Research Article | Open Access
Partial Actions and Power Sets
We consider a partial action with enveloping action . In this work we extend α to a partial action on the ring and find its enveloping action . Finally, we introduce the concept of partial action of finite type to investigate the relationship between and .
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 . Later in 2003, Abadie  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 . 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 . 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 .
Definition 1. A partial action of the group on the set is a collection of subsets , , of and bijections such that for all , the following statements hold.(1) and is the identity of . (2). (3), for all .
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 , for each . Then, it is easy to see that 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 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. (1). (2), for all . (3) is generated by .
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.
In this section, we consider a nonempty set and a partial action the 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 .
The action on can be extended to an action on . Moreover, since , , is a bijective function, we have that and for all and all . Therefore, the group acts on the ring . This action will also be denoted by .
Proposition 4. If acts partially on then acts partially on the ring .
Proof. Let a partial action of on and consider the collection , where , , and is defined by for all and all . It is clear that , , is a well-defined function, and it is a bijection. Now, we must prove that defines a partial action of on the ring . We verify 2 and 3 of Definition 1, since 1 is evident.
(2) If , then . Thus, for each . Hence, , and we conclude that .
(3) For all , we have that . Since (item 2), then . In conclusion, for all .
Finally, for all , we have that and , because each , , is a bijection. Therefore, acts partially on the ring .
The partial action of on will also be denoted by .
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. (1) is an ideal of . (2) is a -invariant ideal of . (3)The enveloping action of is , where each , ,acts on by restriction.
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.
To prove 2, let . Then, . Since and is the enveloping action of , we have that for all and all . Thus, , and we conclude that for all .
To prove 1, let . Then, and thus . Hence, for all .
For the other inclusion, let such that . Then, . Since is the enveloping action of , we have for all . Hence, and thus .
We conclude that is the enveloping action of .
The final result shows that , the enveloping action of , is a subaction of . Thus it is natural to ask in which case or equivalently when is the enveloping action of . To solve this problem, we first define the concept of partial action of finite type.
Definition 6. Let be a partial action of on the set with enveloping action . is said to be of finite type if there exist such that .
A partial action of on the ring is called of finite type [5, Definition 1.1] if there exists a finite subset of , such that, for any . If the partial action has an enveloping action, then it can be characterized as follows [5, Proposition 1.2].
Proposition 7. Let be a partial action of on the ring with enveloping action . The following statements are equivalent. (1)is of finite type. (2)There exist such that . (3) has an identity element.
The following theorem is the main result of this work. Without loss of generality, we can assume that in Definition 6 and Proposition 7. First, we prove the following specialization of [5, Proposition 1.10].
Proposition 8. Under the previous assumptions, if with , then is the identity element of .
Proof. By induction on , it is enough to consider the case with two summands, that is, . Then, . Since the addition is the symmetric difference and the product is the intersection, we obtain that .
Theorem 9. Let be a partial action of on the set with enveloping action . The following statements are equivalent. (1) is of finite type. (2). (3) is of finite type.
Proof. . Suppose that there exist such that . By Proposition 8, the identity element of the ring is . So, , and since is an ideal of , we conclude that .
. If , then . So, is a ring with identity, and by Proposition 7 the result follows.
. If is of finite type, then there exist such that . Thus, for each , there exist such that , which implies that for each . Hence, . In conclusion, is of finite type.
To illustrate the results obtained, we include the following examples.
Example 10. Let be the set of even integers and the group . We define a partial action of on as follows: if is even and if is odd; for , an even integer is defined by for all , and for , an odd integer is the empty function. The enveloping action of is where and is the action of on , defined by for all . Since and the set of odd integers , we have . Hence, , and thus is the enveloping action of .
Example 11. Let where is a fixed integer and is the group . We define a partial action of on as follows: and for ; is the identity and is the empty function in other case. The enveloping action of is , that of the previous example.
Note that each singleton of is an element of for some integer . So, the enveloping action of coincides with the collection of all finite subsets of . Hence, because is an infinite set, and we conclude that .
In  it was proved that if is a ring and is a partial action of a group on with enveloping action , then is right (left) Noetherian (Artinian), if and only if is right (left) Noetherian (Artinian) and is of finite type (Corollary 1.3). Under the same assumptions, they also proved that is semisimple if and only if is semisimple and is of finite type (Corollary 1.8).
By using these results and Theorem 9 we obtain the following result.
Proposition 12. Under the previous assumptions, the following statements are equivalent. (1) is finite, and is of finite type. (2)The ring is Noetherian (Artinian, semisimple), and is of finite type. (3)The ring is Noetherian (Artinian, semisimple).
Proof. It is enough to observe that is finite if and only if the ring is noetherian (Artinian, semisimple) and apply Theorem 9.
The authors are grateful to the referee for the several comments which help to improve the first version of this paper. This work was partially supported by “Oficina de Investigaciones y Desarrollo Científico de la Universidad del Tolima” and by “Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul.”
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Copyright © 2013 Jesús Ávila and João Lazzarin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.