[6, 11916] finitely based
[6, 11916] is a regular semigroup. It has 4 idempotents. It is finitely based: 2 identities define its variety, which 4 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 1 | 2 | 5 | 6 |
| 2 | 2 | 1 | 2 | 1 | 5 | 6 |
| 3 | 3 | 4 | 3 | 4 | 5 | 6 |
| 4 | 4 | 3 | 4 | 3 | 5 | 6 |
| 5 | 5 | 6 | 5 | 6 | 5 | 6 |
| 6 | 6 | 5 | 6 | 5 | 5 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 11916)
- Idempotents
- 1, 3, 5, 6
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 2
- Rank
- 3, generated by {1, 4, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 2 identities irredundant
- x ≈ x³
- xy²zy ≈ xzy
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
- x ≈ x³
- x²y ≈ xy²xy
- x²yxy ≈ xyx²y
- x²y²x ≈ xyxyx
- xy²zy ≈ xzy
- xyzyz ≈ xzy²z
- x²yzxy ≈ xyxzxy
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6L3HeavyRank2.Materialized.S6_11916.representative_basis
BasisFor CoRoots.Order6L3HeavyRank2.Materialized.S6_11916.table.semigroup CoRoots.Order6L3HeavyRank2.basisS3_15S4_96
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- family proof. One basis is proved once to derive every identity of every semigroup of a family; on each table only the identities of the basis are checked, by computation.
- Size
- Checking this class alone compiles 228 Lean files with 82,136 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 132 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_11916 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_11916 is the table, with the elements numbered 0 to 5.
Variety
[6, 11916] generates the variety V[11916]; 4 other semigroups of order six generate it too, and 575 semigroups of order six lie in it. [6, 11916] lies in 3 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[11916] in the inclusion graph, with the varieties above and below it.