SemiBase
← [6, 3050][6, 3052] →

[6, 3051] finitely based

monoidself-dualother generated proof

[6, 3051] is a monoid with 2 idempotents, a zero, 1 and an identity element, 5. It is finitely based: 3 identities define its variety, which 3 other semigroups of order six also generate.

Cayley table

·123456
1111111
2111122
3111133
4112144
5123456
6123465

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, 3051)
Idempotents
1, 5
Zero
1
Identity
5
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
5
Rank
3, generated by {3, 4, 6}
Self-dual
yes: anti-isomorphic to itself

Identity basis

Shortest known basis: 3 identities irredundant

  1. x² ≈ x⁴
  2. x²y ≈ xyx
  3. x³y ≈ yx³

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: 7 identities

  1. x² ≈ x⁴
  2. x²y ≈ xyx
  3. x²y ≈ yx²
  4. x³y ≈ yx³
  5. x²y² ≈ y²x²
  6. x³y² ≈ y²x³
  7. x³y³ ≈ y³x³

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6RouteLocalRootAdapters.S6_3051.representative_basis

BasisFor Generated.Order6RouteLocalRootAdapters.S6_3051.table.semigroup
  Generated.Order6RouteLocalRootAdapters.S6_3051.targetBasis
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 969 Lean files with 685,035 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_3051 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_3051 is the table, with the elements numbered 0 to 5.

Variety

[6, 3051] generates the variety V[1262]; 3 other semigroups of order six generate it too, and 1,542 semigroups of order six lie in it. [6, 3051] lies in 27 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[2860] V[2999] V[4044] V[6856]
Directly below
V[1117] V[1229] V[1236]

V[1262] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 1262] [6, 1348] [6, 1354]

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.