[6, 2708] finitely based
[6, 2708] is a semigroup with 2 idempotents and a zero, 1. It is finitely based: 4 identities define its variety, which no other semigroup of order six generates.
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 | 1 | 2 | 4 |
| 4 | 1 | 1 | 2 | 1 | 2 | 4 |
| 5 | 1 | 1 | 1 | 1 | 2 | 4 |
| 6 | 1 | 2 | 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, 2708)
- Idempotents
- 1, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- 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: 4 identities irredundant
- x³y ≈ x²y
- x²y²x ≈ x²y³
- xyzx ≈ xzyx
- xyzt ≈ yxzt
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: 12 identities
- x³ ≈ x⁴
- x³y ≈ x²y
- x²yx ≈ xyx²
- x²yx ≈ yx³
- xyxy ≈ yx²y
- x²y²x ≈ x²y³
- xyxz ≈ yx²z
- xyzx ≈ xzyx
- xyzx ≈ yxzx
- xyz² ≈ yxz²
- x²y²z ≈ xy²xz
- xyzt ≈ yxzt
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.LeeZhang.S6_2708.Sigma12.representative_basis
BasisFor CoRoots.Order6Day7.LeeZhang.S6_2708.table.semigroup CoRoots.Order6Day7.LeeZhang.S6_2708.sigma12
- 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 33 Lean files with 8,171 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 305 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_2708 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2708 is the table, with the elements numbered 0 to 5.
Variety
[6, 2708] generates the variety V[2708]; no other semigroup of order six generates it, and 4,469 semigroups of order six lie in it. [6, 2708] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[2708] in the inclusion graph, with the varieties above and below it.