Variety V[4061]

The variety generated by [6, 4061]: all semigroups that satisfy every identity of [6, 4061]. 8 semigroups of order six generate it, and 4,902 semigroups of order six lie in it.

Shortest known basis

4 identities irredundant

x² ≈ x⁴
xyx ≈ xyx³
x²yzy ≈ xy²zx
xyzt ≈ xzyt

Irredundant: none of these identities follows from the others; for each, a semigroup satisfies the others but not it. Obtained from a certified basis by removing identities that follow from the others (Vampire). Shortest known, not known to be minimal.

Certified bases

13 identities, certified in Lean for 8 semigroups

Among the 505 varieties

Directly above: V[4091] V[6239] V[8861]
Directly below: V[1247] V[3307] V[4055] V[4058]

A variety lies above another when it contains it. An inclusion holds when the basis of the smaller variety derives that of the larger one, proved by Vampire; it fails when a semigroup of order six lies in the one and not in the other.

Explore V[4061] in the inclusion graph, with everything above and below it.

Generated by 8 semigroups of order six

[6, 4061] [6, 4096] [6, 4166] [6, 4205] [6, 4273] [6, 4302] [6, 6234] [6, 6502]

Contains 4,902 semigroups of order six

The semigroups of order six that satisfy the basis.

Show all 4,902