Residuation algebras with functional duals

TL;DR

This paper investigates residuation algebras with functional duals using canonical extensions, showing that functionality cannot be characterized by any universal first-order sentence in their language.

Contribution

It provides a partial answer to Gehrke's question by proving the non-existence of a universal first-order characterization of functionality in these algebras.

Findings

No universal first-order sentence characterizes functionality.

Canonical extensions are used to analyze the structure.

Relational structures associated with these algebras are not universally definable.

Abstract

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 no universal first-order sentence in the language of residuation algebras is equivalent to the functionality of the associated relational structures.