State maps of general behaviors, their lattice structure and bisimulations

In this paper we study the so called state maps in the behavioral approach to systems theory. The discussion in this paper continues some preliminary development presented in [5] and it also can be thought of as a generalization of the work in [10]. The concept of states or state variables is present in almost all branches of dynamical systems theory. In areas as remotely connected as discrete event systems and linear time invariant systems we can observe that the notion of states is present. One may think that this is a mere coincidence, but this is not true. The different notions of states have something in common. They are all connected by the so called state property or the axiom of state. In short (and perhaps rather inaccurately), one can say that a quantity or variable possesses the state property (or satisfies the axiom of state) if it captures the necessary information about the evolution of the dynamical system. There is of course a more mathematically formal and rigor formulation of this property, for example in [9]. In this paper, we will make use of this concept intensively, as will be revealed in the following sections. It is quite a common view in the field of dynamical systems that states are understood to be internal. When the system is interconnected with other systems, states usually do not appear explicitly in the description of the interconnection. Nevertheless, they play an arguably central role in characterizing the compatibility of the interconnection. We shall not discuss this issue further, and the interested reader is referred to [13] and [5]. Following the earlier development in [5, 10], our point of view, which is based on the behavioral approach, is that states are constructed out of the system trajectories (the behavior). In the behavioral point of view, the behavior (i.e. the collection of all possible