Partial Actions and Power Sets

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Volume 2013, Article ID 376915,4pages http://dx.doi.org/10.1155/2013/376915

Research Article

Partial Actions and Power Sets

Jesús Ávila

1

and João Lazzarin

2

1_{Departamento de Matem´aticas y Estad´ıstica, Universidad del Tolima, Ibagu´e, Colombia}

2_{Departamento de Matem´atica, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil}

Correspondence should be addressed to Jes´us ´Avila; [email protected]

Received 4 October 2012; Accepted 26 December 2012

Academic Editor: Stefaan Caenepeel

Copyright © 2013 J. ´Avila and J. 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 extend𝛼to a partial action on the ring(𝑃(𝑋), Δ, ∩) and find its enveloping action(𝐸, 𝛽). Finally, we introduce the concept of partial action offinite typeto 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 [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 set𝑃(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. InSection 2, we extend a partial action𝛼on the set

𝑋 to a partial action on the ring(𝑃(𝑋), Δ, ∩). In addition, we introduce the concept of partial actionof 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 inSection 2. Other details of this theory can be found in [2,3].

Definition 1. A partial action𝛼of the group𝐺on the set𝑋

is a collection of subsets𝑆_𝑔,𝑔 ∈ 𝐺, of𝑋and bijections𝛼_𝑔 :

𝑆_𝑔−1 → 𝑆_𝑔such that for all𝑔, ℎ ∈ 𝐺, the following statements

hold.

(1)𝑆₁= 𝑋and𝛼₁is the identity of𝑋.

(2)𝑆_(𝑔ℎ)−1 ⊇ 𝛼−1_ℎ (𝑆_ℎ∩ 𝑆_𝑔−1).

(3)𝛼_𝑔∘ 𝛼_ℎ(𝑥) = 𝛼_𝑔ℎ(𝑥), for all𝑥 ∈ 𝛼−1_ℎ (𝑆_ℎ∩ 𝑆_𝑔−1).

The partial action 𝛼 will be denoted by(𝑋, 𝛼) or𝛼 =

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2 International Journal of Mathematics and Mathematical Sciences

𝛼 = {𝑆_𝑔, 𝛼_𝑔}_𝑔∈𝐺is a partial action of𝐺on𝑋. In this case,𝛼is called the restriction of𝛾to𝑋. In fact, for any partial action

(𝑋, 𝛼)there exists a minimal global action(𝑇, 𝛽)(enveloping action of(𝑋, 𝛼)), such that𝛼is the restriction of𝛽to𝑋[2, Theorem 1.1].

To define a partial action of the group𝐺on the ring𝑅, it is enough to assume inDefinition 1that each𝑆_𝑔,𝑔 ∈ 𝐺, is an ideal of𝑅and that every map𝛼_𝑔 : 𝑆_𝑔−1 → 𝑆_𝑔is 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 action𝛽of a group𝐺on the ring𝐸is

said to be an enveloping action for the partial action𝛼of𝐺 on a ring𝑅, if there exists a ring isomorphism𝜑of𝑅onto an ideal of𝐸such that for all𝑔 ∈ 𝐺, the following conditions hold.

(1)𝜑(𝑆_𝑔) = 𝜑(𝑅) ∩ 𝛽_𝑔(𝜑(𝑅)).

(2)𝜑 ∘ 𝛼_𝑔(𝑥) = 𝛽_𝑔∘ 𝜑(𝑥), for all𝑥 ∈ 𝑆_𝑔−1.

(3)𝐸is generated by⋃_𝑔∈𝐺𝛽_𝑔(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. Let𝑅be a unital ring. Then a partial action𝛼of a

group𝐺on𝑅admits an enveloping action𝛽if and only if each

ideal𝑆_𝑔, 𝑔 ∈ 𝐺, 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 set𝑋and a partial action 𝛼 = {𝑋_𝑔, 𝛼_𝑔}_𝑔∈𝐺 be 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 set𝑇and a global action𝛽 = {𝛽_𝑔|

𝑔 ∈ 𝐺}of𝐺on𝑇, where each𝛽_𝑔is a bijection of𝑇, such that the partial action𝛼is given by restriction. Thus, we can assume that𝑋 ⊆ 𝑇,𝑇is the orbit of𝑋,𝑆_𝑔 = 𝑋 ∩ 𝛽_𝑔(𝑋)for each𝑔 ∈ 𝐺and𝛼_𝑔(𝑥) = 𝛽_𝑔(𝑥)for all𝑔 ∈ 𝐺and all𝑥 ∈ 𝑆_𝑔−1.

