[6, 2854] finitely based
[6, 2854] is a semigroup with 2 idempotents. It is finitely based: 3 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 | 4 | 1 | 1 |
| 2 | 1 | 1 | 1 | 4 | 1 | 1 |
| 3 | 1 | 1 | 1 | 4 | 2 | 3 |
| 4 | 4 | 4 | 4 | 1 | 4 | 4 |
| 5 | 1 | 1 | 1 | 4 | 2 | 3 |
| 6 | 1 | 2 | 3 | 4 | 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, 2854)
- Idempotents
- 1, 6
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 5
- Rank
- 3, generated by {4, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x⁴y ≈ x²y
- x³y²x ≈ x²y⁴
- xyz ≈ yxz
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: 8 identities
- x³ ≈ x⁵
- xyx ≈ yx²
- x⁴y ≈ x²y
- x³y²x ≈ x²y⁴
- x³y³ ≈ x²y³x
- x³y³x ≈ x²y³
- x³y⁴ ≈ x²y²x
- xyz ≈ yxz
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.S2_2.Rank009.CyclicRecordedSplice.s6_2854_representative_basis
BasisFor CoRoots.Order6Day7.S2_2.Rank009.S6_2854.table.semigroup CoRoots.Order6Day7.S2_2.Rank009.CyclicRecordedSplice.targetBasis
- 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 192 Lean files with 110,138 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_2854 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2854 is the table, with the elements numbered 0 to 5.
Variety
[6, 2854] generates the variety V[2854]; 2 other semigroups of order six generate it too, and 5,854 semigroups of order six lie in it. [6, 2854] 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[2854] in the inclusion graph, with the varieties above and below it.