[6, 3122] finitely based
[6, 3122] is a semigroup with 2 idempotents. It is finitely based: 2 identities define its variety, which 298 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 3 | 3 | 1 | 3 |
| 2 | 1 | 1 | 3 | 3 | 1 | 3 |
| 3 | 3 | 3 | 1 | 1 | 3 | 1 |
| 4 | 3 | 3 | 1 | 1 | 3 | 1 |
| 5 | 1 | 2 | 3 | 4 | 5 | 6 |
| 6 | 3 | 4 | 1 | 2 | 6 | 5 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 3122)
- Idempotents
- 1, 5
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 2, generated by {2, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 2 identities irredundant
- x³y ≈ xy
- x²y² ≈ y²x²
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
- x³y ≈ xy
- x²y² ≈ y²x²
- xy³ ≈ yx³
- xyz ≈ yxz
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6ExtensionTransfersV7.S6_3122.representative_basis
BasisFor Generated.Order6ExtensionTransfersV7.S6_3122.table.semigroup Generated.Order6ExtensionTransfersV7.S6_3122.targetBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- transfer from order ≤ 5. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup; S has order at most five, and this semigroup satisfies the basis of S. Then every identity of this semigroup holds in S and follows from the basis, so the basis of S is a basis here too. An earlier wave of such transfers.
- Size
- Checking this class alone compiles 30 Lean files with 29,614 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_3122 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_3122 is the table, with the elements numbered 0 to 5.
Variety
[6, 3122] generates the variety V[1223]; 298 other semigroups of order six generate it too, and 1,136 semigroups of order six lie in it. [6, 3122] lies in 55 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[1223] in the inclusion graph, with the varieties above and below it.