[6, 7016] finitely based
[6, 7016] is a semigroup with 3 idempotents and a zero, 4. It is finitely based: 3 identities define its variety, which 44 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 4 | 4 | 4 |
| 2 | 2 | 1 | 1 | 4 | 4 | 4 |
| 3 | 2 | 1 | 1 | 4 | 4 | 4 |
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| 5 | 5 | 5 | 5 | 4 | 4 | 4 |
| 6 | 5 | 5 | 5 | 4 | 5 | 6 |
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, 7016)
- Idempotents
- 1, 4, 6
- Zero
- 4
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 5
- Rank
- 2, generated by {3, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x² ≈ x⁴
- x²yx ≈ yxy²
- x²yxy ≈ yxy
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: 20 identities
- x² ≈ x⁴
- x³yx ≈ xyx
- x²yx ≈ xyx²
- x²yx ≈ yxy²
- x²yx² ≈ xyx
- x²yxy ≈ yxy
- x²y²x ≈ yxy
- xyx ≈ xyx³
- xyx ≈ xyxy²
- xyx ≈ xy²xy
- xyx ≈ xy³x
- xyx ≈ yx²y²
- xyx ≈ yxyxy
- xyx ≈ yxy²x
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- xyzx ≈ xzyx
- xyxzx ≈ zyxz²
- xyzxz ≈ xyz²x
- xyzxz ≈ zyx²z
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6FinalL5TransferV3.S6_7016.representative_basis
BasisFor Generated.Order6FinalL5TransferV3.S6_7016.table.semigroup Generated.Order6FinalL5TransferV3.S6_7016.targetBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- transfer from order ≤ 5. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup; S has order at most five, and this semigroup satisfies the basis of S. Then every identity of this semigroup holds in S and follows from the basis, so the basis of S is a basis here too. This is the final sweep of such transfers.
- Size
- Checking this class alone compiles 149 Lean files with 72,360 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_7016 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_7016 is the table, with the elements numbered 0 to 5.
Variety
[6, 7016] generates the variety V[4042]; 44 other semigroups of order six generate it too, and 2,200 semigroups of order six lie in it. [6, 7016] lies in 9 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[4042] in the inclusion graph, with the varieties above and below it.