Duality theory and representations for distributive quasi relation algebras and DInFL-algebras

TL;DR

This paper establishes duality theories for certain distributive quasi relation and involutive FL-algebras, connecting algebraic structures with structured frames, and explores their representability as lattices of binary relations.

Contribution

It develops duality frameworks for distributive quasi relation and involutive FL-algebras, extending to all algebras and analyzing their representability as relation lattices.

Findings

Duality for complete perfect distributive quasi relation algebras

Duality for complete perfect distributive involutive FL-algebras

Characterization of algebraic representability as relation lattices

Abstract

We develop dualities for complete perfect distributive quasi relation algebras and complete perfect distributive involutive FL-algebras. The duals are partially ordered frames with additional structure. These frames are analogous to the atom structures used to study relation algebras. We also extend the duality from complete perfect algebras to all algebras, using so-called doubly-pointed frames with a Priestley topology. We then turn to the representability of these algebras as lattices of binary relations. Some algebras can be realised as term subreducts of representable relation algebras and are hence representable. We provide a detailed account of known representations for all algebras up to size six.