Minimal Realization of an Input-Output L-fuzzy Multiset Behavior

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 NMA(Set)(O?) and LMA(Set)(O?) of non-deterministic and L -fuzzy multiset automata over category Set and determine the functorial relationship between them, in which L is a distributive lattice. Next, we introduce other concrete categories LMAO(Set)(O?, t?) and DLMAO(Set)(O?, t?) of L-fuzzy (crisp deterministic L-fuzzy) multi set automata with output over category Set and their input-output L-fuzzy multiset behavior. Further, we minimize the introduced category DLMAO(Set)(O?, 23?) using functorial maps. Finally, we introduce the concept of realization of an input-output L -fuzzy multiset behavior and using Myhill-Nerode's theory, we construct a minimal crisp deterministic L -fuzzy multiset automata with output in Set which realizes the given input- output L -fuzzy multiset behavior.