Variety V[3368]

The variety generated by [6, 3368]: all semigroups that satisfy every identity of [6, 3368]. 6 semigroups of order six generate it, and 1,566 semigroups of order six lie in it.

Shortest known basis

4 identities irredundant

x² ≈ x³
xyx ≈ xy²
x²yz ≈ xyz
xyzx ≈ xzyx

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

5 identities, certified in Lean for 6 semigroups

Among the 505 varieties

Directly above: V[3373] V[3812] V[5577] V[5661] V[6165] V[7390] V[7404]
Directly below: V[993] V[3191] V[3362]

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[3368] in the inclusion graph, with everything above and below it.

Generated by 6 semigroups of order six

[6, 3368] [6, 3370] [6, 3374] [6, 3376] [6, 6175] [6, 6176]

Contains 1,566 semigroups of order six

The semigroups of order six that satisfy the basis.

Show all 1,566