[6, 8562] finitely based
[6, 8562] is a regular semigroup. It has 4 idempotents. It is finitely based: 5 identities define its variety, which 1 other semigroup of order six also generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 4 | 1 | 1 | 2 |
| 3 | 1 | 6 | 1 | 3 | 1 | 1 |
| 4 | 1 | 2 | 1 | 4 | 1 | 1 |
| 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| 6 | 1 | 1 | 3 | 1 | 1 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 8562)
- Idempotents
- 1, 4, 5, 6
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 2
- Rank
- 3, generated by {2, 3, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 5 identities reduced
- x² ≈ x³
- xyx ≈ xyxyx
- xyxz² ≈ xz²yx
- xy²zy ≈ xzy²z
- xyxyzx ≈ xyz²x
Reduced as far as the search went: some identity may still follow from the others. 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 24 identities; the shortest known basis above is equivalent to it.
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6L2DRank22.S6_8562.representative_basis
BasisFor CoRoots.Order6L2DRank22.S6_8562.table.semigroup CoRoots.Order6L2DRank22.basis
- 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 91 Lean files with 45,353 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_8562 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_8562 is the table, with the elements numbered 0 to 5.
Variety
[6, 8562] generates the variety V[8562]; 1 other semigroup of order six generates it too, and 2,325 semigroups of order six lie in it. [6, 8562] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[8562] in the inclusion graph, with the varieties above and below it.