The finite representation property for composition, intersection, domain and range

TL;DR

This paper proves that finite algebras with certain operations can be represented by partial functions with a bounded base size, ensuring decidability of their representability, and provides a counterexample for a different signature.

Contribution

It establishes the finite representation property for a broad class of algebraic signatures involving composition, intersection, domain, range, and additional operations, with explicit bounds.

Findings

Finite representation property holds for these signatures.

Base size needed is double-exponential in algebra size.

Representability is decidable for all these signatures.

Abstract

We prove that the finite representation property holds for representation by partial functions for the signature consisting of composition, intersection, domain and range and for any expansion of this signature by the antidomain, fixset, preferential union, maximum iterate and opposite operations. The proof shows that, for all these signatures, the size of base required is bounded by a double-exponential function of the size of the algebra. This establishes that representability of finite algebras is decidable for all these signatures. We also give an example of a signature for which the finite representation property fails to hold for representation by partial functions.