SemiBase
← [6, 6698][6, 6700] →

[6, 6699] finitely based

not self-dualembedding transfer

[6, 6699] is a semigroup with 3 idempotents and a zero, 4. It is finitely based: 3 identities define its variety, which 8 other semigroups of order six also generate.

Cayley table

·123456
1111441
2111442
3111443
4444444
5555445
6111446

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, 6699)
Idempotents
1, 4, 6
Zero
4
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
6
Rank
4, generated by {2, 3, 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. xy ≈ xy²
  2. x²y ≈ xyx
  3. x²y ≈ yxy

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: 4 identities

  1. x² ≈ x³
  2. xy ≈ xy²
  3. x²y ≈ xyx
  4. x²y ≈ yxy

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6EmbeddingTransfers.S6_6699.representative_basis

BasisFor Generated.Order6EmbeddingTransfers.S6_6699.table.semigroup (reversedBasis Examples.finalMarkerThreeBasis)
Table
The theorem is about the semigroup with exactly this table.
Method
embedding transfer. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup, and this semigroup satisfies the basis of S. Then every identity of this semigroup holds in S and follows from the basis, so the basis of S is a basis here too.
Size
Checking this class alone compiles 14 Lean files with 6,878 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_6699 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_6699 is the table, with the elements numbered 0 to 5.

Variety

[6, 6699] generates the variety V[3057]; 8 other semigroups of order six generate it too, and 182 semigroups of order six lie in it. [6, 6699] lies in 238 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[3059] V[3097] V[3301] V[6723] V[14973]
Directly below
V[962]

V[3057] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 3057] [6, 6656] [6, 6675] [6, 6690] [6, 6713] [6, 6721] [6, 6724] [6, 6728]

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.