The action𝛽on𝑇can be extended to an action on𝑃(𝑇). Moreover, since𝛽_𝑔,𝑔 ∈ 𝐺, is a bijective function, we have that

𝛽_𝑔(𝐴Δ𝐵) = 𝛽_𝑔(𝐴)Δ𝛽_𝑔(𝐵)and𝛽_𝑔(𝐴∩𝐵) = 𝛽_𝑔(𝐴)∩ 𝛽_𝑔(𝐵)for all𝐴, 𝐵 ∈ 𝑃(𝑇)and all𝑔 ∈ 𝐺. Therefore, the group𝐺acts on the ring(𝑃(𝑇), Δ, ∩). This action will also be denoted by𝛽.

Proposition 4. If𝐺acts partially on𝑋then𝐺acts partially

on the ring(𝑃(𝑋), Δ, ∩).

Proof. Let𝛼 = {𝑋_𝑔, 𝛼_𝑔}_𝑔∈𝐺a partial action of𝐺on𝑋 and

consider the collection𝛼󸀠 = {𝑆_𝑔, 𝛼󸀠_𝑔}_𝑔∈𝐺, where𝑆_𝑔 = 𝑃(𝑋_𝑔),

𝑔 ∈ 𝐺, and𝛼󸀠_𝑔: 𝑆_𝑔−1 → 𝑆_𝑔is defined by𝛼󸀠_𝑔(𝐴) = {𝛼_𝑔(𝑎) : 𝑎 ∈

𝐴}for all𝑔 ∈ 𝐺and all𝐴 ∈ 𝑃(𝑋_𝑔). It is clear that𝛼󸀠_𝑔,𝑔 ∈ 𝐺, is a well-defined function, and it is a bijection. Now, we must

prove that𝛼󸀠defines a partial action of𝐺on the ring𝑃(𝑋). We verify 2 and 3 ofDefinition 1, since 1 is evident.

(2) If𝐴 ∈ 𝛼󸀠−1_ℎ (𝑆_ℎ∩ 𝑆_𝑔−1), then𝛼󸀠_ℎ(𝐴) ∈ 𝑃(𝑋_ℎ∩ 𝑋_𝑔−1).

Thus,𝑎 ∈ 𝛼−1_ℎ (𝑋_ℎ∩𝑋_𝑔−1) ⊆ 𝑋_(𝑔ℎ)−1for each𝑎 ∈ 𝐴. Hence,𝐴 ∈

𝑃(𝑋_(𝑔ℎ)−1) = 𝑆_(𝑔ℎ)−1, and we conclude that𝛼_ℎ󸀠−1(𝑆_ℎ∩ 𝑆_𝑔−1) ⊆

𝑆_(𝑔ℎ)−1.

(3) For all𝐴 ∈ 𝛼󸀠−1_ℎ (𝑆_ℎ∩ 𝑆_𝑔−1), we have that(𝛼󸀠_𝑔∘ 𝛼󸀠_ℎ)(𝐴) =

{(𝛼_𝑔∘ 𝛼_ℎ)(𝑎) : 𝑎 ∈ 𝐴}. Since𝐴 ∈ 𝑆_(𝑔ℎ)−1 (item 2), then(𝛼󸀠_𝑔∘

𝛼󸀠

ℎ)(𝐴) = {𝛼𝑔ℎ(𝑎) : 𝑎 ∈ 𝐴} = 𝛼󸀠𝑔ℎ(𝐴). In conclusion,(𝛼𝑔󸀠 ∘

𝛼󸀠

ℎ)(𝐴) = 𝛼󸀠𝑔ℎ(𝐴)for all𝐴 ∈ 𝛼ℎ󸀠−1(𝑆ℎ∩ 𝑆𝑔−1).

Finally, for all𝐴, 𝐵 ∈ 𝑃(𝑋_𝑔−1), we have that𝛼󸀠_𝑔(𝐴Δ𝐵) =

𝛼󸀠

𝑔(𝐴)Δ𝛼𝑔󸀠(𝐵)and𝛼𝑔󸀠(𝐴 ∩ 𝐵) = 𝛼𝑔󸀠(𝐴) ∩ 𝛼𝑔󸀠(𝐵), because each

𝛼󸀠_𝑔,𝑔 ∈ 𝐺, is a bijection. Therefore,𝐺acts partially on the ring

(𝑃(𝑋), Δ, ∩).

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

In the previous proposition, note that each ideal 𝑆_𝑔 =

𝑃(𝑋_𝑔), 𝑔 ∈ 𝐺, has the identity element 𝑋_𝑔. 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. Let𝛼be a partial action of𝐺on the nonempty

set𝑋. The following statements hold.

(1)𝑃(𝑋)is an ideal of𝑃(𝑇).

(2)𝐸 = ∑_𝑔∈𝐺𝛽_𝑔(𝑃(𝑋))is a𝛽-invariant ideal of𝑃(𝑇).

