SemiBase
← [6, 8226][6, 8228] →

[6, 8227] finitely based

not self-dualfamily proof

[6, 8227] is a semigroup with 4 idempotents. It is finitely based: 7 identities define its variety, which 5 other semigroups of order six also generate.

Cayley table

·123456
1111111
2111112
3111142
4444444
5555555
6123416

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

Structure

Smallsemi
SmallSemigroup(6, 8227)
Idempotents
1, 4, 5, 6
Zero
none
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
4
Rank
3, generated by {3, 5, 6}
Self-dual
no: the class also stands for the opposite semigroup, with the transposed table

Identity basis

Shortest known basis: 7 identities reduced

  1. x² ≈ x³
  2. xyx ≈ xyx²
  3. xyxy ≈ xy²x
  4. xyzxty ≈ xyzytx
  5. xyzxtz ≈ xyz²tx
  6. xyxzt² ≈ xyxzt²x
  7. xyztxuz ≈ xyztzux

Reduced as far as the search went: some identity may still follow from the others. 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: 14 identities

  1. x² ≈ x³
  2. xyx ≈ xyx²
  3. xyxy ≈ xy²x
  4. x²y² ≈ x²y²x
  5. xyxzy ≈ xy²zx
  6. xyzxy ≈ xyzyx
  7. xyzxz ≈ xyz²x
  8. x²yz² ≈ x²yz²x
  9. xyxz² ≈ xyxz²x
  10. xyzxty ≈ xyzytx
  11. xyzxtz ≈ xyz²tx
  12. xyztxz ≈ xyztzx
  13. xyxzt² ≈ xyxzt²x
  14. xyztxuz ≈ xyztzux

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6Hull21_1ClassEndpoints.s6_8227_basis

BasisFor Order6Subdirect.S6_8227.table.semigroup Order6Subdirect.Hull21_1_S5_831_S3_8.publishedBasis
Table
The theorem is about the semigroup with exactly this table.
Method
family proof. One basis is proved once to derive every identity of every semigroup of a family; on each table only the identities of the basis are checked, by computation.
Size
Checking this class alone compiles 73 Lean files with 29,576 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_8227 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_8227 is the table, with the elements numbered 0 to 5.

Variety

[6, 8227] generates the variety V[8222]; 5 other semigroups of order six generate it too, and 5,936 semigroups of order six lie in it. [6, 8227] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[13562]
Directly below
V[3813] V[6512] V[8219] V[8221]

V[8222] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 8222] [6, 8223] [6, 8228] [6, 10983] [6, 10984]

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.