[6, 12456] finitely based
[6, 12456] is a semigroup with 5 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which 26 other semigroups of order six also generate.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 1 | 1 | 1 | 1 |
| 3 | 1 | 2 | 3 | 3 | 5 | 5 |
| 4 | 1 | 2 | 4 | 4 | 6 | 6 |
| 5 | 1 | 2 | 3 | 3 | 5 | 5 |
| 6 | 1 | 2 | 4 | 4 | 6 | 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, 12456)
- Idempotents
- 1, 3, 4, 5, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 3
- Rank
- 3, generated by {2, 3, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x²y ≈ xy
- xyxz ≈ xyz
- xyzx ≈ xzyx
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: 3 identities
- x²y ≈ xy
- xyxz ≈ xyz
- xyzx ≈ xzyx
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6DirectPowerAvailableV3.S6_7183_Ad4b303415746_Part01.S6_12456.representative_basis
BasisFor Generated.Order6DirectPowerAvailableV3.S6_7183_Ad4b303415746_Part01.S6_12456.table.semigroup Generated.Order6DirectPowerAvailableV3Sources.S6_7183.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 73 Lean files with 37,352 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_12456 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_12456 is the table, with the elements numbered 0 to 5.
Variety
[6, 12456] generates the variety V[7183]; 26 other semigroups of order six generate it too, and 1,467 semigroups of order six lie in it. [6, 12456] lies in 33 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[7183] in the inclusion graph, with the varieties above and below it.