Residuation algebras with functional duals. (English) Zbl 1480.03012
Summary: We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras.
MSC:
| 03B47 | Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) |
| 06D50 | Lattices and duality |
| 06E25 | Boolean algebras with additional operations (diagonalizable algebras, etc.) |
| 03G25 | Other algebras related to logic |
References:
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