[6, 10822] finitely based
[6, 10822] is a semigroup with 4 idempotents and a zero, 1. It is finitely based: 3 identities define its variety, which 44 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 | 2 |
| 3 | 1 | 1 | 3 | 4 | 1 | 3 |
| 4 | 1 | 1 | 4 | 3 | 1 | 4 |
| 5 | 1 | 2 | 1 | 1 | 5 | 2 |
| 6 | 1 | 1 | 3 | 4 | 1 | 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, 10822)
- Idempotents
- 1, 3, 5, 6
- Zero
- 1
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- no
- Group
- no
- 𝒥-classes
- 5
- Rank
- 3, generated by {4, 5, 6}
- Self-dual
- no: the class also stands for the opposite semigroup, with the transposed table
Identity basis
Shortest known basis: 3 identities irredundant
- x² ≈ x⁴
- x²yx ≈ yxy²
- x²yxy ≈ yxy
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: 20 identities
- x² ≈ x⁴
- x³yx ≈ xyx
- x²yx ≈ xyx²
- x²yx ≈ yxy²
- x²yx² ≈ xyx
- x²yxy ≈ yxy
- x²y²x ≈ yxy
- xyx ≈ xyx³
- xyx ≈ xyxy²
- xyx ≈ xy²xy
- xyx ≈ xy³x
- xyx ≈ yx²y²
- xyx ≈ yxyxy
- xyx ≈ yxy²x
- xyxy ≈ xy²x
- xyxy ≈ yx²y
- xyzx ≈ xzyx
- x²yzx ≈ yxyzy
- xyxzy ≈ xy²zx
- xyxzy ≈ yx²zy
Lean proof
Endpoint theorem: SemigroupBasis.Generated.Order6DirectPowerAvailableV3.S6_4042_Ad4b303415746_Part01.S6_10822.representative_basis
BasisFor Generated.Order6DirectPowerAvailableV3.S6_4042_Ad4b303415746_Part01.S6_10822.table.semigroup Generated.Order6DirectPowerAvailableV3Sources.S6_4042.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 106 Lean files with 59,549 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_10822 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_10822 is the table, with the elements numbered 0 to 5.
Variety
[6, 10822] generates the variety V[4042]; 44 other semigroups of order six generate it too, and 2,200 semigroups of order six lie in it. [6, 10822] lies in 9 of the 505 varieties of the census, those whose basis it satisfies.
- Directly above
- Directly below
V[4042] 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, 10821]: 1 entry (5, 6), variety V[4040]
- [6, 10190]: 2 entries (2, 6), (5, 6), variety V[1223]
- [6, 10730]: 2 entries (3, 4), (4, 3), variety V[3467]