Transition operators assigned to physical systems

By a physical system we recognize a set of propositions about a given system with their truth-values depending on the states of the system. Since every physical system can go from one state in another one, there exists a binary relation on the set of states describing this transition. Our aim is to assign to every such system an operator on the set of propositions which is fully determined by the mentioned relation. We establish conditions under which the given relation can be recovered by means of this transition operator.