[6, 8878] A₂ᵍ no finite basis
[6, 8878] is the semigroup A₂ᵍ, obtained from A₂ by adjoining an element g, one of the four semigroups of order six without a finite identity basis.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 3 | 1 | 1 | 1 |
| 2 | 1 | 1 | 3 | 1 | 2 | 4 |
| 3 | 3 | 3 | 1 | 3 | 3 | 3 |
| 4 | 1 | 2 | 3 | 4 | 2 | 4 |
| 5 | 1 | 1 | 3 | 1 | 5 | 6 |
| 6 | 1 | 5 | 3 | 6 | 5 | 6 |
The product of the row element and the column element, numbered as in Smallsemi. Blue: idempotents on the diagonal.
Structure
- Smallsemi
- SmallSemigroup(6, 8878)
- Idempotents
- 1, 4, 5, 6
- Zero
- none
- Identity
- none
- Nilpotent
- no
- Commutative
- no
- Regular
- yes
- Group
- no
- 𝒥-classes
- 2
- Rank
- 3, generated by {2, 3, 6}
- Self-dual
- yes: anti-isomorphic to itself
No finite identity basis
For every n, the identity
(x₁²⋯xₙ²)² ≈ (x₁²⋯xₙ²)³
holds in A₂ᵍ. The identities of A₂ᵍ are read off a directed graph of the word together with a parity; the parity with which a word winds around the cycle x₁ → ⋯ → xₙ → x₁ survives every derivation step with an identity in fewer variables. A finite set of identities uses boundedly many variables, so it misses one of these identities and is not a basis.
Argument: a winding-parity argument on cycle words; that A₂ᵍ has no finite basis is in Lee, Li and Zhang (2012). That L, B₂¹, A₂ᵍ and A₂¹ are the only semigroups of order six without a finite basis was shown by Lee, Li and Zhang (2012). See the four nonfinitely based semigroups.
Lean proof
Endpoint theorem: SemigroupBasis.Examples.AC2.s6_8878_nonfinitelyBased
theorem s6_8878_nonfinitelyBased : NonfinitelyBased catalogueTable.semigroup
- Size
- Checking this class alone compiles 10 Lean files with 3,988 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_8878 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_8878 is the table, with the elements numbered 0 to 5.
Varieties
[6, 8878] generates a variety without a finite basis, so not one of the 505 varieties of this census. It lies in none of them: no finitely based semigroup of order six generates a variety that contains it, as evaluating their bases in this table shows.