[6, 15854] finitely based
[6, 15854] is a left zero semigroup: xy = x for all x and y. It is finitely based: 1 identity defines its variety, which no other semigroup of order six generates.
Cayley table
| ยท | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| 6 | 6 | 6 | 6 | 6 | 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, 15854)
- Idempotents
- 1, 2, 3, 4, 5, 6
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- ๐ฅ-classes
- 1
- Rank
- 6, generated by {1, 2, 3, 4, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 1 identity irredundant
- x โ xy
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: 1 identity
- x โ xy
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6EmbeddingTransfers.S6_15854.representative_basis
BasisFor Generated.Order6EmbeddingTransfers.S6_15854.table.semigroup Examples.leftZeroBasis
- Table
- The theorem is about the semigroup with exactly this table.
- Method
- embedding transfer. A semigroup S with a certified basis lies in the variety of this one, as a subsemigroup, 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 11 Lean files with 2,107 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_15854 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_15854 is the table, with the elements numbered 0 to 5.
Variety
[6, 15854] generates the variety V[15854]; no other semigroup of order six generates it, and 1 semigroup of order six lies in it. [6, 15854] lies in 232 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
- none: a minimal variety of the census
V[15854] in the inclusion graph, with the varieties above and below it.