SemiBase
[6, 2] →

[6, 1] finitely based

nilpotent of class 2commutativeself-dualembedding transfer

[6, 1] is a null semigroup: every product is equal to the zero. It is finitely based: 1 identity defines its variety, which no other semigroup of order six generates.

Cayley table

·123456
1111111
2111111
3111111
4111111
5111111
6111111

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, 1)
Idempotents
1
Zero
1
Identity
none
Nilpotent
yes, of class 2
Commutative
yes
Regular
no
Group
no
𝒥-classes
6
Rank
5, generated by {2, 3, 4, 5, 6}
Self-dual
yes: anti-isomorphic to itself

Identity basis

Shortest known basis: 1 identity irredundant

  1. xy ≈ zt

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: 1 identity

  1. xy ≈ zt

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6EmbeddingTransfers.S6_1.representative_basis

BasisFor Generated.Order6EmbeddingTransfers.S6_1.table.semigroup Examples.nullBasis
Table
The theorem is about the semigroup with exactly this table.
Method
embedding transfer. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup, and this semigroup satisfies the basis of S. Then every identity of this semigroup holds in S and follows from the basis, so the basis of S is a basis here too.
Size
Checking this class alone compiles 24 Lean files with 9,119 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_1 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_1 is the table, with the elements numbered 0 to 5.

Variety

[6, 1] generates the variety V[1]; no other semigroup of order six generates it, and 1 semigroup of order six lies in it. [6, 1] lies in 468 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[6] V[186] V[962] V[974] V[4334] V[14999]
Directly below
none: a minimal variety of the census

V[1] 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.