The theory of finite automata provides a formal approach to the design of sequential circuits, assuming the sequential aspect of the realisation to be in the form of bistables. No formal approach has been developed, however, to take advantage of the various sequential units available in MSI (Medium Scale Integration) form. The problem can be viewed as that of "decomposing" the objective automaton into an interconnection of MSI sequential units, and this is the approach adopted in the present study. However the study of such ''composite realisations'' raises fundamental problems, for example what does an objective automaton represent? Moreover, how is an objective automaton to be formulated? It is also essential to clarify what is meant by a "realisation" of an objective automaton, so that in forming a "composite realisation" the basic aim is clearly understood. The initial aim in the present study, however, is to consider even more fundamental problems. It would seem that finite-automata theory can be developed from just a few essential concepts, furthermore these concepts are closely interrelated so a unified appreciation can be gained. By adopting this approach, the theory of finite automata can be developed in close association with more general abstract algebra, and can be developed with regard to axiomatic set theory and universal algebra.