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-membership, 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.