Variety V[12949]
The variety generated by [6, 12949]: all semigroups that satisfy every identity of [6, 12949]. 6 semigroups of order six generate it, and 2,658 semigroups of order six lie in it.
Shortest known basis
6 identities irredundant
- x² ≈ x³
- xyx ≈ xy²x
- xyx ≈ xyxyx
- x²y² ≈ x²y²x²y²
- x²y²x²y²z ≈ x²y²x²z
- xy²z²y² ≈ xz²y²z²y²
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
- 7 identities, certified in Lean for 6 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[12949] in the inclusion graph, with everything above and below it.
Generated by 6 semigroups of order six
Contains 2,658 semigroups of order six
The semigroups of order six that satisfy the basis.