SemiBase
← [6, 10403][6, 10405] →

[6, 10404] finitely based

not self-dualother generated proof

[6, 10404] is a semigroup with 4 idempotents and a zero, 1. It is finitely based: 2 identities define its variety, which 8 other semigroups of order six also generate.

Cayley table

·123456
1111111
2111111
3123456
4124356
5125556
6126656

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, 10404)
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: 2 identities irredundant

  1. x³y ≈ xy
  2. xyx ≈ yx²

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: 3 identities

  1. x² ≈ x⁴
  2. x³y ≈ xy
  3. xyx ≈ yx²

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6CASExplicitSourceWrappersV3.S6_10404.representative_basis

BasisFor Generated.Order6CASExplicitSourceWrappersV3.S6_10404.table.semigroup
  Generated.Order6CASExplicitSourceAdapters.S6_6197Opposite.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 137 Lean files with 65,743 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 93 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_10404 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_10404 is the table, with the elements numbered 0 to 5.

Variety

[6, 10404] generates the variety V[8827]; 8 other semigroups of order six generate it too, and 1,380 semigroups of order six lie in it. [6, 10404] lies in 4 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[8867] V[8920]
Directly below
V[4025] V[7157]

V[8827] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 8827] [6, 8836] [6, 8919] [6, 8977] [6, 8986] [6, 9060] [6, 10397] [6, 11749]

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.