[6, 5625] finitely based
[6, 5625] is a semigroup with 3 idempotents and a zero, 1. It is finitely based: 12 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 2 |
| 3 | 1 | 1 | 1 | 1 | 1 | 3 |
| 4 | 1 | 1 | 1 | 2 | 1 | 4 |
| 5 | 1 | 1 | 3 | 1 | 5 | 1 |
| 6 | 1 | 2 | 1 | 4 | 1 | 6 |
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, 5625)
- Idempotents
- 1, 5, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 6
- Rank
- 4, generated by {3, 4, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 12 identities reduced
- x³ ≈ x⁴
- x²y² ≈ xyxy
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- x³yx ≈ x²yx
- xyzx ≈ xzyx
- x²yzy ≈ xyxzy
- xyxzy ≈ yx²zy
- xyxz² ≈ xyzxz
- xyzxy ≈ xyzyx
- xyztx ≈ xytzx
- xyxztz ≈ xyzxtz
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: 18 identities
- x³ ≈ x⁴
- x²y² ≈ xyxy
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- x³yx ≈ x²yx
- xyzx ≈ xzyx
- x²yzy ≈ xyxzy
- xyxzy ≈ yx²zy
- xyxz² ≈ xyzxz
- xyzxy ≈ xyzyx
- xyzxy ≈ yxzxy
- xyzxz ≈ xyz²x
- xyztx ≈ xytzx
- xyztx ≈ xzytx
- xyxztz ≈ xyzxtz
- xyzxty ≈ yxzxty
- xyztxz ≈ xyztzx
- xyztux ≈ xytzux
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6RouteLocalRootAdapters.S6_5625.representative_basis
BasisFor Generated.Order6RouteLocalRootAdapters.S6_5625.table.semigroup Generated.Order6RouteLocalRootAdapters.S6_5625.targetBasis
- 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 969 Lean files with 685,035 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_5625 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_5625 is the table, with the elements numbered 0 to 5.
Variety
[6, 5625] generates the variety V[5625]; no other semigroup of order six generates it, and 5,075 semigroups of order six lie in it. [6, 5625] lies in 2 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[5625] 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.
- [6, 5609]: 1 entry (3, 6), variety V[2727]
- [6, 5626]: 1 entry (5, 6), variety V[5626]
- [6, 7902]: 1 entry (4, 4), variety V[3464]
- [6, 5605]: 2 entries (3, 6), (5, 3), variety V[2726]
- [6, 5623]: 2 entries (5, 3), (6, 3), variety V[2726]
- [6, 5644]: 2 entries (3, 5), (3, 6), variety V[2726]
- [6, 7323]: 2 entries (2, 6), (4, 4), variety V[3258]
- [6, 7903]: 2 entries (4, 4), (5, 6), variety V[3467]