Modules with fusion and implication based over distributive lattices: Representation and Duality

TL;DR

This paper introduces FIDL-modules, a new class of modules over distributive lattices with fusion and implication, providing duality theory and characterizations of their structural properties.

Contribution

It develops the theory of FIDL-modules, including duality and topological descriptions, which were not previously established for this class of modules.

Findings

Bi-sorted Priestley-like duality for FIDL-modules

Topological description of FIDL-congruences

Characterization of simple and subdirectly irreducible FIDL-modules

Abstract

In this paper we study the class of modules with fusion and implication based over distributive lattices, or FIDL-modules, for short. We introduce the concepts of FIDL-subalgebra and FIDL-congruence as well as the notions of simple and subdirectly irreducible FIDL-modules. We give a bi-sorted Priestley-like duality for FIDL-modules and moreover, as an application of such a duality, we provide a topological bi-spaced description of the FIDL-congruences. This result will allows us to characterize the simple and subdirectly irreducible FIDL-modules.