On Minimal Realizations of Neutrosophic Languages: A Categorical Approach

This paper develops a categorical framework for deterministic neutrosophic automata with neutrosophic outputs and their associated neutrosophic languages. Building upon neutrosophic set theory, which independently captures truth-membe rship, indeterminacy-membership, and falsity-membership, we introduce and systematically investigate the structural notions of run maps, reachability, and observability within this framework. Furthermore, two distinct approaches for constructing minimal realizations of a given neutrosophic language are presented and proven to be equivalent. These constructions yield canonical minimal automata and extend the classical Myhill-Nerode paradigm to the neutrosophic setting. The functorial relationships established between the categories of automata and languages provide a unified and robust theoretical foundation for modeling systems involving uncertainty and indeterminacy.