SemiBase
← [6, 6549][6, 6551] →

[6, 6550] finitely based

monoidself-dualother generated proof

[6, 6550] is a monoid with 3 idempotents, a zero, 1 and an identity element, 4. It is finitely based: 5 identities define its variety, which 8 other semigroups of order six also generate.

Cayley table

·123456
1111111
2111221
3111333
4123456
5123546
6121666

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, 6550)
Idempotents
1, 4, 6
Zero
1
Identity
4
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
5
Rank
4, generated by {2, 3, 5, 6}
Self-dual
yes: anti-isomorphic to itself

Identity basis

Shortest known basis: 5 identities irredundant

  1. x² ≈ x⁴
  2. x²y² ≈ yx²y
  3. xyx ≈ xyx³
  4. x²yzx ≈ xyxzx
  5. xyxz² ≈ xyz²x

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: 16 identities

  1. x² ≈ x⁴
  2. x³yx ≈ xyx
  3. x²yx ≈ xyx²
  4. x²yx² ≈ xyx
  5. x²y² ≈ xyxy
  6. x²y² ≈ xy²x
  7. x²y² ≈ yx²y
  8. xyx ≈ xyx³
  9. x³y² ≈ yx³y
  10. x²yzx ≈ xyxzx
  11. x²yzy ≈ xyxzy
  12. x²yzy ≈ yx²zy
  13. xyxz² ≈ xyzxz
  14. xyxz² ≈ xyz²x
  15. xyzxy ≈ xyzyx
  16. xyzxy ≈ yxzxy

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6OneLocalExisting15.S6_6550.representative_basis

BasisFor Generated.Order6OneLocalExisting15.S6_6550.table.semigroup Generated.Order6OneLocalExisting15.S6_6550.basis
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 180 Lean files with 108,101 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_6550 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_6550 is the table, with the elements numbered 0 to 5.

Variety

[6, 6550] generates the variety V[4089]; 8 other semigroups of order six generate it too, and 4,070 semigroups of order six lie in it. [6, 6550] lies in 4 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[8862] V[8874]
Directly below
V[3541] V[4053] V[4076] V[4079]

V[4089] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 4089] [6, 4199] [6, 4295] [6, 6471] [6, 8873] [6, 9017] [6, 11230] [6, 11521]

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.