  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.
