[6, 15959] finitely based
[6, 15959] is a regular semigroup. It has 2 idempotents. It is finitely based: 2 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 3 | 4 | 3 | 4 |
| 2 | 2 | 2 | 5 | 6 | 5 | 6 |
| 3 | 3 | 3 | 4 | 1 | 4 | 1 |
| 4 | 4 | 4 | 1 | 3 | 1 | 3 |
| 5 | 5 | 5 | 6 | 2 | 6 | 2 |
| 6 | 6 | 6 | 2 | 5 | 2 | 5 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 15959)
- Idempotents
- 1, 2
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 1
- Rank
- 2, generated by {1, 5}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 2 identities irredundant
- x ≈ x⁴
- x² ≈ xyxy²
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
- x ≈ x⁴
- x ≈ xy³
- x²y ≈ xyx
- x² ≈ xyxy²
- x² ≈ xy²xy
- x³yx ≈ xy
- x²yx² ≈ xy
- xyz ≈ xzy
- xy ≈ xzyz²
- xy ≈ xz²yz
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.S2_4.Rank070.Seed.s6_15959_representative_basis
BasisFor CoRoots.Order6Day7.S2_4.Rank070.S6_15959.table.semigroup CoRoots.Order6Day7.S2_4.Rank070.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 4,629 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 853 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_15959 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_15959 is the table, with the elements numbered 0 to 5.
Variety
[6, 15959] generates the variety V[15959]; no other semigroup of order six generates it, and 2 semigroups of order six lie in it. [6, 15959] lies in 12 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[15959] in the inclusion graph, with the varieties above and below it.