[6, 14996] finitely based
[6, 14996] is the cyclic group of order six. It is finitely based: 2 identities define its variety, which no other semigroup of order six generates.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 1 | 6 | 5 | 4 | 3 |
| 3 | 3 | 6 | 4 | 1 | 2 | 5 |
| 4 | 4 | 5 | 1 | 3 | 6 | 2 |
| 5 | 5 | 4 | 2 | 6 | 3 | 1 |
| 6 | 6 | 3 | 5 | 2 | 1 | 4 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 14996)
- Idempotents
- 1
- Zero
- none
- Identity
- 1
- Nilpotent
- no
- Commutative
- yes
- Regular
- yes, inverse
- Group
- yes
- đ„-classes
- 1
- Rank
- 1, generated by {5}
- Self-dual
- yes: anti-isomorphic to itself
Identity basis
Shortest known basis: 2 identities irredundant
- xy â yx
- x â xyâ¶
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
- xy â yx
- x â xyâ¶
Lean proof
Endpoint theorem: SemigroupBasis.CoRoots.Order6CyclicSix.basis_complete
BasisFor CoRoots.Order6CyclicSix.table.semigroup CoRoots.Order6CyclicSix.basis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- family proof. One basis is proved once to derive every identity of every semigroup of a family; on each table only the identities of the basis are checked, by computation.
- Size
- Checking this class alone compiles 7 Lean files with 1,759 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 595 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_14996 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_14996 is the table, with the elements numbered 0 to 5.
Variety
[6, 14996] generates the variety V[14996]; no other semigroup of order six generates it, and 1 semigroup of order six lies in it. [6, 14996] lies in 7 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
- none: a minimal variety of the census
V[14996] in the inclusion graph, with the varieties above and below it.