[6, 5421] finitely based
[6, 5421] is a semigroup with 2 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which 9 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 | 1 | 1 | 1 | 3 | 3 |
| 4 | 1 | 1 | 1 | 2 | 1 | 1 |
| 5 | 1 | 1 | 3 | 1 | 5 | 5 |
| 6 | 1 | 1 | 3 | 2 | 5 | 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, 5421)
- Idempotents
- 1, 5
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 3, generated by {3, 4, 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 ≈ x²y
- xyz ≈ xzy
- xyz ≈ yxz
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: 6 identities
- x³ ≈ x⁴
- x²y ≈ xyx
- x²y ≈ yx²
- x³y ≈ x²y
- xyz ≈ xzy
- xyz ≈ yxz
Lean proof
Endpoint theorem: SemigroupBasis.Order6.S6_5421.representative_basis
BasisFor Order6.S6_5421.table.semigroup Order6.S6_5421.targetBasis
- 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 21 Lean files with 6,743 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 169 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_5421 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5421 is the table, with the elements numbered 0 to 5.
Variety
[6, 5421] generates the variety V[2612]; 9 other semigroups of order six generate it too, and 3,626 semigroups of order six lie in it. [6, 5421] lies in 53 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[2612] 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.