SemiBase
← [6, 6446][6, 6448] →

[6, 6447] finitely based

monoidnot self-dualfamily proof

[6, 6447] is a monoid with 3 idempotents, a zero, 1 and an identity element, 6. It is finitely based: 4 identities define its variety, which no other semigroup of order six generates.

Cayley table

·123456
1111111
2111112
3111123
4123444
5123445
6123456

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, 6447)
Idempotents
1, 4, 6
Zero
1
Identity
6
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: 4 identities irredundant

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx² ≈ yx²
  4. xyztxz ≈ xyztzx

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: 12 identities

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx² ≈ yx²
  4. x²y² ≈ xyxy
  5. x²y² ≈ xy²x
  6. x²yzy ≈ yx²zy
  7. xyxz² ≈ xyzxz
  8. xyxz² ≈ xyz²x
  9. xyzxy ≈ xyzyx
  10. xyxztz ≈ xyzxtz
  11. xyxztz ≈ xyzxtz
  12. xyztxz ≈ xyztzx

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.LeeZhang.Recursive6447.representative_basis6447

BasisFor CoRoots.Order6Day7.LeeZhang.Recursive6447.table.semigroup CoRoots.Order6Day7.LeeZhang.Recursive6447.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 28 Lean files with 3,666 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 1,071 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_6447 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_6447 is the table, with the elements numbered 0 to 5.

Variety

[6, 6447] generates the variety V[6447]; no other semigroup of order six generates it, and 1,930 semigroups of order six lie in it. [6, 6447] 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[2980] V[3266]

V[6447] 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.