SemiBase
← [6, 11227][6, 11229] →

[6, 11228] finitely based

not self-dualfamily proof

[6, 11228] is a semigroup with 4 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which 2 other semigroups of order six also generate.

Cayley table

·123456
1111111
2111122
3123411
4124311
5111156
6111156

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, 11228)
Idempotents
1, 3, 5, 6
Zero
1
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: 3 identities irredundant

  1. x² ≈ x⁴
  2. xyx ≈ yxy²x
  3. x²y²z ≈ xy²xz

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. xyzx ≈ xzyx
  13. x²yxz ≈ yxy²z
  14. x²y²z ≈ xy²xz
  15. x²yzy ≈ xyxzy
  16. x²yzy ≈ xzxy²
  17. x²yzy ≈ yx²zy

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.S3_11.SeedS5_796Opposite.representative_basis_S6_11228

BasisFor CoRoots.Order6Day7.S3_11.Rank001.S6_11228.table.semigroup CoRoots.Order6Day7.S3_11.Rank001.basis
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 84 Lean files with 31,998 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_11228 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_11228 is the table, with the elements numbered 0 to 5.

Variety

[6, 11228] generates the variety V[8863]; 2 other semigroups of order six generate it too, and 2,193 semigroups of order six lie in it. [6, 11228] lies in 3 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[8865]
Directly below
V[4040] V[4164] V[7428]

V[8863] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 8863] [6, 9009]

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.