SemiBase
← [6, 2885][6, 2887] →

[6, 2886] finitely based

not self-dualsubdirect product

[6, 2886] is a semigroup with 2 idempotents. It is finitely based: 3 identities define its variety, which 9 other semigroups of order six also generate.

Cayley table

·123456
1113331
2113331
3331113
4331113
5331123
6113446

The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.

Structure

Smallsemi
SmallSemigroup(6, 2886)
Idempotents
1, 6
Zero
none
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
5
Rank
2, generated by {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. x²yx ≈ xy³
  2. xyz ≈ yxz
  3. x³yz ≈ xyz

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

  1. x³ ≈ x⁵
  2. xyx ≈ yx²
  3. x⁴y ≈ x²y
  4. x³yx ≈ xyx
  5. x²yx ≈ xy³
  6. x²y² ≈ xy²x
  7. x²y³ ≈ xyx
  8. xyx ≈ xy³x
  9. xyz ≈ yxz
  10. x³yz ≈ xyz

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6FactorPairResidualV3.S6_2885_A47f45831945d_Part01.S6_2886.representative_basis

BasisFor Generated.Order6FactorPairResidualV3.S6_2885_A47f45831945d_Part01.S6_2886.table.semigroup
  Generated.Order6FactorPairResidualV3Sources.S6_2885.sourceBasis
Table
The theorem is about the semigroup with exactly this table.
Method
subdirect product. This semigroup is a subdirect product of two factors, and the proof combines the bases of the factors.
Size
Checking this class alone compiles 35 Lean files with 16,226 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 192 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_2886 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2886 is the table, with the elements numbered 0 to 5.

Variety

[6, 2886] generates the variety V[2847]; 9 other semigroups of order six generate it too, and 4,561 semigroups of order six lie in it. [6, 2886] lies in 4 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[2852] V[2854] V[2856]
Directly below
V[1223] V[2586] V[2849]

V[2847] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 2847] [6, 2848] [6, 2869] [6, 2878] [6, 2885] [6, 5380] [6, 5382] [6, 5872] [6, 5879]

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.