[6, 3842] C₈ finitely based
[6, 3842] is the semigroup C₈ of Section 19 of Lee and Zhang (2015): a semigroup with 3 idempotents and a zero, 1. It is finitely based: 5 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 | 2 | 1 | 4 | 1 |
| 5 | 1 | 1 | 3 | 1 | 5 | 1 |
| 6 | 1 | 2 | 2 | 4 | 4 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal; grey: the zero. Shaded: the two entries in which this table differs from that of [6, 3843] L.
Structure
- Smallsemi
- SmallSemigroup(6, 3842)
- Idempotents
- 1, 5, 6
- Zero
- 1
- Identity
- none
- 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: 5 identities irredundant
- x² ≈ x³
- x²yx ≈ xyx
- xyxzx ≈ xzxyx
- xyztxz ≈ zyxtxz
- xyxzxtu²x ≈ xyuzxtu
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
The Lean theorem certifies a longer basis, of 38 identities; the shortest known basis above is equivalent to it.
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6Astra.C8Catalogue.basisFor3842
BasisFor CoRoots.Order6Astra.C8Catalogue.table.semigroup CoRoots.Order6Astra.C8Catalogue.basis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- individual proof. A proof written for this semigroup or a few semigroups, outside the generated and the family proofs.
- Size
- Checking this class alone compiles 93 Lean files with 21,702 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 16,729 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_3842 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_3842 is the table, with the elements numbered 0 to 5.
Variety
[6, 3842] generates the variety V[3842]; no other semigroup of order six generates it, and 2,230 semigroups of order six lie in it. [6, 3842] 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[3842] 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, 3783]: 2 entries (4, 3), (6, 3), variety V[3467]
- [6, 3841]: 2 entries (6, 3), (6, 5), variety V[3841]
- [6, 3843] L: 2 entries (4, 6), (5, 6), no finite basis