Difference-restriction algebras of partial functions: axiomatisations and representations

TL;DR

This paper provides a finite axiomatization for algebras of partial functions with specific operations, characterizes complete representations, and establishes their properties and limitations.

Contribution

It introduces a finite equational axiomatisation for representable algebras of partial functions and characterizes complete representations in this context.

Findings

Finite axiomatization for representable algebras

Complete representations are characterized by atomicity and representability

The axiomatization for complete representations is proven to be minimal and optimal

Abstract

We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational axiomatisation for the class of algebras representable by partial functions. As a corollary, the same equations axiomatise the algebras representable as injective partial functions. For complete representations, we show that a representation is meet complete if and only if it is join complete. Then we show that the class of completely representable algebras is precisely the class of atomic and representable algebras. As a corollary, the same properties axiomatise the class of algebras completely representable by injective partial functions. The universal-existential-universal axiomatisation this yields for these complete representation classes is the simplest possible, in the sense that no existential-universal-existential axiomatisation exists.