[6, 2561] finitely based
[6, 2561] is a nilpotent semigroup: every product of 4 elements is equal to the zero. It is finitely based: 4 identities define its variety, which 81 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 | 4 | 1 |
| 3 | 1 | 1 | 1 | 1 | 1 | 4 |
| 4 | 1 | 1 | 1 | 1 | 1 | 1 |
| 5 | 1 | 4 | 1 | 1 | 2 | 1 |
| 6 | 1 | 1 | 4 | 1 | 4 | 3 |
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, 2561)
- Idempotents
- 1
- Zero
- 1
- Identity
- none
- Nilpotent
- yes, of class 4
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 2, generated by {5, 6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 4 identities irredundant
- x²y ≈ xy²
- xyz ≈ xzy
- xyz ≈ yxz
- x⁴ ≈ xyzt
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: 6 identities
- x²y ≈ xyx
- x²y ≈ xy²
- x²y ≈ yx²
- xyz ≈ xzy
- xyz ≈ yxz
- x⁴ ≈ xyzt
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6CASExplicitSourceWrappersV3.S6_2561.opposite_basis
BasisFor Generated.Order6CASExplicitSourceWrappersV3.S6_2561.table.semigroup (reversedBasis Generated.Order6CASExplicitSourceAdapters.S6_852Representative.basis)
- 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
- other generated proof. A machine-generated proof that builds on semigroups certified earlier: an adapter or wrapper of a transfer found by the campaign's search.
- Size
- Checking this class alone compiles 46 Lean files with 35,362 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 7,711 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_2561 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2561 is the table, with the elements numbered 0 to 5.
Variety
[6, 2561] generates the variety V[852]; 81 other semigroups of order six generate it too, and 2,780 semigroups of order six lie in it. [6, 2561] lies in 17 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[852] in the inclusion graph, with the varieties above and below it.
Generating the same variety
Nearby tables
Semigroups whose Smallsemi table differs from this one in one or two entries (row, column), as the tables are listed, not up to renumbering.