IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 376915 10.1155/2013/376915 376915 Research Article Partial Actions and Power Sets 0000-0002-8713-2449 Ávila Jesús 1 Lazzarin João 2 Caenepeel Stefaan 1 Departamento de Matemáticas y Estadística Universidad del Tolima, Ibagué Colombia ut.edu.co 2 Departamento de Matemática Universidade Federal de Santa Maria, Santa Maria, RS Brazil ufsm.br 2013 3 2 2013 2013 04 10 2012 26 12 2012 2013 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.

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).

1. 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 . 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 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).

2. 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 α of the group G on the set X is a collection of subsets Sg, gG, of X and bijections αg:Sg-1Sg such that for all g,hG, the following statements hold.

S1=X and α1 is the identity of X.

S(gh)-1αh-1(ShSg-1).

αgαh(x)=αgh(x), for all xαh-1(ShSg-1).

The partial action α will be denoted by (X,α) or α={Sg,αg}gG. Examples of partial actions can be obtained by restricting a global action to a subset. More exactly, suppose that G acts on Y by bijections γg:YY and let X be a subset of Y. Set Sg=Xγg(X) and let αg be the restriction of γg to Sg-1, for each gG. Then, it is easy to see that α={Sg,αg}gG is a partial action of G on X. In this case, α is called the restriction of γ to X. In fact, for any partial action (X,α) there exists a minimal global action (T,β) (enveloping action of (X,α)), such that α is the restriction of β to X [2, Theorem 1.1].

To define a partial action of the group G on the ring R, it is enough to assume in Definition 1 that each Sg, gG, is an ideal of R and that every map αg:Sg-1Sg 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 β of a group G on the ring E is said to be an enveloping action for the partial action α of G on a ring R, if there exists a ring isomorphism φ of R onto an ideal of E such that for all gG, the following conditions hold.

φ(Sg)=φ(R)βg(φ(R)).

φαg(x)=βgφ(x), for all xSg-1.

E is generated by gGβg(φ(R)).

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 R be a unital ring. Then a partial action α of a group G on R admits an enveloping action β if and only if each ideal Sg,gG, is a unital ring. Moreover, if such an enveloping action exists, it is unique up equivalence.

3. Results

In this section, we consider a nonempty set X and a partial action α={Xg,αg}gG the of the group G on X. By [2, Theorem 1.1] there exists an enveloping action (T,β) for (X,α). That is, there exist a set T and a global action β={βggG} of G on T, where each βg is a bijection of T, such that the partial action α is given by restriction. Thus, we can assume that XT, T is the orbit of X, Sg=Xβg(X) for each gG and αg(x)=βg(x) for all gG and all xSg-1.

The action β on T can be extended to an action on P(T). Moreover, since βg, gG, is a bijective function, we have that βg(AΔB)=βg(A)Δβg(B) and βg(AB)=βg(A)  βg(B) for all A,BP(T) and all gG. Therefore, the group G acts on the ring (P(T),Δ,). This action will also be denoted by β.

Proposition 4.

If G acts partially on X then G acts partially on the ring (P(X),Δ,).

Proof.

Let α={Xg,αg}gG a partial action of G on X and consider the collection α={Sg,αg}gG, where Sg=P(Xg), gG, and αg:Sg-1Sg is defined by αg(A)={αg(a):aA} for all gG and all AP(Xg). It is clear that αg, gG, is a well-defined function, and it is a bijection. Now, we must prove that α defines a partial action of G on the ring P(X). We verify 2 and 3 of Definition 1, since 1 is evident.

(2) If Aαh-1(ShSg-1), then αh(A)P(XhXg-1). Thus, aαh-1(XhXg-1)X(gh)-1 for each aA. Hence, AP(X(gh)-1)=S(gh)-1, and we conclude that αh-1(ShSg-1)S(gh)-1.

(3) For all Aαh-1(ShSg-1), we have that (αgαh)(A)={(αgαh)(a):aA}. Since AS(gh)-1 (item 2), then (αgαh)(A)={αgh(a):aA}=αgh(A). In conclusion, (αgαh)(A)=αgh(A) for all Aαh-1(ShSg-1).

Finally, for all A,BP(Xg-1), we have that αg(AΔB)=αg(A)Δαg(B) and αg(AB)=αg(A)αg(B), because each αg, gG, is a bijection. Therefore, G acts partially on the ring (P(X),Δ,).

The partial action of G on (P(X),Δ,) will also be denoted by α.

In the previous proposition, note that each ideal Sg=P(Xg), gG, has the identity element Xg. Thus, by Theorem 3, we conclude that there exists an enveloping action for the partial action (P(X),α). In the following result, we find this enveloping action and show its relationship with (P(T),β).

Proposition 5.

Let α be a partial action of G on the nonempty set X. The following statements hold.

P(X)  is an ideal of P(T).

E=gGβg(P(X))  is a β-invariant ideal of P(T).

The enveloping action of  (P(X),α)  is  (E,β), where each  βg, gG,acts on  E  by restriction.

Proof.

(1) It is a direct consequence of the inclusion XT.

(2) Since P(X) is an ideal of P(T), we have that E=gGβg(P(X)) is an ideal of P(T), and it is clear that E is β-invariant.

(3) We must prove 1, 2, and 3 of Definition 2. Note that by item 2, the action β on E is global. Moreover, we can identify P(X) with φ(P(X)) because P(X) is an ideal of E. The item 3 is consequence of 2.

