Universal realization

A study is made of input-process machines, in the sense of Arbib and Manes, and their behavior. For a given input-process X: K → K the categories Mach(X) of machines and Beh(X) of behaviors are constructed, also a functor E: Mach(X) → Beh(X) which assigns to each machine its behavior. It is shown that E has a left adjoint and that abstract Nerode realization is universal. A consequence is a characterization of minimal realization functors: a result similar to those arrived at by Goguen for machines in closed categories. We then show that by restricting machine and behavior morphisms, realization is universal for the most general type of Nerode realization, i.e., reflexive Nerode realization.