[6, 4337] finitely based
[6, 4337] is the symmetric group S₃. It is finitely based: 3 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 | 5 | 6 | 3 | 4 |
| 3 | 3 | 6 | 1 | 5 | 4 | 2 |
| 4 | 4 | 5 | 6 | 1 | 2 | 3 |
| 5 | 5 | 4 | 2 | 3 | 6 | 1 |
| 6 | 6 | 3 | 4 | 2 | 1 | 5 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 4337)
- Idempotents
- 1
- Zero
- none
- Identity
- 1
- Nilpotent
- no
- Commutative
- no
- Regular
- yes, inverse
- Group
- yes
- 𝒥-classes
- 1
- Rank
- 2, generated by {2, 3}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 3 identities reduced
- x ≈ x⁷
- x²y² ≈ y²x²
- x⁶ ≈ y⁶
Reduced as far as the search went: some identity may still follow from the others. 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: 3 identities
- x ≈ x⁷
- x²y² ≈ y²x²
- x⁶ ≈ y⁶
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6RouteLocalRootAdapters.S6_4337.representative_basis
BasisFor Generated.Order6RouteLocalRootAdapters.S6_4337.table.semigroup Generated.Order6RouteLocalRootAdapters.S6_4337.targetBasis
- 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 969 Lean files with 685,035 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_4337 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_4337 is the table, with the elements numbered 0 to 5.
Variety
[6, 4337] generates the variety V[4337]; no other semigroup of order six generates it, and 2 semigroups of order six lie in it. [6, 4337] 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[4337] in the inclusion graph, with the varieties above and below it.