[6, 9117] finitely based
[6, 9117] is a semigroup with 2 idempotents. It is finitely based: 3 identities define its variety, which 2 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 1 | 1 | 1 |
| 2 | 2 | 1 | 1 | 2 | 2 | 2 |
| 3 | 2 | 1 | 1 | 2 | 2 | 2 |
| 4 | 1 | 2 | 3 | 4 | 5 | 6 |
| 5 | 1 | 2 | 3 | 5 | 6 | 4 |
| 6 | 1 | 2 | 3 | 6 | 4 | 5 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 9117)
- Idempotents
- 1, 4
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 2, generated by {3, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x²y² ≈ xy²x
- x⁷y ≈ xy
- xyz ≈ yxz
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: 4 identities
- xyx ≈ yx²
- x²y² ≈ xy²x
- x⁷y ≈ xy
- xyz ≈ yxz
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6FactorIntersectionIdentification.S6_9117.representative_basis
BasisFor Generated.Order6FactorIntersectionIdentification.S6_9117.table.semigroup Generated.Order6FactorIntersectionIdentification.S6_9117.basis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- subdirect product. This semigroup is a subdirect product of two factors, and the proof combines the bases of the factors.
- Size
- Checking this class alone compiles 69 Lean files with 38,182 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_9117 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9117 is the table, with the elements numbered 0 to 5.
Variety
[6, 9117] generates the variety V[9113]; 2 other semigroups of order six generate it too, and 1,237 semigroups of order six lie in it. [6, 9117] lies in 1 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- none: a maximal variety of the census
- Directly below
V[9113] in the inclusion graph, with the varieties above and below it.