Cascade connections and triangular products of linear automata

In this note, we want to resume attention to the basics of triangular product of automata construction and to introduce the notion of linear automata complexity. It contains three main results. (1) For any two pure automata we consider the category of their cascade connections. It possesses the universal terminal object. This object is the wreath product of the automata. Hence, every cascade connection admits a natural embedding into wreath product of automata. (2) A similar theory is built for linear automata, where we also consider the category of cascade connections. It also has the terminal object. This object is the triangular product of linear automata. (3) Triangular products have various applications. This construction is used in linear automata decomposition theory, in the definition of complexity of a linear automaton. We consider a special linear complexity and give the rule for its calculation.