SemiBase
← [6, 13665][6, 13667] →

[6, 13666] finitely based

not self-dualindividual proof

[6, 13666] is a semigroup with 5 idempotents. It is finitely based: 5 identities define its variety, which no other semigroup of order six generates.

Cayley table

·123456
1111111
2111222
3333333
4111446
5121456
6111466

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

Structure

Smallsemi
SmallSemigroup(6, 13666)
Idempotents
1, 3, 4, 5, 6
Zero
none
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
4
Rank
5, generated by {2, 3, 4, 5, 6}
Self-dual
no: the class also stands for the opposite semigroup, with the transposed table

Identity basis

Shortest known basis: 5 identities irredundant

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx ≈ xyx²
  4. xyxzx ≈ xyzx
  5. xy²z² ≈ xzy²z

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: 8 identities

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx ≈ xyx²
  4. x²y² ≈ xyxy
  5. xyxzx ≈ xyzx
  6. x²yzy ≈ xyxzy
  7. xyxz² ≈ xyzxz
  8. xy²z² ≈ xzy²z

Lean proof

Endpoint theorem: SemigroupBasis.Order6.S6_13666.representative_basis

BasisFor Order6.S6_13666.table.semigroup Order6.S6_13666.targetBasis
Table
The theorem is about the semigroup with exactly this table.
Method
individual proof. A proof written for this semigroup or a few semigroups, outside the generated and the family proofs.
Size
Checking this class alone compiles 71 Lean files with 27,096 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 145 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_13666 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_13666 is the table, with the elements numbered 0 to 5.

Variety

[6, 13666] generates the variety V[13666]; no other semigroup of order six generates it, and 5,031 semigroups of order six lie in it. [6, 13666] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[12776]
Directly below
V[7691] V[8206] V[12775] V[12949] V[13654]

V[13666] in the inclusion graph, with the varieties above and below it.

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.