[6, 13689] finitely based
[6, 13689] is a monoid with 5 idempotents and an identity element, 5. It is finitely based: 5 identities define its variety, which 12 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 | 2 | 2 | 2 |
| 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| 4 | 1 | 1 | 3 | 4 | 4 | 6 |
| 5 | 1 | 2 | 3 | 4 | 5 | 6 |
| 6 | 1 | 1 | 3 | 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, 13689)
- Idempotents
- 1, 3, 4, 5, 6
- Zero
- none
- Identity
- 5
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 4
- Rank
- 5, generated by {2, 3, 4, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 5 identities irredundant
- x² ≈ x³
- x²yx ≈ xyx
- xyx ≈ xyx²
- xyxzx ≈ xyzx
- xyxz² ≈ xyzxz
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: 7 identities
- x² ≈ x³
- x²yx ≈ xyx
- xyx ≈ xyx²
- x²y² ≈ xyxy
- xyxzx ≈ xyzx
- x²yzy ≈ xyxzy
- xyxz² ≈ xyzxz
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6OverlookedExisting63FanoutV1.S6_13689.representative_basis
BasisFor Generated.Order6OverlookedExisting63FanoutV1.S6_13689.table.semigroup Generated.Order6FactorPairS3_16SharedSevenLawTargets.S6_13668.basis
- 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 74 Lean files with 53,247 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 68 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_13689 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_13689 is the table, with the elements numbered 0 to 5.
Variety
[6, 13689] generates the variety V[12776]; 12 other semigroups of order six generate it too, and 7,820 semigroups of order six lie in it. [6, 13689] 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[12776] in the inclusion graph, with the varieties above and below it.
Generating the same variety
Nearby tables
Semigroups whose Smallsemi table differs from this one in one or two entries (row, column), as the tables are listed, not up to renumbering.
- [6, 13685]: 1 entry (5, 2), variety V[12774]
- [6, 13688]: 1 entry (6, 5), variety V[7648]
- [6, 13668]: 2 entries (4, 3), (6, 3), the same variety
- [6, 13684]: 2 entries (5, 2), (6, 5), variety V[7637]
- [6, 13726]: 2 entries (4, 2), (6, 2), the same variety