Shifts of hereditarily finite type, a subclass of shifts of finite type.

We introduce the class of hereditarily finite type shift spaces. Hereditarily finite type shift spaces are the subclass of shift spaces such that all of their projections are of finite type. We provide a method to construct a graph for projections of these spaces. We establish that the question of whether a shift space is of hereditarily finite type is answerable for mixing shifts of finite type. We then present a family of shifts of finite type, derived from a family of labeled graphs, and classify these spaces as hereditarily finite type or not, under the assumption that only a single edge label in the graph of said space is used more than once.