 Every history preserving function and every future preserving function is also order preserving, but not vice versa.In the theory of dynamical systems, history preserving maps capture the intuition that the behavior in one system is a "refinement" of the behavior in another.Furthermore, functions that are history and future preserving surjections capture the notion of bisimulation between systems, and thus the intuition that a given refinement is "correct" with respect to a specification.