Extended Contact Algebras: Algebraic analysis and duality theory

TL;DR

This paper develops an algebraic and duality-theoretic framework for Extended Contact Algebras, introducing Pseudo-Inference Algebras and extending Stone duality to ternary relations, thereby unifying spatial and logical semantics.

Contribution

It introduces Pseudo-Inference Algebras as algebraic structures for ECAs and extends Stone duality to ternary relations, providing a unified semantic framework.

Findings

Relational Pseudo-Inference Algebras correspond exactly to ECAs.

The subclass of relational ECAs generates a discriminator variety.

Extended Stone duality is established for ternary contact relations.

Abstract

The ternary extended contact relation was introduced in (Ivanova, 2020) as a more expressive counterpart of the standard binary contact relation. The class of Boolean algebras expanded with the relation was named Extended Contact Algebras (ECAs). In this work, we take an algebraic perspective on ECAs, interpreting the ternary relation as a form of entailment. We introduce Pseudo-Inference Algebras, purely algebraic tructures where the ternary relation is replaced by a monotone ternary operator, capturing the logical character of extended contact. We show that the subclass of relational Pseudo-Inference Algebras corresponds precisely to ECAs and generates a subvariety of strict PSI-Algebras, which forms a discriminator variety. Furthermore, we extend Stone duality to this ternary context, introducing descriptive PSI-frames and establishing three interrelated dualities that differ in their morphisms while sharing the same class of topological objects. The framework developed in the paper provides a nified relational semantics for Boolean algebras equipped with monotone ternary operators, connecting spatial and logical notions within a categorical and topological setting.