Categorical Representations of Nice Automata and Their Subclasses

This paper establishes a categorical framework for the study of nice automata. We prove that the class of nice automata, equipped with suitable morphisms, constitutes a category, and we demonstrate the existence of a functor from this category to the category of biordered sets. Furthermore, we identify the categories of synchronizing, trapped, and B-band automata as full subcategories of the category of nice automata. Finally, we establish functorial correspondences between these specific automata classes and the categories of right zero semigroups, left zero semigroups, and biordered subsets of bands, respectively.