[6, 14914] finitely based
[6, 14914] is a semigroup with 3 idempotents and a zero, 1. It is finitely based: 4 identities define its variety, which 2 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 | 2 | 1 | 1 |
| 3 | 1 | 2 | 3 | 1 | 5 | 6 |
| 4 | 1 | 1 | 1 | 4 | 1 | 1 |
| 5 | 1 | 2 | 5 | 1 | 6 | 3 |
| 6 | 1 | 2 | 6 | 1 | 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, 14914)
- Idempotents
- 1, 3, 4
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 4
- Rank
- 3, generated by {2, 4, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 4 identities irredundant
- x² ≈ x⁵
- x⁴yx ≈ xyx
- x³yx ≈ yxy³
- x²yzy ≈ yx²zy
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: 13 identities
- x² ≈ x⁵
- x²y² ≈ yx²y
- x²y² ≈ yxyx
- x²y² ≈ y²x²
- x⁴yx ≈ xyx
- x³yx ≈ yxy³
- x³y² ≈ yx³y
- xyzx ≈ xzyx
- x²yzy ≈ xyxzy
- x²yzy ≈ xy²zx
- x²yzy ≈ xzxy²
- x²yzy ≈ yx²zy
- xyztx ≈ xytzx
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.LeeZhang.Msg0443Group14914.group14914_representative_basis
BasisFor CoRoots.Order6Day7.LeeZhang.Msg0443Group14914.table14914.semigroup CoRoots.Order6Day7.LeeZhang.Msg0443Group14914.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 68 Lean files with 19,640 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 345 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_14914 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_14914 is the table, with the elements numbered 0 to 5.
Variety
[6, 14914] generates the variety V[14914]; 2 other semigroups of order six generate it too, and 1,245 semigroups of order six lie in it. [6, 14914] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[14914] in the inclusion graph, with the varieties above and below it.