[6, 9937] finitely based
[6, 9937] is a monoid with 3 idempotents and an identity element, 5. It is finitely based: 3 identities define its variety, which 4 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 | 2 | 2 |
| 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| 4 | 3 | 3 | 3 | 3 | 4 | 4 |
| 5 | 1 | 2 | 3 | 4 | 5 | 6 |
| 6 | 3 | 4 | 1 | 2 | 6 | 5 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 9937)
- Idempotents
- 1, 3, 5
- Zero
- none
- Identity
- 5
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 2, generated by {2, 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
- x³y ≈ xyx²
- xyx ≈ xyx³
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: 10 identities
- x² ≈ x⁴
- x³y ≈ xyx²
- x³yx ≈ xyx
- x²y ≈ x²yx²
- xyx ≈ xyx³
- xyxy ≈ xy²x
- xyx²zx ≈ xyzx
- xyzxy ≈ xyzyx
- xyzxz ≈ xyz²x
- xyztxz ≈ xyztzx
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6LeeLiCondition9Roots.S6_9937.representative_basis
BasisFor CoRoots.Order6LeeLiCondition9Roots.S6_9937.table.semigroup.opposite CoRoots.Order6LeeLiCondition9.leeLiCondition9Basis
- Table
- The theorem is about the opposite semigroup, with the transposed table; read backwards, its basis is the basis of this table shown above.
- 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 29 Lean files with 16,800 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_9937 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9937 is the table, with the elements numbered 0 to 5.
Variety
[6, 9937] generates the variety V[9937]; 4 other semigroups of order six generate it too, and 4,967 semigroups of order six lie in it. [6, 9937] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[9937] in the inclusion graph, with the varieties above and below it.