In the last fifteen years many researchers working in such diverse areas as pattern recognition, parallel computation, self-reproducing automata, pattern recognition, evolution theory, information retrieval, and integrated circuit technology have found it convenient to use an array of identical intercommunicating finite-state machines as their underlying model. This naturally suggests that it would be worthwhile to develop a comprehensive theory of such arrays which would then provide a common framework for all these areas. Yamada and Amoroso have taken up this suggestion, and this paper is the first in a proposed series of papers reporting their results. However, as they readily admit, they have not attempted to produce a truly comprehensive model (an ill-defined, if not impossible, task), but rather they have been concerned with a "fairly general" model based largely on the earlier work of Moore.