SemiBase
← [6, 11261][6, 11263] →

[6, 11262] finitely based

not self-dualfamily proof

[6, 11262] is a semigroup with 4 idempotents. It is finitely based: 4 identities define its variety, which 1 other semigroup of order six also generates.

Cayley table

·123456
1111111
2111122
3333333
4333344
5121156
6121256

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

Structure

Smallsemi
SmallSemigroup(6, 11262)
Idempotents
1, 3, 5, 6
Zero
none
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
4
Rank
3, generated by {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 ≈ x³
  2. x²yx ≈ xyx
  3. xyxzx ≈ xzxyx
  4. xy²zt²uv² ≈ xt²uy²zv²

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: 20 identities

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx ≈ xyx²
  4. xyxyx ≈ xy²x
  5. xyxyx ≈ xy²x
  6. x²y²z ≈ xyxyz
  7. x²yz² ≈ xyxz²
  8. x²yz² ≈ xyzxz
  9. x²yz² ≈ xzxyz
  10. x²yz² ≈ xzxyz
  11. x²yz² ≈ xzxyz
  12. xyxzx ≈ xzxyx
  13. xyzyz ≈ xzy²z
  14. xyzyz ≈ xzy²z
  15. xyxyxz ≈ xy²xz
  16. xyxzxy ≈ xzyxy
  17. x²yzyz ≈ xyxyz²
  18. xyxyzx ≈ xyzxyx
  19. xy²xz² ≈ xy²zxz
  20. xy²zt²uv² ≈ xt²uy²zv²

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6LeeZhang23_9Endpoints.s6_11262_basis

BasisFor Order6Subdirect.S6_11262.table.semigroup CoRoots.Order6LeeZhang23_9B20Data.b20Basis
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 62 Lean files with 27,660 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_11262 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_11262 is the table, with the elements numbered 0 to 5.

Variety

[6, 11262] generates the variety V[8448]; 1 other semigroup of order six generates it too, and 3,799 semigroups of order six lie in it. [6, 11262] lies in 3 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[8453]
Directly below
V[3608] V[3812] V[7184] V[7644]

V[8448] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 8448]

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.