Extended Contact Algebras: Algebraic analysis and duality theory
Rafał Gruszczyński, Paula Menchón, William Zuluaga
TL;DR
The paper develops an algebraic and topological framework for Extended Contact Algebras by replacing the ternary relation $\vdash$ with a monotone ternary operator $\Diamond$, producing PSI-Algebras. It establishes a one-to-one correspondence between ECAs and relational PSI-Algebras, and shows that relational PSI-Algebras generate a discriminator subvariety $\mathcal{SPSI}$ with CEP, EDPC, and arithmetical properties. By extending Stone duality to the ternary setting, the authors introduce descriptive PSI-frames and PSI-spaces, yielding three related dualities that connect algebraic and relational/topological semantics for monotone ternary operators. The resulting framework unifies spatial and logical entailment notions within a categorical/topological context and provides a robust semantic toolkit for Boolean algebras endowed with ternary operators, with precise morphism correspondences and dualities.
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.