[6, 8564] B₂¹ no finite basis
[6, 8564] is the Brandt monoid B₂¹, one of the four semigroups of order six without a finite identity basis.
Cayley table
| · | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | 4 | 1 | 2 | 2 |
| 3 | 1 | 5 | 1 | 3 | 1 | 3 |
| 4 | 1 | 2 | 1 | 4 | 1 | 4 |
| 5 | 1 | 1 | 3 | 1 | 5 | 5 |
| 6 | 1 | 2 | 3 | 4 | 5 | 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, 8564)
- Idempotents
- 1, 4, 5, 6
- Zero
- 1
- Identity
- 6
- Nilpotent
- no
- Commutative
- no
- Regular
- yes, inverse
- Group
- no
- 𝒥-classes
- 3
- Rank
- 3, generated by {2, 3, 6}
- Self-dual
- yes: anti-isomorphic to itself
No finite identity basis
For every n, the identity
x y₁⋯yₙ x yₙ⋯y₁ ≈ x yₙ⋯y₁ x y₁⋯yₙ
holds in B₂¹. A pattern in the occurrences of the letters survives every derivation step with an identity in fewer variables; the argument uses that xytyx and xtyxy are isoterms, equal in B₂¹ only to themselves. A finite set of identities uses boundedly many variables, so it misses one of these identities and is not a basis.
Argument: P. Perkins (1969), in the reconstruction of M. V. Sapir (1988). 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.Nonfinite.B2One.s6_8564_nonfinitelyBased
theorem s6_8564_nonfinitelyBased : NonfinitelyBased catalogueTable.semigroup
- Size
- Checking this class alone compiles 14 Lean files with 13,559 lines: the endpoint theorem and everything it imports, the shared library included. Of these, 12,670 lines are used by the proof of this class and of no other.
- Census
- SemiBase.Census.S6_8564 checks that the theorem is about the table of this class and concludes Classified; SemiBase.Catalogue.Order6.S6_8564 is the table, with the elements numbered 0 to 5.
Varieties
[6, 8564] 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.
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.