[6, 13756] finitely based
[6, 13756] is a monoid with 5 idempotents and an identity element, 4. It is finitely based: 7 identities define its variety, which 30 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 | 2 | 3 |
| 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| 4 | 1 | 2 | 3 | 4 | 5 | 6 |
| 5 | 3 | 3 | 3 | 5 | 5 | 6 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 13756)
- Idempotents
- 1, 3, 4, 5, 6
- Zero
- none
- Identity
- 4
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 4
- Rank
- 4, generated by {2, 4, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 7 identities reduced
- x² ≈ x³
- xyx ≈ xyx²
- xyxy ≈ xy²x
- xyxz ≈ xyxzx
- xyzxy ≈ xyzyx
- xyzxz ≈ xyz²x
- xyztxz ≈ xyztzx
Reduced as far as the search went: some identity may still follow from the others. 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
- x² ≈ x³
- x²y ≈ x²yx
- xyx ≈ xyx²
- xyxy ≈ xy²x
- xyxz ≈ xyxzx
- xyzxy ≈ xyzyx
- xyzxz ≈ xyz²x
- xyztxz ≈ xyztzx
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6FinalL5TransferV3.S6_13756.representative_basis
BasisFor Generated.Order6FinalL5TransferV3.S6_13756.table.semigroup Generated.Order6FinalL5TransferV3.S6_13756.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 59 Lean files with 38,882 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_13756 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_13756 is the table, with the elements numbered 0 to 5.
Variety
[6, 13756] generates the variety V[7674]; 30 other semigroups of order six generate it too, and 3,840 semigroups of order six lie in it. [6, 13756] lies in 6 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[7674] in the inclusion graph, with the varieties above and below it.