Variety V[42]
The variety generated by [6, 42]: all semigroups that satisfy every identity of [6, 42]. 1 semigroup of order six generates it, and 185 semigroups of order six lie in it.
Shortest known basis
3 identities irredundant
- x² ≈ x³
- xyz ≈ zyx
- x²y ≈ ztuv
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
- 39 identities, certified in Lean for 1 semigroup
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[42] in the inclusion graph, with everything above and below it.
Generated by 1 semigroup of order six
Contains 185 semigroups of order six
The semigroups of order six that satisfy the basis.