SemiBase
← [6, 10909][6, 10911] →

[6, 10910] finitely based

monoidnot self-dualdirect-power transfer

[6, 10910] is a monoid with 4 idempotents, a zero, 1 and an identity element, 6. It is finitely based: 7 identities define its variety, which 62 other semigroups of order six also generate.

Cayley table

·123456
1111111
2111112
3123333
4123344
5123355
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, 10910)
Idempotents
1, 3, 5, 6
Zero
1
Identity
6
Nilpotent
no
Commutative
no
Regular
no
Group
no
𝒥-classes
6
Rank
4, generated by {2, 4, 5, 6}
Self-dual
no: the class also stands for the opposite semigroup, with the transposed table

Identity basis

Shortest known basis: 7 identities irredundant

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx ≈ xyx²
  4. xyxzx ≈ xyzx
  5. xyxz² ≈ xyzxz
  6. xyzxty ≈ yxzxty
  7. 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: 16 identities

  1. x² ≈ x³
  2. x²yx ≈ xyx
  3. xyx ≈ xyx²
  4. x²y² ≈ xyxy
  5. xyxy ≈ xy²x
  6. xyxy ≈ yx²y
  7. xyxzx ≈ xyzx
  8. x²yzy ≈ xyxzy
  9. xyxzy ≈ yx²zy
  10. xyxz² ≈ xyzxz
  11. xyzxy ≈ xyzyx
  12. xyzxy ≈ yxzxy
  13. xyzxz ≈ xyz²x
  14. xyxztz ≈ xyzxtz
  15. xyzxty ≈ yxzxty
  16. xyztxz ≈ xyztzx

Lean proof

Endpoint theorem: SemigroupBasis.Generated.Order6DirectPowerAvailableV3.S6_3541_Ad4b303415746_Part01.S6_10910.representative_basis

BasisFor Generated.Order6DirectPowerAvailableV3.S6_3541_Ad4b303415746_Part01.S6_10910.table.semigroup
  Generated.Order6DirectPowerAvailableV3Sources.S6_3541.sourceBasis
Table
The theorem is about the semigroup with exactly this table.
Method
direct-power transfer. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup of a direct power of this one, 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 124 Lean files with 83,853 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_10910 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_10910 is the table, with the elements numbered 0 to 5.

Variety

[6, 10910] generates the variety V[3541]; 62 other semigroups of order six generate it too, and 2,560 semigroups of order six lie in it. [6, 10910] lies in 20 of the 505 varieties of the census, those whose basis it satisfies.

Directly above
V[3944] V[4089] V[7585] V[8206] V[13411]
Directly below
V[3319] V[7943]

V[3541] in the inclusion graph, with the varieties above and below it.

Generating the same variety

[6, 3541] [6, 3543] [6, 3551] [6, 3739] [6, 3740] [6, 3743] [6, 3792] [6, 3794] [6, 3795] [6, 3798] [6, 3844] [6, 3858] [6, 3863] [6, 3865] [6, 3928] [6, 6468] [6, 6469] [6, 6470] [6, 6547] [6, 7578] [6, 7579] [6, 7580] [6, 7613] [6, 8161] [6, 8164] [6, 8167] [6, 8170] [6, 8171] [6, 8179] [6, 8181] [6, 8183] [6, 8255] [6, 8262] [6, 8263] [6, 8266] [6, 8267] [6, 8311] [6, 8312] [6, 8322] [6, 8370] [6, 8375] [6, 8407] [6, 8414] [6, 8419] [6, 8480] [6, 8485] [6, 8679] [6, 10788]

All 62 on the variety page

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.