[6, 14897] finitely based
[6, 14897] is a regular monoid. It has 5 idempotents and an identity element, 1. It is finitely based: 2 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 1 | 4 | 3 | 6 | 5 |
| 3 | 3 | 5 | 3 | 3 | 5 | 5 |
| 4 | 4 | 6 | 4 | 4 | 6 | 6 |
| 5 | 5 | 3 | 3 | 3 | 5 | 5 |
| 6 | 6 | 4 | 4 | 4 | 6 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 14897)
- Idempotents
- 1, 3, 4, 5, 6
- Zero
- none
- Identity
- 1
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 2
- Rank
- 2, generated by {2, 3}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 2 identities irredundant
- x ≈ x³
- xyx²zx ≈ xyzx
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: 2 identities
- x ≈ x³
- xyx²zx ≈ xyzx
Lean proof
Endpoint theorem: SemigroupBasis.Order6.S6_14897.representative_basis
BasisFor Order6.S6_14897.table.semigroup Order6.S6_14897.basis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- individual proof. A proof written for this semigroup or a few semigroups, outside the generated and the family proofs.
- Size
- Checking this class alone compiles 31 Lean files with 8,846 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 929 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_14897 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_14897 is the table, with the elements numbered 0 to 5.
Variety
[6, 14897] generates the variety V[14897]; no other semigroup of order six generates it, and 1,257 semigroups of order six lie in it. [6, 14897] 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[14897] in the inclusion graph, with the varieties above and below it.