Complete representation by partial functions for composition, intersection and antidomain

TL;DR

This paper investigates the properties of algebraic representations by partial functions involving intersection, composition, and antidomain, establishing conditions for completeness and atomicity, and exploring their axiomatizability.

Contribution

It characterizes complete and atomic representations in algebras with partial functions and analyzes their logical definability and axiomatization.

Findings

Complete representations are meet and join complete.

Not all atomic algebras are completely representable.

The class of completely representable algebras is not axiomatisable by any existential-universal-existential first-order theory.

Abstract

For representation by partial functions in the signature with intersection, composition and antidomain, we show that a representation is meet complete if and only if it is join complete. We show that a representation is complete if and only if it is atomic, but that not all atomic representable algebras are completely representable. We show that the class of completely representable algebras is not axiomatisable by any existential-universal-existential first-order theory. By giving an explicit representation, we show that the completely representable algebras form a basic elementary class, axiomatisable by a universal-existential-universal sentence.