The algebra of functions with antidomain and range

TL;DR

This paper provides a complete set of axioms for algebras of unary partial functions with operations like composition, domain, antidomain, range, and intersection, analyzing their complexity and finite representability.

Contribution

It completes the classification of algebras of unary partial functions with these operations and studies their complexity and finite representability.

Findings

Complete finite quasiequational axiomatisations established.

Complexity of the equational theories is nondeterministic polynomial.

Finite algebras can be represented over finite sets without intersection in the signature.

Abstract

We give complete, finite quasiequational axiomatisations for algebras of unary partial functions under the operations of composition, domain, antidomain, range and intersection. This completes the extensive programme of classifying algebras of unary partial functions under combinations of these operations. We look at the complexity of the equational theories and provide a nondeterministic polynomial upper bound. Finally we look at the problem of finite representability and show that finite algebras can be represented as a collection of unary functions over a finite base set provided that intersection is not in the signature.