SemiBase
← [6, 5853][6, 5855] →

[6, 5854] finitely based

not self-dualfamily proof

[6, 5854] is a semigroup with 2 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which no other semigroup of order six generates.

Cayley table

·123456
1111111
2111111
3111112
4123444
5123444
6123445

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, 5854)
Idempotents
1, 4
Zero
1
Identity
none
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
6
Rank
2, generated by {3, 6}
Self-dual
no: the class also stands for the opposite semigroup, with the transposed table

Identity basis

Shortest known basis: 3 identities irredundant

  1. x²yz ≈ xyz
  2. xyxzy ≈ yxzy
  3. xyz² ≈ xyz²y

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: 15 identities

  1. x³ ≈ x⁴
  2. xy² ≈ yx²
  3. x³y ≈ x²y
  4. x²yx ≈ xyx
  5. x²y² ≈ xy²
  6. x²yz ≈ xyz
  7. xyxyz ≈ xyxz
  8. xyxzx ≈ xyzx
  9. xyxzy ≈ yxzy
  10. xyxz² ≈ xyz²
  11. xyzxy ≈ xyz²
  12. xyzxz ≈ xyz²
  13. xyzyx ≈ xyz²
  14. xyz² ≈ xyz²x
  15. xyz² ≈ xyz²y

Lean proof

Endpoint theorem: SemigroupBasis.CoRoots.Order6Day7.S3_4.SeedS5_240.representative_basis_S6_5854

BasisFor CoRoots.Order6Day7.S3_4.Rank085.S6_5854.table.semigroup CoRoots.Order6Day7.S3_4.Rank085.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 33 Lean files with 9,896 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 1,491 lines are used by the proof of this class and of no other.
Census
SemiBase.Census.S6_5854 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5854 is the table, with the elements numbered 0 to 5.

Variety

[6, 5854] generates the variety V[5854]; no other semigroup of order six generates it, and 3,645 semigroups of order six lie in it. [6, 5854] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[5441]
Directly below
V[2638] V[2933]

V[5854] 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.