SemiBase
← [6, 9725][6, 9727] →

[6, 9726] finitely based

monoidnot self-dualfamily proof

[6, 9726] is a monoid with 4 idempotents and an identity element, 6. It is finitely based: 8 identities define its variety, which no other semigroup of order six generates.

Cayley table

·123456
1111111
2111112
3112143
4444444
5555555
6123456

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

Structure

Smallsemi
SmallSemigroup(6, 9726)
Idempotents
1, 4, 5, 6
Zero
none
Identity
6
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: 8 identities reduced

  1. x³ ≈ x⁴
  2. x³y ≈ x²yx
  3. xyxy ≈ xy²x
  4. xyx² ≈ xyx³
  5. xyx²z ≈ xyxzx
  6. xyzxy ≈ xyzyx
  7. xyzxz ≈ xyz²x
  8. xyztxz ≈ xyztzx

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: 8 identities

  1. x³ ≈ x⁴
  2. x³y ≈ x²yx
  3. xyxy ≈ xy²x
  4. xyx² ≈ xyx³
  5. xyx²z ≈ xyxzx
  6. xyzxy ≈ xyzyx
  7. xyzxz ≈ xyz²x
  8. xyztxz ≈ xyztzx

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.LeeZhang.Recursive9726.representative_basis9726

BasisFor CoRoots.Order6Day7.LeeZhang.Recursive9726.table.semigroup CoRoots.Order6Day7.LeeZhang.Recursive9726.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 22 Lean files with 3,065 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 470 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_9726 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9726 is the table, with the elements numbered 0 to 5.

Variety

[6, 9726] generates the variety V[9726]; no other semigroup of order six generates it, and 7,292 semigroups of order six lie in it. [6, 9726] lies in 1 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
none: a maximal variety of the census
Directly below
V[338] V[583] V[5612] V[5636] V[7674] V[9699]

V[9726] 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.