To prove 2, let AP(Sg-1). Then, αg(A)={αg(x)xA}. Since ASg-1X and (T,β) is the enveloping action of (X,α), we have that αg(x)=βg(x) for all xA and all gG. Thus, αg(A)={βg(x)xA}=βg(A), and we conclude that αg(A)=βg(A) for all AP(Sg-1).

To prove 1, let AP(Sg). Then, βg-1(A)=αg-1(A) and thus A=βg(αg-1(A))βg(P(X)). Hence, P(Sg)P(X)βg(P(X)) for all gG.

For the other inclusion, let A,BP(X) such that A=βg(B). Then, A=Xβg(XB)=Xβg(X)βg(B). Since (T,β) is the enveloping action of (X,α), we have Sg=Xβg(X) for all gG. Hence, A=Sgβg(B)Sg and thus AP(Sg).

We conclude that (E,β) is the enveloping action of (P(X),α).

The final result shows that (E,β), the enveloping action of (P(X),α), is a subaction of (P(T),β). Thus it is natural to ask in which case E=P(T) or equivalently when (P(T),β) is the enveloping action of (P(X),α). To solve this problem, we first define the concept of partial action of finite type.

Definition 6.

Let (X,α) be a partial action of G on the set X with enveloping action (T,β). (X,α) is said to be of finite type if there exist g1,,gnG such that T=i=1nβgi(X).

A partial action α={Dg,αg}gG of G on the ring R is called of finite type [5, Definition 1.1] if there exists a finite subset {g1,,gn} of G, such that, i=1nDggi=R for any gG. If the partial action has an enveloping action, then it can be characterized as follows [5, Proposition 1.2].

Proposition 7.

Let α be a partial action of G on the ring R with enveloping action (E,β). The following statements are equivalent.

αis of finite type.

There exist g1,,gnG such that E=i=1nβgi(R).

E has an identity element.

The following theorem is the main result of this work. Without loss of generality, we can assume that g1=1 in Definition 6 and Proposition 7. First, we prove the following specialization of [5, Proposition 1.10].

Proposition 8.

Under the previous assumptions, if S=i=1nβgi(P(X)) with g1=1,,gnG, then 1S=i=1nβgi(X) is the identity element of S.

Proof.

By induction on n, it is enough to consider the case with two summands, that is, S=P(X)+βg2(P(X)). Then, 1S=1P(X)+1βg2(P(X))-1P(X)1βg2(P(X)). Since the addition is the symmetric difference and the product is the intersection, we obtain that 1S=XΔβg2(X)Δ(Xβg2(X))=Xβg2(X).

Theorem 9.

Let (X,α) be a partial action of G on the set X with enveloping action (T,β). The following statements are equivalent.

(X,α) is of finite type.

E=P(T).

(P(X),α) is of finite type.

Proof.

1 2 . Suppose that there exist g1=1,,gnG such that T=i=1nβgi(X). By Proposition 8, the identity element of the ring S=i=1nβgi(P(X))E is 1S=i=1nβgi(X)=T. So, TE, and since E is an ideal of P(T), we conclude that E=P(T).

2 3 . If E=P(T), then TE. So, E is a ring with identity, and by Proposition 7 the result follows.

3 1 . If (P(X),α) is of finite type, then there exist g1=1,,gnG such that E=i=1nβgi(P(X)). Thus, for each gG, there exist A1g,,AngX such that βg(X)=i=1nβgi(Aig), which implies that βg(X)i=1nβgi(X) for each gG. Hence, T=gGβg(X)i=1nβgi(X). In conclusion, (X,α) is of finite type.

To illustrate the results obtained, we include the following examples.

Example 10.

Let X be the set of even integers and G the group . We define a partial action α of G on X as follows: Sn=X if n is even and Sn= if n is odd; for n, an even integer αn:S-nSn is defined by αn(k)=k+n for all kS-n, and for n, an odd integer αn is the empty function. The enveloping action of (X,α) is (T,β) where T= and β is the action of on T, defined by βn(k)=k+n for all kT. Since Xβ0(P(X)) and the set of odd integers Yβ1(P(X)), we have X+Y=TE. Hence, E=P(T), and thus (P(T),β) is the enveloping action of (P(X),α).

Example 11.

Let X={n0} where n0 is a fixed integer and G is the group . We define a partial action α of G on X as follows: S0=X and Sn= for n0; α0:S0S0 is the identity and αn is the empty function in other case. The enveloping action of (X,α) is (T,β), that of the previous example.

Note that each singleton of T is an element of βn(P(X)) for some integer n. So, E=nβg(P(X)) the enveloping action of (P(X),α) coincides with the collection of all finite subsets of T. Hence, TE because T is an infinite set, and we conclude that EP(T).

In  it was proved that if R is a ring and α is a partial action of a group G on R with enveloping action (T,β), then T is right (left) Noetherian (Artinian), if and only if R is right (left) Noetherian (Artinian) and α is of finite type (Corollary 1.3). Under the same assumptions, they also proved that T is semisimple if and only if R is semisimple and α 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.

X is finite, and (X,α) is of finite type.

The ring P(X) is Noetherian (Artinian, semisimple), and (P(X),α) is of finite type.

The ring E is Noetherian (Artinian, semisimple).

Proof.

It is enough to observe that X is finite if and only if the ring P(X) is noetherian (Artinian, semisimple) and apply Theorem 9.

Acknowledgments

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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