[6, 15857] finitely based
[6, 15857] is a band: every element is idempotent. It is finitely based: 2 identities define its variety, which 43 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 5 | 5 |
| 2 | 1 | 2 | 1 | 1 | 5 | 5 |
| 3 | 1 | 1 | 3 | 4 | 5 | 6 |
| 4 | 4 | 4 | 4 | 4 | 6 | 6 |
| 5 | 1 | 1 | 5 | 1 | 5 | 5 |
| 6 | 4 | 4 | 6 | 4 | 6 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 15857)
- Idempotents
- 1, 2, 3, 4, 5, 6
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 3
- Rank
- 3, generated by {2, 3, 6}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 2 identities irredundant
- x ≈ x²
- xyxzx ≈ xyzx
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: 2 identities
- x ≈ x²
- xyxzx ≈ xyzx
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6DirectPowerAvailableV3Sources.S6_15857.basis_complete
BasisFor Generated.Order6DirectPowerAvailableV3Sources.S6_15857.sourceSemigroup Generated.Order6DirectPowerAvailableV3Sources.S6_15857.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 50 Lean files with 25,823 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_15857 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_15857 is the table, with the elements numbered 0 to 5.
Variety
[6, 15857] generates the variety V[15091]; 43 other semigroups of order six generate it too, and 864 semigroups of order six lie in it. [6, 15857] lies in 9 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[15091] in the inclusion graph, with the varieties above and below it.
Generating the same variety
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, 14052]: 1 entry (2, 2), variety V[12178]
- [6, 14080]: 2 entries (2, 2), (3, 2), variety V[12185]
- [6, 15856]: 2 entries (3, 4), (3, 6), variety V[15088]