[6, 414] finitely based
[6, 414] is a nilpotent semigroup: every product of 4 elements is equal to the zero. It is finitely based: 2 identities define its variety, which 3 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 | 1 | 1 |
| 4 | 1 | 1 | 1 | 1 | 1 | 3 |
| 5 | 1 | 3 | 1 | 1 | 1 | 4 |
| 6 | 1 | 1 | 1 | 1 | 3 | 2 |
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, 414)
- Idempotents
- 1
- Zero
- 1
- Identity
- none
- Nilpotent
- yes, of class 4
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 2, generated by {5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 2 identities irredundant
- xyz ≈ xzy
- x²y ≈ ztuv
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
The Lean theorem certifies a longer basis, of 29 identities; the shortest known basis above is equivalent to it.
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6Nilpotent.Certificates.Cutoff4.Signaturea18638907361.S6_414.representative_basis
BasisFor Generated.Order6Nilpotent.Certificates.Cutoff4.Signaturea18638907361.S6_414.table.semigroup Generated.Order6Nilpotent.Signaturea18638907361.representativeBasis
- Table
- The theorem is about the opposite semigroup, with the transposed table; read backwards, its basis is the basis of this table shown above.
- Method
- nilpotent certificate. This semigroup is nilpotent: all products of a fixed number of elements are equal to the zero. The generated certificate lists the identities between short words, and the basis consists of them.
- Size
- Checking this class alone compiles 20 Lean files with 8,388 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_414 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_414 is the table, with the elements numbered 0 to 5.
Variety
[6, 414] generates the variety V[413]; 3 other semigroups of order six generate it too, and 2,665 semigroups of order six lie in it. [6, 414] lies in 33 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[413] 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.