Homomorphisms of Finite Automata

A cyclic finite automaton is a finite automaton which is generated by one single state, and each finite automata is the union of some cyclic finite automata. In this paper, in order to further investigate more insightful properties underlying cyclic finite automata and reveal the relationship between general finite automata and cyclic finite automata, by using the algebraic fashion we analyze the endomorphisms of cyclic finite automata, and propose an algorithm for finding the endomorphism semigroup and the automorphism group of a cyclic finite automaton. Moreover, the homomorphisms of general finite automata are discussed, and it is proved that every finite automaton is a homomorphism image of a direct sum of cyclic finite automata.