On the Existence of Finite Isomorphically Complete Systems of Automata

In the theory of compositions of finite automata it is a central problem to study systems from which every automaton can be built under a given composition and representation. Such systems are called complete with respect to the considered composition and representation. From practical and theoretical points of view, those compositions and representations have great importance for which there are finite complete systems. Under the isomorphic embedding as representation, a graph theoretical necessary condition of the existence of finite complete systems is presented in [3]. In this paper, we give a sufficient condition for a composition to admit a finite complete system under the isomorphic representation, which reduces the question to a graph coloring problem.