Group actions on sets and automata theory

Abstract

In this work we deal with group actions on sets. We would like to establish some relations between isotropy subgroups G x , x∈ X of G where G is any group and act on X ( X being a given set). The action of G on X generate an equivalence relation on X, denoted by ∼. In the first part of this work we establish concretly some relations between isotropy subgroups G x , G x ′ when x, x ′ are in the same class of ∼. In the second and third part we proceed to applications. Namely, in the second part we establish some relations between group actions and category theory (that is we mention two categories and a functor between them) and in the part three we will look at the set X like the set of all states of a given physical system (or automaton) and we will define a certain way of interaction for our system and we will state a principle for this interaction. The basic idea for this interaction is following: the system (automaton) takes part into interaction if the value of some physical quantity f (viewed like a function defined on “states-set”) is changing (that is we mention a transition between orbits of Ker f , the equivalence relation generated by f on the “states-set”).