[6, 5115] finitely based
[6, 5115] is a commutative semigroup with 1 idempotent. 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 | 4 | 5 | 4 |
| 2 | 2 | 1 | 1 | 5 | 4 | 5 |
| 3 | 2 | 1 | 1 | 5 | 4 | 5 |
| 4 | 4 | 5 | 5 | 2 | 1 | 2 |
| 5 | 5 | 4 | 4 | 1 | 2 | 1 |
| 6 | 4 | 5 | 5 | 2 | 1 | 3 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 5115)
- Idempotents
- 1
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- yes
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 1, generated by {6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 3 identities irredundant
- xy ≈ yx
- x⁵yz ≈ xyz
- x²y ≈ x²yz⁴
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: 8 identities
- x³ ≈ x⁷
- xy ≈ yx
- x⁴ ≈ y⁴
- x³ ≈ x³y⁴
- x⁶y ≈ x²y
- x²y ≈ x²y⁵
- x⁵yz ≈ xyz
- x²y ≈ x²yz⁴
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6GenericCASRootWrappers.S6_5115.opposite_basis
BasisFor Generated.Order6GenericCASRootWrappers.S6_5115.table.semigroup.opposite Generated.Order6GenericCASRootData.S6_5113.basis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- other generated proof. A machine-generated proof that builds on semigroups certified earlier: an adapter or wrapper of a transfer found by the campaign's search.
- Size
- Checking this class alone compiles 28 Lean files with 8,262 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 749 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_5115 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5115 is the table, with the elements numbered 0 to 5.
Variety
[6, 5115] generates the variety V[5113]; 2 other semigroups of order six generate it too, and 253 semigroups of order six lie in it. [6, 5115] 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[5113] in the inclusion graph, with the varieties above and below it.