[6, 9504] finitely based
[6, 9504] is a regular monoid. It has 3 idempotents and an identity element, 1. It is finitely based: 3 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 1 | 4 | 3 | 5 | 6 |
| 3 | 3 | 4 | 2 | 1 | 5 | 6 |
| 4 | 4 | 3 | 1 | 2 | 5 | 6 |
| 5 | 5 | 5 | 6 | 6 | 5 | 6 |
| 6 | 6 | 6 | 5 | 5 | 5 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 9504)
- Idempotents
- 1, 5, 6
- Zero
- none
- Identity
- 1
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 2
- Rank
- 2, generated by {3, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x ≈ x⁵
- x²yx ≈ yx³
- xyxy ≈ yx²y
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: 5 identities
- x ≈ x⁵
- x²yx ≈ yx³
- xyxy ≈ yx²y
- xyxzy ≈ yx²zy
- xyzxy ≈ yxzxy
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day10.S4_96.Rank095ClassEndpoints.S6_9504.basisFor
BasisFor CoRoots.Order6Day10.S4_96.Rank095ClassEndpoints.S6_9504.table.semigroup CoRoots.Order6Day10.S4_96.Rank095Intersection.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 26 Lean files with 9,267 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 681 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_9504 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9504 is the table, with the elements numbered 0 to 5.
Variety
[6, 9504] generates the variety V[9504]; no other semigroup of order six generates it, and 201 semigroups of order six lie in it. [6, 9504] lies in 1 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- none: a maximal variety of the census
- Directly below
V[9504] in the inclusion graph, with the varieties above and below it.