Preservation theorems for Tarski's relation algebra

TL;DR

This paper explores the finite generation of various semantically defined fragments of Tarski's algebra of binary relations, revealing which are finitely generated and which are not, with implications for their expressibility.

Contribution

It provides new results on the finite generation and expressibility of different relation algebra fragments, including positive and negative findings.

Findings

Homomorphism-safe fragment is finitely generated.

Function-preserving fragment is not finitely generated.

Forward-looking function-preserving fragment is finitely generated.

Abstract

We investigate a number of semantically defined fragments of Tarski's algebra of binary relations, including the function-preserving fragment. We address the question whether they are generated by a finite set of operations. We obtain several positive and negative results along these lines. Specifically, the homomorphism-safe fragment is finitely generated (both over finite and over arbitrary structures). The function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable function-preserving operations). Similarly, the total-function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable total-function-preserving operations). In contrast, the forward-looking function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an `injective union' operation.