SemiBase
← [6, 5357][6, 5359] →

[6, 5358] finitely based

monoidnot self-dualother generated proof

[6, 5358] is a monoid with 2 idempotents, a zero, 1 and an identity element, 6. It is finitely based: 3 identities define its variety, which 1 other semigroup of order six also generates.

Cayley table

·123456
1111111
2111112
3111123
4111214
5112235
6123456

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, 5358)
Idempotents
1, 6
Zero
1
Identity
6
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
6
Rank
3, generated by {4, 5, 6}
Self-dual
no: the class also stands for the opposite semigroup, with the transposed table

Identity basis

Shortest known basis: 3 identities irredundant

  1. x⁴ ≈ x⁵
  2. x²y ≈ yx²
  3. xyx ≈ yx²

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: 11 identities

  1. x⁴ ≈ x⁵
  2. x²y ≈ yx²
  3. xyx ≈ yx²
  4. x³y ≈ yx³
  5. x²y² ≈ y²x²
  6. x⁴y ≈ yx⁴
  7. x³y² ≈ y²x³
  8. x⁴y² ≈ y²x⁴
  9. x³y³ ≈ y³x³
  10. x⁴y³ ≈ y³x⁴
  11. x⁴y⁴ ≈ y⁴x⁴

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6RouteLocalRootAdapters.S6_5358.representative_basis

BasisFor Generated.Order6RouteLocalRootAdapters.S6_5358.table.semigroup
  Generated.Order6RouteLocalRootAdapters.S6_5358.targetBasis
Table
The theorem is about the semigroup with exactly this table.
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 969 Lean files with 685,035 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_5358 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5358 is the table, with the elements numbered 0 to 5.

Variety

[6, 5358] generates the variety V[2812]; 1 other semigroup of order six generates it too, and 3,997 semigroups of order six lie in it. [6, 5358] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[5690]
Directly below
V[2740] V[2800] V[2807]

V[2812] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 2812]

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.