Variety V[2771]
The variety generated by [6, 2771]: all semigroups that satisfy every identity of [6, 2771]. 6 semigroups of order six generate it, and 4,929 semigroups of order six lie in it.
Shortest known basis
31 identities reduced
- x³ ≈ x⁴
- x²y² ≈ xyxy
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- x³yx ≈ x²yx
- x²yx² ≈ xyx²
- x²yzy ≈ x²zy²
- x²yzy ≈ xyxzy
- x²yzy ≈ xyzxy
- x²yz² ≈ xyxz²
- xyxzy ≈ xy²zx
- xyxzy ≈ yx²zy
- xyzxy ≈ xyzyx
- xyzxz ≈ xyz²x
- xyzyx ≈ xzy²x
- x²yxzx ≈ xyxzx
- x²yztz ≈ x²tzyz
- x²yztz ≈ xyxztz
- x²yztz ≈ xyztxz
- xyxztz ≈ xyzxtz
- xyxzt² ≈ xytzxt
- xyxzt² ≈ xtyxzt
- xyzxty ≈ yxzxty
- xyzxtz ≈ zyx²tz
- xyz²tx ≈ xyztzx
- xyztxz ≈ xyztzx
- xyxztut ≈ xyxutzt
- xyxztut ≈ xytzxut
- xyxztut ≈ xzxytut
- xyztxuz ≈ xyztzux
- xyztxuz ≈ zyxtxuz
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
- 57 identities, certified in Lean for 6 semigroups
Among the 505 varieties
- Directly above
- Directly below
- Undecided
- whether V[1110] lies in V[2771]: 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[2771] in the inclusion graph, with everything above and below it.
Generated by 6 semigroups of order six
Contains 4,929 semigroups of order six
The semigroups of order six that satisfy the basis.