Finite Representation Property for Relation Algebra Reducts

TL;DR

This paper investigates the finite representation property (FRP) in relation algebra reducts, conjecturing conditions under which FRP fails, and provides preliminary evidence supporting this conjecture.

Contribution

It proposes a conjecture linking FRP failure to specific algebraic operations and offers initial results supporting this link.

Findings

Right-to-left implication of the conjecture holds

Preliminary results suggest the left-to-right implication

FRP remains largely unknown for many reduct languages

Abstract

The decision problem of membership in the Representation Class of Relation Algebras (RRA) for finite structures is undecidable. However, this does not hold for many Relation Algebra reduct languages. Two well known properties that are sufficient for decidability are the Finite Axiomatisability (FA) of the representation class and the Finite Representation Property (FRP). Furthermore, neither of the properties is stronger that the other, and thus, neither is also a necessary condition. Although many results are known in the area of FA, the FRP remains unknown for the majority of the reduct languages. Here we conjecture that the FRP fails for a Relation Algebra reduct if and only if it contains both composition and negation, or both composition and meet. We then show the right-to-left implication of the conjecture holds and present preliminary results that suggest the left-to-right implication.