[6, 9882] finitely based
[6, 9882] is a semigroup with 3 idempotents. It is finitely based: 2 identities define its variety, which 6 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 | 2 | 3 | 4 | 5 | 6 |
| 4 | 5 | 6 | 4 | 3 | 1 | 2 |
| 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| 6 | 5 | 5 | 5 | 5 | 5 | 5 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 9882)
- Idempotents
- 1, 3, 5
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 2, generated by {2, 4}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 2 identities irredundant
- x³y ≈ xy
- xyzxz ≈ xyz²x
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: 8 identities
- x² ≈ x⁴
- x³y ≈ xy
- xyx ≈ xyx³
- xyxy ≈ xy²x
- xyx²z ≈ xyz
- xyzxy ≈ xyzyx
- xyzxz ≈ xyz²x
- xyztxz ≈ xyztzx
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6RouteLocalRootAdapters.S6_9882.representative_basis
BasisFor Generated.Order6RouteLocalRootAdapters.S6_9882.table.semigroup Generated.Order6RouteLocalRootAdapters.S6_9882.targetBasis
- 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 969 Lean files with 685,035 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_9882 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9882 is the table, with the elements numbered 0 to 5.
Variety
[6, 9882] generates the variety V[8921]; 6 other semigroups of order six generate it too, and 3,320 semigroups of order six lie in it. [6, 9882] lies in 4 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[8921] in the inclusion graph, with the varieties above and below it.