[6, 9877] finitely based
[6, 9877] is a semigroup with 3 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which 58 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 | 1 |
| 3 | 1 | 2 | 3 | 4 | 3 | 4 |
| 4 | 1 | 2 | 4 | 3 | 4 | 3 |
| 5 | 1 | 2 | 5 | 6 | 5 | 6 |
| 6 | 1 | 2 | 6 | 5 | 6 | 5 |
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, 9877)
- Idempotents
- 1, 3, 5
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 3, generated by {2, 3, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x³y ≈ xy
- xyx ≈ xy²xy
- xyzx ≈ xzyx
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
The Lean theorem certifies a longer basis, of 23 identities; the shortest known basis above is equivalent to it.
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6DirectPowerAvailableV3.S6_4022_Ad4b303415746_Part02.S6_9877.representative_basis
BasisFor Generated.Order6DirectPowerAvailableV3.S6_4022_Ad4b303415746_Part02.S6_9877.table.semigroup Generated.Order6DirectPowerAvailableV3Sources.S6_4022.sourceBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- direct-power transfer. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup of a direct power of this one, 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.
- Size
- Checking this class alone compiles 93 Lean files with 68,315 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_9877 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9877 is the table, with the elements numbered 0 to 5.
Variety
[6, 9877] generates the variety V[4022]; 58 other semigroups of order six generate it too, and 2,065 semigroups of order six lie in it. [6, 9877] lies in 19 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[4022] in the inclusion graph, with the varieties above and below it.