Variety V[5689]
The variety generated by [6, 5689]: all semigroups that satisfy every identity of [6, 5689]. 2 semigroups of order six generate it, and 5,335 semigroups of order six lie in it.
Shortest known basis
2 identities irredundant
- x⁴ ≈ x⁵
- xyz ≈ xzy
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
- 3 identities, certified in Lean for 2 semigroups
Among the 505 varieties
- Directly above
- Directly below
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[5689] in the inclusion graph, with everything above and below it.
Generated by 2 semigroups of order six
Contains 5,335 semigroups of order six
The semigroups of order six that satisfy the basis.