[6, 2222] finitely based
[6, 2222] is a commutative nilpotent semigroup: every product of 4 elements is equal to the zero. It is finitely based: 2 identities define its variety, which 2 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 3 |
| 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| 4 | 1 | 1 | 1 | 1 | 3 | 1 |
| 5 | 1 | 1 | 1 | 3 | 2 | 4 |
| 6 | 1 | 3 | 1 | 1 | 4 | 2 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal; grey: the zero.
Structure
- Smallsemi
- SmallSemigroup(6, 2222)
- Idempotents
- 1
- Zero
- 1
- Identity
- none
- Nilpotent
- yes, of class 4
- Commutative
- yes
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 2, generated by {5, 6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 2 identities irredundant
- xy ≈ yx
- xyzt ≈ uvws
Irredundant: none of these identities follows from the others; for each, a semigroup satisfies the others but not it. Obtained from a basis certified in Lean by removing identities that follow from the others, with the prover Vampire; the equivalence rests on Vampire's proofs, not on Lean. Shortest known, not known to be minimal.
Basis certified in Lean: 9 identities
- xy ≈ yx
- x²y ≈ xyx
- x²y ≈ yx²
- xyx ≈ yx²
- xyz ≈ xzy
- xyz ≈ yxz
- xyz ≈ yzx
- xyz ≈ zyx
- xyzt ≈ uvws
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6Nilpotent.Certificates.Cutoff4.Signature0cb0f3893ed9.S6_2222.representative_basis
BasisFor Generated.Order6Nilpotent.Certificates.Cutoff4.Signature0cb0f3893ed9.S6_2222.table.semigroup Generated.Order6Nilpotent.Signature0cb0f3893ed9.representativeBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- nilpotent certificate. This semigroup is nilpotent: all products of a fixed number of elements are equal to the zero. The generated certificate lists the identities between short words, and the basis consists of them.
- Size
- Checking this class alone compiles 20 Lean files with 7,367 lines: the endpoint theorem and everything it imports, the shared library included. All of it is shared with the proofs of other classes.
- Census
- SemiBase.Census.S6_2222 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2222 is the table, with the elements numbered 0 to 5.
Variety
[6, 2222] generates the variety V[881]; 2 other semigroups of order six generate it too, and 198 semigroups of order six lie in it. [6, 2222] lies in 19 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[881] in the inclusion graph, with the varieties above and below it.