Variety V[1229]
The variety generated by [6, 1229]: all semigroups that satisfy every identity of [6, 1229]. 322 semigroups of order six generate it, and 1,246 semigroups of order six lie in it.
Shortest known basis
2 identities irredundant
- x² ≈ x⁴
- xy ≈ yx
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
- 2 identities, certified in Lean for 322 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[1229] in the inclusion graph, with everything above and below it.
Generated by 322 semigroups of order six
Show all 322
Contains 1,246 semigroups of order six
The semigroups of order six that satisfy the basis.