Publisher Summary This chapter presents several characteristics of connectedness and of separation in automata and discusses their relationships to other connectivity properties and to the basic structure of automata. An automaton is finite only if its set of states is finite. A primary of a nonempty finite automaton is a maximal singly generated subautomaton. The union and the intersection of subautomata of A are themselves subautomata of A. The chapter illustrates a method for finding the minimal separated subautomata of a finite automaton and explains its use in determining the homomorphisms, endomorphisms, isomorphisms, and automorphisms of finite automata. The separated parts of an automaton can be regarded as unrelated automata with the same input alphabet. Every nonempty automaton is made entirely of blocks. Many problems are profitably reducible to the natural building blocks. Two blocks of a nonempty automaton are either identical or disjoint. A nonempty subautomaton C of A is separated only if C is the union of blocks of A.
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