Variety V[5661]
The variety generated by [6, 5661]: all semigroups that satisfy every identity of [6, 5661]. 1 semigroup of order six generates it, and 6,804 semigroups of order six lie in it.
Shortest known basis
28 identities reduced
- x³ ≈ x⁴
- x²y² ≈ xyxy
- x²y² ≈ xy²x
- x²yx ≈ x²yx²
- xyx² ≈ xyx³
- x²yzy ≈ x²zy²
- x²yzy ≈ xyxzy
- x²yzy ≈ xyzyx
- x²yz² ≈ xyzxz
- x²yz² ≈ xyz²x
- xyxz² ≈ xzyxz
- xyxz² ≈ xz²yx
- xyxzx ≈ xyxzx²
- x²yztz ≈ x²tzyz
- x²yztz ≈ xyzxtz
- x²yztz ≈ xyztzx
- xyxztz ≈ xyxtz²
- xyxztz ≈ xzyxtz
- xyxztz ≈ xztzyx
- xyxzt² ≈ xztyxt
- xyxzt² ≈ xzt²yx
- xyxztut ≈ xyxutzt
- xyxztut ≈ xztyxut
- xyxztut ≈ xztutyx
- x²yzytuvu ≈ x²tuvuyzy
- x²yztzuvwv ≈ x²uvwvyztz
- xyxzt²uvwv ≈ xyxuvwvzt²
- xyxztutvwsw ≈ xyxvwswztut
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
- 64 identities, certified in Lean for 1 semigroup
Among the 505 varieties
- Directly above
- none: a maximal variety of the census
- Directly below
- Undecided
- whether V[3553] lies in V[5661]; whether V[3812] lies in V[5661]; whether V[1110] lies in V[5661]: no proof was found, and no semigroup of order six refutes it.
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[5661] in the inclusion graph, with everything above and below it.
Generated by 1 semigroup of order six
Contains 6,804 semigroups of order six
The semigroups of order six that satisfy the basis.