Contractions of quasi relation algebras and applications to representability

TL;DR

This paper introduces a method to construct contractions of quasi relation algebras using positive symmetric idempotent elements, demonstrating that certain contractions preserve representability and identifying classes that are not finitely representable.

Contribution

It develops a new approach to constructing contractions of qRAs and analyzes their representability properties, advancing understanding of their algebraic structure.

Findings

Contractions of distributive qRAs are representable if the original algebra is.

Identified a class of distributive qRAs that are not finitely representable.

Provided a method to generate new qRAs from existing ones.

Abstract

Quasi relation algebras (qRAs) were first described by Galatos and Jipsen in 2013. They are generalisations of relation algebras and can also be viewed as certain residuated lattice expansions. We identify positive symmetric idempotent elements in qRAs and show that they can be used to construct new qRAs, so-called contractions of the original algebra. We then show that the contraction of a distributive qRA will be representable when the original algebra is representable. Further, we identify a class of distributive qRAs that are not finitely representable.