503] [6, 3544] [6, 3545] [6, 3552] [6, 3587] [6, 3588] [6, 3593] [6, 3638] [6, 3639] [6, 3640] [6, 3641] [6, 3644] [6, 3645] [6, 3648] [6, 3649] [6, 3652] [6, 3674] [6, 3675] [6, 3678] [6, 3679] [6, 3681] [6, 3683] [6, 3687] [6, 3688] [6, 3691] [6, 3695] [6, 3699] [6, 3700] [6, 3701] [6, 3741] [6, 3742] [6, 3744] [6, 3749] [6, 3753] [6, 3756] [6, 3760] [6, 3762] [6, 3765] [6, 3793] [6, 3797] [6, 3799] [6, 3834] [6, 3836] [6, 3845] [6, 3853] [6, 3854] [6, 3855] [6, 3857] [6, 3860] [6, 3861] [6, 3866] [6, 3901] [6, 3902] [6, 3905] [6, 3921] [6, 3922] [6, 3924] [6, 3925] [6, 3926] [6, 3929] [6, 6078] [6, 6079] [6, 6080] [6, 6081] [6, 6083] [6, 6085] [6, 6110] [6, 6111] [6, 6113] [6, 6204] [6, 6229] [6, 6231] [6, 6305] [6, 6306] [6, 6307] [6, 6308] [6, 6310] [6, 6312] [6, 6319] [6, 6320] [6, 6322] [6, 6365] [6, 6366] [6, 6368] [6, 6377] [6, 6378] [6, 6382] [6, 6389] [6, 6390] [6, 6392] [6, 6472] [6, 6473] [6, 6475] [6, 6488] [6, 6498] [6, 6499] [6, 6529] [6, 6530] [6, 6531] [6, 6548] [6, 6580] [6, 6581] [6, 6583] [6, 6586] [6, 6756] [6, 7301] [6, 7302] [6, 7309] [6, 7326] [6, 7327] [6, 7334] [6, 7368] [6, 7369] [6, 7375] [6, 7402] [6, 7403] [6, 7418] [6, 7419] [6, 7420] [6, 7421] [6, 7426] [6, 7427] [6, 7469] [6, 7493] [6, 7494] [6, 7588] [6, 7590] [6, 7592] [6, 7618] [6, 7738] [6, 7739] [6, 7742] [6, 7754] [6, 7755] [6, 7758] [6, 7780] [6, 7781] [6, 7783] [6, 7810] [6, 7811] [6, 7812] [6, 7813] [6, 7818] [6, 7819] [6, 7844] [6, 7845] [6, 7886] [6, 7888] [6, 7891] [6, 7904] [6, 7906] [6, 7909] [6, 7927] [6, 7929] [6, 7931] [6, 7956] [6, 7968] [6, 7969] [6, 7973] [6, 8013] [6, 8015] [6, 8021] [6, 8048] [6, 8049] [6, 8115] [6, 8118] [6, 8122] [6, 8162] [6, 8165] [6, 8168] [6, 8175] [6, 8176] [6, 8180] [6, 8182] [6, 8185] [6, 8261] [6, 8268] [6, 8269] [6, 8315] [6, 8317] [6, 8319] [6, 8324] [6, 8365] [6, 8367] [6, 8369] [6, 8373] [6, 8374] [6, 8377] [6, 8409] [6, 8416] [6, 8421] [6, 8487] [6, 8540] [6, 8541] [6, 8542] [6, 8549] [6, 8608] [6, 8609] [6, 8616] [6, 8643] [6, 8644] [6, 8647] [6, 8659] [6, 8661] [6, 8664] [6, 8680] [6, 8690] [6, 9910] [6, 9912] [6, 9916] [6, 10100] [6, 10101] [6, 10118] [6, 10292] [6, 10691] [6, 10701] [6, 10715] [6, 10716] [6, 10732] [6, 10735] [6, 10763] [6, 10764] [6, 10766] [6, 10771] [6, 10779] [6, 10782] [6, 10789] [6, 10856] [6, 10857] [6, 10858] [6, 10859] [6, 10865] [6, 10867] [6, 10882] [6, 10883] [6, 10884] [6, 10885] [6, 10886] [6, 10887] [6, 10894] [6, 10896] [6, 10899] [6, 10908] [6, 10911] [6, 10922] [6, 10923] [6, 10928] [6, 10929] [6, 10933] [6, 10939] [6, 10945] [6, 11186] [6, 11209] [6, 11211] [6, 11213] [6, 11217] [6, 11218] [6, 11223] [6, 11333] [6, 11340] [6, 11345] [6, 11349] [6, 11352] [6, 11486] [6, 11487] [6, 11494] [6, 11502] [6, 11573] [6, 11670] [6, 11671] [6, 11673] [6, 11692] [6, 12533] [6, 12542] [6, 12554] [6, 12572] [6, 12573] [6, 12606] [6, 12620] [6, 12635] [6, 12636] [6, 12667] [6, 12671] [6, 12700] [6, 12709] [6, 12710] [6, 12714] [6, 12803] [6, 12808] [6, 12816] [6, 12831] [6, 12832] [6, 12985] [6, 12986] [6, 12990] [6, 12993] [6, 12995] [6, 12996] [6, 13208] [6, 13238] [6, 13254] [6, 13297] [6, 13387] [6, 13390] [6, 13391] [6, 13613] [6, 13615] [6, 13893] [6, 13902] [6, 13914] [6, 13932] [6, 13933] [6, 13975] [6, 14121] [6, 14123] [6, 14125] [6, 14241]

The identities

x² ≈ x³
xyx ≈ yx²
x²y² ≈ y²x²