[6, 14907] finitely based
[6, 14907] is a semigroup with 3 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 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3 | 1 | 2 | 3 | 1 | 5 | 6 |
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| 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.
Structure
- Smallsemi
- SmallSemigroup(6, 14907)
- Idempotents
- 1, 3, 4
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- 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: 3 identities irredundant
- x⁴y ≈ xy
- x²y² ≈ xy²x
- xyzx ≈ xzyx
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: 18 identities
- x² ≈ x⁵
- x⁴y ≈ xy
- x²yx ≈ xyx²
- x²y² ≈ xyxy
- x²y² ≈ xy²x
- x³yx ≈ xy⁴
- x²y³ ≈ xy³x
- x²yz ≈ xyxz
- xyzx ≈ xzyx
- xyzy ≈ xzy²
- x²yzy ≈ xy²zx
- x³yzx ≈ xy³zy
- xyx²zx ≈ xy³zy
- xy³zy ≈ xyzx³
- xyztx ≈ xytzx
- xyztx ≈ xytzx
- xyztx ≈ xytzx
- xyztx ≈ xytzx
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.S2_4.Rank005.Seed.s6_14907_representative_basis
BasisFor CoRoots.Order6Day7.S2_4.Rank005.S6_14907.table.semigroup CoRoots.Order6Day7.S2_4.Rank005.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 73 Lean files with 34,729 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 1,341 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_14907 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_14907 is the table, with the elements numbered 0 to 5.
Variety
[6, 14907] generates the variety V[14907]; 2 other semigroups of order six generate it too, and 1,311 semigroups of order six lie in it. [6, 14907] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[14907] in the inclusion graph, with the varieties above and below it.