Products of Automata and the Problem of Covering
Transactions of the American Mathematical Society
Introduction. The first part of this paper (Theorems 1-3) gives a short, unified treatment of (Mealy type  ) automata (sequential machines). By associating with every input two binary relations ("next-state" and "output" relations) we obtain an easy and concise algebraic method for the description and study of complete or partial, finite or infinite automata. In the second part (Theorems 4-7) we develop further the algebraic decomposition theory of automata, continuing previous work by J.
... evious work by J. Hartmanis - and M. Yoeli -. To make the exposition self-contained, we repeat some of the material contained in . For other approaches to automata decompositions the reader is referred to [l], , , , . In  the concept of semi-automaton (see Section I) was introduced and methods for its decomposition by means of overlapping partitions were derived. In the present paper these investigations are extended to (Mealy type) automata and the problems of covering specified automata by direct and cascade products are studied. This approach leads to an interesting new algebraic concept, namely that of a weak (i.e., generalized) homomorphism denned by overlapping partitions. Recently this concept and its applicability to partial algebras has been further investigated  and generalizations of well-known results on homomorphisms and subdirect products of partial algebras have