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This research aims to introduce and study the minimal realization problem of a new class of automaton with input and output as multisets. We begin by introducing the concrete categories [Formula: see text] and [Formula: see text] of non-deterministic and [Formula: see text]-fuzzy multiset automata over category Set and determine the functorial relationship between them, in which [Formula: see text] is a distributive lattice. Next, we introduce other concrete categories [Formula: see text] and [Formula: see text] of [Formula: see text]-fuzzy (crisp deterministic [Formula: see text]-fuzzy) multiset automata with output over category Set and their input–output [Formula: see text]-fuzzy multiset behavior. Further, we minimize the introduced category [Formula: see text] using functorial maps. Finally, we introduce the concept of realization of an input–output [Formula: see text]-fuzzy multiset behavior and using Myhill–Nerode’s theory, we construct a minimal crisp deterministic [Formula: see text]-fuzzy multiset automata with output in Set which realizes the given input–output [Formula: see text]-fuzzy multiset behavior.

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