SemiBase
← [6, 9622][6, 9624] →

[6, 9623] finitely based

not self-dualother generated proof

[6, 9623] is a semigroup with 3 idempotents and a zero, 4. It is finitely based: 3 identities define its variety, which no other semigroup of order six generates.

Cayley table

·123456
1111441
2111442
3112443
4444444
5555445
6123446

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, 9623)
Idempotents
1, 4, 6
Zero
4
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
6
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: 3 identities irredundant

  1. x³ ≈ x⁴
  2. x²y² ≈ yx²y
  3. xyz ≈ xzy

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: 4 identities

  1. x³ ≈ x⁴
  2. x²y ≈ xyx
  3. x²y² ≈ yx²y
  4. xyz ≈ xzy

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6CASExplicitSourceWrappersV3.S6_9623.representative_basis

BasisFor Generated.Order6CASExplicitSourceWrappersV3.S6_9623.table.semigroup
  Generated.Order6CASExplicitSourceAdapters.S6_2727Opposite.sourceBasis
Table
The theorem is about the semigroup with exactly this table.
Method
other generated proof. A machine-generated proof that builds on semigroups certified earlier: an adapter or wrapper of a transfer found by the campaign's search.
Size
Checking this class alone compiles 46 Lean files with 28,198 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 142 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_9623 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_9623 is the table, with the elements numbered 0 to 5.

Variety

[6, 9623] generates the variety V[9623]; no other semigroup of order six generates it, and 3,780 semigroups of order six lie in it. [6, 9623] lies in 20 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[2737] V[2764] V[5611]
Directly below
V[413] V[2611] V[2612] V[2726] V[6677]

V[9623] 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.