Partial Actions and Power Sets

We consider a partial action (𝑋,𝛼) with enveloping action (𝑇,𝛽) . In this work we extend 𝛼 to a partial action on the ring (𝑃(𝑋),Δ,∩) and find its enveloping action (𝐸,𝛽) . Finally, we introduce the concept of partial action of finite type to investigate the relationship between (𝐸,𝛽) and (𝑃(𝑇),𝛽)


Introduction
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 () arising from partial actions on the set  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  on the set  to a partial action on the ring ((), Δ, ∩).In addition, we introduce the concept of partial action of finite type to investigate the relationship between the enveloping action of ((), ) and ((), ), the power set of the enveloping action of (, ).

Preliminaries
In this section, we present some results related to the partial actions, which will be used in Section 2. Other details of this theory can be found in [2,3].Definition 1.A partial action  of the group  on the set  is a collection of subsets   ,  ∈ , of  and bijections   :   −1 →   such that for all , ℎ ∈ , the following statements hold.
(1)  1 =  and  1 is the identity of . (2) The partial action  will be denoted by (, ) or  = {  ,   } ∈ .Examples of partial actions can be obtained by restricting a global action to a subset.More exactly, suppose that  acts on  by bijections   :  →  and let  be a subset of .Set   =  ∩   () and let   be the restriction of   to   −1 , for each  ∈ .Then, it is easy to see that International Journal of Mathematics and Mathematical Sciences  = {  ,   } ∈ is a partial action of  on .In this case,  is called the restriction of  to .In fact, for any partial action (, ) there exists a minimal global action (, ) (enveloping action of (, )), such that  is the restriction of  to  [2, Theorem 1.1].
To define a partial action of the group  on the ring , it is enough to assume in Definition 1 that each   ,  ∈ , is an ideal of  and that every map   :   −1 →   is an isomorphism of ideals.Natural examples of partial actions on rings can be obtained by restricting a global action to an ideal.In this case, the notion of enveloping action is the following ([3, Definition 4.2]).Definition 2. A global action  of a group  on the ring  is said to be an enveloping action for the partial action  of  on a ring , if there exists a ring isomorphism  of  onto an ideal of  such that for all  ∈ , the following conditions hold.
In general, there exist partial actions on rings which do not have an enveloping action [3,Example 3.5].The conditions that guarantee the existence of such an enveloping action are given in the following result [3, Theorem 4.5].

Theorem 3.
Let  be a unital ring.Then a partial action  of a group  on  admits an enveloping action  if and only if each ideal   ,  ∈ , is a unital ring.Moreover, if such an enveloping action exists, it is unique up equivalence.

Results
In this section, we consider a nonempty set  and a partial action  = {  ,   } ∈ be of the group  on .By [2, Theorem 1.1] there exists an enveloping action (, ) for (, ).That is, there exist a set  and a global action  = {  |  ∈ } of  on , where each   is a bijection of , such that the partial action  is given by restriction.Thus, we can assume that  ⊆ ,  is the orbit of ,   =  ∩   () for each  ∈  and   () =   () for all  ∈  and all  ∈   −1 .
In the previous proposition, note that each ideal   = (  ),  ∈ , has the identity element   .Thus, by Theorem 3, we conclude that there exists an enveloping action for the partial action ((), ).In the following result, we find this enveloping action and show its relationship with ((), ).Proposition 5. Let  be a partial action of  on the nonempty set .The following statements hold.
Proof.(1) It is a direct consequence of the inclusion  ⊆ .
(2) Since () is an ideal of (), we have that  = ∑ ∈   (()) is an ideal of (), and it is clear that  is invariant.
(3) We must prove 1, 2, and 3 of Definition 2. Note that by item 2, the action  on  is global.Moreover, we can identify () with (()) because () is an ideal of .The item 3 is consequence of 2.