On Minimal Fuzzy Realization in Category Theoretic Setting

Abstract

This paper aims to study the minimal fuzzy realization for a fuzzy language with membership values in a complete residuated lattice by using category theory. Specifically, we introduce the concept of a category [Formula: see text], whose object-class is complete transition residuated lattices corresponding to deterministic [Formula: see text]-semiautomata. We give the categorical characterization of reachability and observability maps for a given deterministic fuzzy automaton. In another direction, we demonstrate that the category [Formula: see text] is a subcategory of the categories [Formula: see text] of [Formula: see text]-coalgebras and [Formula: see text] of [Formula: see text]-dialgebras. Also, we discuss the concept of bisimulation between [Formula: see text]-coalgebras. Next, we introduce a general theory of minimal fuzzy realization for a given fuzzy language in a category theory setting. Strikingly, we demonstrate that all minimal fuzzy realization for a given fuzzy language is one of a kind up to isomorphism.