[6, 9095] finitely based
[6, 9095] is a semigroup with 2 idempotents and a zero, 1. It is finitely based: 4 identities define its variety, which 4 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 | 1 | 1 | 3 | 3 | 3 |
| 4 | 1 | 2 | 1 | 4 | 5 | 6 |
| 5 | 1 | 2 | 1 | 5 | 6 | 4 |
| 6 | 1 | 2 | 1 | 6 | 4 | 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, 9095)
- Idempotents
- 1, 4
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 4
- Rank
- 3, generated by {2, 3, 5}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 4 identities irredundant
- x² ≈ x⁵
- xy⁴z ≈ xyz
- x²yzy ≈ xy²zx
- 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: 14 identities
- x² ≈ x⁵
- x²y² ≈ yx²y
- x²y² ≈ y²x²
- x⁴yx ≈ xyx
- xyx ≈ xy⁴x
- x³yx ≈ yxy³
- x³y² ≈ yx³y
- x²yz ≈ xyxz
- xyzx ≈ xzyx
- xyzy ≈ xzy²
- xy⁴z ≈ xyz
- x²yzy ≈ xy²zx
- x²yzy ≈ yx²zy
- xyzt ≈ xzyt
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.LeeZhang.Msg0443Single9095.singleton9095_representative_basis
BasisFor CoRoots.Order6Day7.LeeZhang.Msg0443Single9095.table9095.semigroup CoRoots.Order6Day7.LeeZhang.Msg0443Single9095.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 98 Lean files with 63,455 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 288 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_9095 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9095 is the table, with the elements numbered 0 to 5.
Variety
[6, 9095] generates the variety V[9095]; 4 other semigroups of order six generate it too, and 1,048 semigroups of order six lie in it. [6, 9095] lies in 3 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[9095] in the inclusion graph, with the varieties above and below it.