SemiBase
← [6, 4163][6, 4165] →

[6, 4164] finitely based

not self-dualsubdirect product

[6, 4164] is a semigroup with 3 idempotents. It is finitely based: 4 identities define its variety, which no other semigroup of order six generates.

Cayley table

·123456
1113311
2113311
3331133
4331144
5123356
6123356

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

Structure

Smallsemi
SmallSemigroup(6, 4164)
Idempotents
1, 5, 6
Zero
none
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
4
Rank
4, generated by {2, 4, 5, 6}
Self-dual
no: the class also stands for the opposite semigroup, with the transposed table

Identity basis

Shortest known basis: 4 identities irredundant

  1. x² ≈ x⁴
  2. xyx ≈ yxy²x
  3. x²yz ≈ xyxz
  4. xy³z ≈ 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: 17 identities

  1. x² ≈ x⁴
  2. x³yx ≈ xyx
  3. x³y² ≈ yxy
  4. x²yx ≈ xyx²
  5. x²yx² ≈ xyx
  6. x²yxy ≈ yxy
  7. x²y² ≈ xyxy
  8. x²y² ≈ yx²y
  9. xyx ≈ xyx³
  10. xyx ≈ xy³x
  11. xyx ≈ yxy²x
  12. x²yz ≈ xyxz
  13. xy³z ≈ xyz
  14. xyzx ≈ xzyx
  15. xyzy ≈ xzy²
  16. x³yz ≈ yxy²z
  17. xyxz² ≈ zyx²z

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6FactorPairSharedFamilies.S6_4164.representative_basis

BasisFor Generated.Order6FactorPairSharedFamilies.S6_4164.table.semigroup
  Generated.Order6FactorPairSharedFamilies.S6_4164.basis
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 222 Lean files with 135,184 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_4164 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_4164 is the table, with the elements numbered 0 to 5.

Variety

[6, 4164] generates the variety V[4164]; no other semigroup of order six generates it, and 1,902 semigroups of order six lie in it. [6, 4164] lies in 4 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[8863]
Directly below
V[1230] V[3346] V[4024] V[6882]

V[4164] 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.