Partial Actions and Power Sets

Partial actions of groups appeared independently in various areas of mathematics, in particular, in the study of operator algebras. The formal definition of this concept was given by Exel in 1998 [1]. Later in 2003, Abadie [2] introduced the notion of enveloping action and found that any partial action possesses an enveloping action. The study of partial actions on arbitrary rings was initiated by Dokuchaev and Exel in 2005 [3]. Among other results, they prove that there exist partial actions without an enveloping action and give sufficient conditions to guarantee the existence of enveloping actions. Many studies have shown that partial actions are a powerful tool to generalize many well-known results of global actions (see [3, 4] and the literature quoted therein). The theory of partial actions of groups has taken several directions over the past thirteen years. One way is to consider actions of monoids and groupoids rather than group actions. Another is to consider sets with some additional structure such as rings, topological spaces, ordered sets, or metric spaces. Partial actions on the power set and its compatibility with its ring structure have not been considered. This work is devoted to study some topics related to partial actions on the power set P(X) arising from partial actions on the set X and its enveloping actions. In Section 1, we present some theoretical results of partial actions and enveloping actions. In Section 2, we extend a partial action α on the set X to a partial action on the ring (P(X), Δ, ∩). In addition, we introduce the concept of partial action of finite type to investigate the relationship between the enveloping action of (P(X), α) and (P(T), β), the power set of the enveloping action of (X, α).