[6, 14915] finitely based
[6, 14915] is a semigroup with 3 idempotents and a zero, 1. It is finitely based: 3 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 | 1 | 2 | 1 | 1 |
| 3 | 1 | 2 | 3 | 2 | 5 | 6 |
| 4 | 1 | 1 | 1 | 4 | 1 | 1 |
| 5 | 1 | 2 | 5 | 2 | 6 | 3 |
| 6 | 1 | 2 | 6 | 2 | 3 | 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, 14915)
- Idempotents
- 1, 3, 4
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 4
- Rank
- 2, generated by {4, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x² ≈ x⁵
- x³yxy ≈ yxy
- xyx²zx ≈ zyxz³
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: 16 identities
- x² ≈ x⁵
- x²yx ≈ xyx²
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- x⁴yx ≈ xyx
- x³yxy ≈ yxy
- x³yx ≈ yxy³
- x²yxy ≈ yx³y
- xyzx ≈ xzyx
- xyxzy ≈ xy²zx
- xyxzy ≈ yx²zy
- xyx²zx ≈ zyxz³
- xyztx ≈ xytzx
- xyztx ≈ xytzx
- xyztx ≈ xytzx
- xyztx ≈ xytzx
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6L2DD029Targets.S6_14915.representative_basis
BasisFor Generated.Order6L2DD029Targets.S6_14915.table.semigroup Generated.Order6L2DD029Targets.S6_14915.basis
- 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 98 Lean files with 47,873 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_14915 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_14915 is the table, with the elements numbered 0 to 5.
Variety
[6, 14915] generates the variety V[14915]; 1 other semigroup of order six generates it too, and 1,411 semigroups of order six lie in it. [6, 14915] 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[14915] in the inclusion graph, with the varieties above and below it.