[6, 2582] finitely based
[6, 2582] is a commutative nilpotent semigroup: every product of 6 elements is equal to the zero. 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 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 3 |
| 3 | 1 | 1 | 1 | 1 | 1 | 1 |
| 4 | 1 | 1 | 1 | 1 | 3 | 2 |
| 5 | 1 | 1 | 1 | 3 | 2 | 4 |
| 6 | 1 | 3 | 1 | 2 | 4 | 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, 2582)
- Idempotents
- 1
- Zero
- 1
- Identity
- none
- Nilpotent
- yes, of class 6
- Commutative
- yes
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 1, generated by {6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 3 identities irredundant
- xy ≈ yx
- xyzt² ≈ tzyx²
- xyztuv ≈ wsrpqa
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 3,513 identities; the shortest known basis above is equivalent to it.
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6Nilpotent.Certificates.Cutoff6.Signature419f45a9b5bc.S6_2582.representative_basis
BasisFor Generated.Order6Nilpotent.Certificates.Cutoff6.Signature419f45a9b5bc.S6_2582.table.semigroup Generated.Order6Nilpotent.Signature419f45a9b5bc.representativeBasis
- Table
- The theorem is about the semigroup with exactly this table.
- 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 54 Lean files with 270,429 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_2582 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2582 is the table, with the elements numbered 0 to 5.
Variety
[6, 2582] generates the variety V[2582]; no other semigroup of order six generates it, and 206 semigroups of order six lie in it. [6, 2582] 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[2582] in the inclusion graph, with the varieties above and below it.