Variety V[5625]
The variety generated by [6, 5625]: all semigroups that satisfy every identity of [6, 5625]. 1 semigroup of order six generates it, and 5,075 semigroups of order six lie in it.
Shortest known basis
12 identities reduced
- x³ ≈ x⁴
- x²y² ≈ xyxy
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- x³yx ≈ x²yx
- xyzx ≈ xzyx
- x²yzy ≈ xyxzy
- xyxzy ≈ yx²zy
- xyxz² ≈ xyzxz
- xyzxy ≈ xyzyx
- xyztx ≈ xytzx
- xyxztz ≈ xyzxtz
Reduced as far as the search went: some identity may still follow from the others. Obtained from a certified basis by removing identities that follow from the others (Vampire). Shortest known, not known to be minimal.
Certified bases
- 18 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[5625] in the inclusion graph, with everything above and below it.
Generated by 1 semigroup of order six
Contains 5,075 semigroups of order six
The semigroups of order six that satisfy the basis.