SemiBase
← [6, 2778][6, 2780] →

[6, 2779] finitely based

self-dualfamily proof

[6, 2779] is a semigroup with 2 idempotents and a zero, 1. It is finitely based: 6 identities define its variety, which 1 other semigroup of order six also generates.

Cayley table

·123456
1111111
2111112
3111113
4112122
5112123
6122446

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, 2779)
Idempotents
1, 6
Zero
1
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
6
Rank
2, generated by {5, 6}
Self-dual
yes: anti-isomorphic to itself

Identity basis

Shortest known basis: 6 identities reduced

  1. x²yx ≈ xyx
  2. xyx ≈ xyx²
  3. x²y² ≈ y²x²
  4. x²yz² ≈ z²yx²
  5. x²y²zt² ≈ x²zy²t²
  6. x²yz²tu² ≈ x²tz²yu²

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: 13 identities

  1. x²yx ≈ xyx
  2. xyx ≈ xyx²
  3. x²y² ≈ y²x²
  4. x³y² ≈ y²x³
  5. x³y³ ≈ y³x³
  6. x²yz² ≈ z²yx²
  7. xyxzx ≈ xzxyx
  8. x³yz² ≈ z²yx³
  9. x³yz³ ≈ z³yx³
  10. x³yx³z² ≈ x³yz²
  11. x²y³zy³ ≈ x²zy³
  12. x²y²zt² ≈ x²zy²t²
  13. x²yz²tu² ≈ x²tz²yu²

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6SporadicSection17.C5C6.S6_2779.representative_basis

BasisFor CoRoots.Order6SporadicSection17.C5C6.S6_2779.table.semigroup
  CoRoots.Order6SporadicSection17.C5C6.Displayed.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 43 Lean files with 7,144 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_2779 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_2779 is the table, with the elements numbered 0 to 5.

Variety

[6, 2779] generates the variety V[2778]; 1 other semigroup of order six generates it too, and 4,733 semigroups of order six lie in it. [6, 2779] 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[413] V[1110] V[2604] V[2606] V[2638]

V[2778] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 2778]

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.