On the structure of free finite state machines

Abstract

The finite state machine U m, n ( M) freely generated by a set consisting of m states and n inputs subjects to the relations holding in the finite state machine M was considered by Birkhoff and Lipson in [1, 2]. In this paper, necessary and sufficient conditions for U m, n ( M) to consist of m disjoint copies of U 1, n ( M) are established. The relationship between U 1, n ( M) and the transition monoid of M, and a representation of U 1, n ( M) as a transition monoid machine are described. The characterization of machines of type U 1, n ( M) is in this way reduced to the characterization of finite monoids possessing a ‘universal presentation’. Some general results concerning finite semigroups and groups with a universal presentation, and precise characterizations of finite semilattices and Abelian groups admitting a universal presentation are described.