[6, 5368] finitely based
[6, 5368] is a commutative semigroup with 2 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which 1 other semigroup of order six also generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3 | 1 | 1 | 1 | 1 | 2 | 1 |
| 4 | 1 | 1 | 1 | 2 | 3 | 1 |
| 5 | 1 | 1 | 2 | 3 | 4 | 1 |
| 6 | 1 | 1 | 1 | 1 | 1 | 6 |
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, 5368)
- Idempotents
- 1, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- yes
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 2, generated by {5, 6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 3 identities reduced
- xy ≈ yx
- x²yz ≈ xy²z
- xyztu ≈ xyztu²
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: 4 identities
- xy ≈ yx
- x³y ≈ x²y²
- x²yz ≈ xy²z
- xyztu ≈ xyztu²
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6GenericCASSubdirectS6_5368.basisFor
BasisFor CoRoots.Order6GenericCASSubdirectS6_5368.rootSemigroup CoRoots.Order6GenericCASSubdirectS6_5368.rootBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- family proof. One basis is proved once to derive every identity of every semigroup of a family; on each table only the identities of the basis are checked, by computation.
- Size
- Checking this class alone compiles 27 Lean files with 9,096 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_5368 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5368 is the table, with the elements numbered 0 to 5.
Variety
[6, 5368] generates the variety V[5368]; 1 other semigroup of order six generates it too, and 536 semigroups of order six lie in it. [6, 5368] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[5368] in the inclusion graph, with the varieties above and below it.