(3)The enveloping action of(𝑃(𝑋), 𝛼)is(𝐸, 𝛽), where each

𝛽_𝑔,𝑔 ∈ 𝐺, acts on𝐸by restriction.

Proof. (1) It is a direct consequence of the inclusion𝑋 ⊆ 𝑇. (2) Since 𝑃(𝑋) is an ideal of 𝑃(𝑇), we have that 𝐸 =

∑_𝑔∈𝐺𝛽_𝑔(𝑃(𝑋))is an ideal of𝑃(𝑇), and it is clear that𝐸is𝛽 -invariant.

(3) We must prove 1, 2, and 3 ofDefinition 2. Note that by item 2, the action𝛽on𝐸is global. Moreover, we can identify

𝑃(𝑋)with𝜑(𝑃(𝑋))because𝑃(𝑋)is an ideal of𝐸. The item 3 is consequence of 2.

To prove 2, let𝐴 ∈ 𝑃(𝑆_𝑔−1). Then,𝛼_𝑔󸀠(𝐴) = {𝛼_𝑔(𝑥) | 𝑥 ∈

𝐴}. Since𝐴 ⊆ 𝑆_𝑔−1 ⊆ 𝑋and(𝑇, 𝛽)is the enveloping action of

(𝑋, 𝛼), we have that𝛼_𝑔(𝑥) = 𝛽_𝑔(𝑥)for all𝑥 ∈ 𝐴and all𝑔 ∈ 𝐺. Thus,𝛼_𝑔󸀠(𝐴) = {𝛽_𝑔(𝑥)𝑥 ∈ 𝐴} = 𝛽_𝑔(𝐴), and we conclude that

𝛼󸀠

𝑔(𝐴) = 𝛽𝑔(𝐴)for all𝐴 ∈ 𝑃(𝑆𝑔−1).

To prove 1, let𝐴 ∈ 𝑃(𝑆_𝑔). Then,𝛽_𝑔−1(𝐴) = 𝛼󸀠_𝑔−1(𝐴)and

thus𝐴 = 𝛽_𝑔(𝛼󸀠_𝑔−1(𝐴)) ∈ 𝛽_𝑔(𝑃(𝑋)). Hence,𝑃(𝑆_𝑔) ⊆ 𝑃(𝑋) ∩

𝛽_𝑔(𝑃(𝑋))for all𝑔 ∈ 𝐺.

For the other inclusion, let𝐴, 𝐵 ∈ 𝑃(𝑋)such that𝐴 =

𝛽_𝑔(𝐵). Then, 𝐴 = 𝑋 ∩ 𝛽_𝑔(𝑋 ∩ 𝐵) = 𝑋 ∩ 𝛽_𝑔(𝑋) ∩ 𝛽_𝑔(𝐵). Since(𝑇, 𝛽)is the enveloping action of(𝑋, 𝛼), we have𝑆_𝑔 =

𝑋∩𝛽_𝑔(𝑋)for all𝑔 ∈ 𝐺. Hence,𝐴 = 𝑆_𝑔∩𝛽_𝑔(𝐵) ⊆ 𝑆_𝑔and thus

𝐴 ∈ 𝑃(𝑆_𝑔).

We conclude that (𝐸, 𝛽) is the enveloping action of

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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 case𝐸 = 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 actionof finite type.

Definition 6. Let(𝑋, 𝛼)be a partial action of𝐺on the set𝑋

with enveloping action(𝑇, 𝛽).(𝑋, 𝛼)is said to be of finite type if there exist𝑔₁, . . . , 𝑔_𝑛∈ 𝐺such that𝑇 = ⋃𝑛_𝑖=1𝛽_𝑔_𝑖(𝑋).

A partial action𝛼󸀠 = {𝐷_𝑔, 𝛼󸀠_𝑔}_𝑔∈𝐺of𝐺on the ring𝑅is calledof finite type[5, Definition 1.1] if there exists a finite subset{𝑔₁, . . . , 𝑔_𝑛}of𝐺, such that,∑𝑛_𝑖=1𝐷_𝑔𝑔_𝑖 = 𝑅for any𝑔 ∈

𝐺. If the partial action has an enveloping action, then it can be characterized as follows [5, Proposition 1.2].

Proposition 7. Let𝛼󸀠be a partial action of𝐺on the ring𝑅

with enveloping action(𝐸󸀠, 𝛽󸀠). The following statements are equivalent.

(1)𝛼󸀠is of finite type.

(2)There exist𝑔₁, . . . , 𝑔_𝑛 ∈ 𝐺such that𝐸󸀠= ∑𝑛_𝑖=1𝛽󸀠_𝑔

𝑖(𝑅).

(3)𝐸󸀠has an identity element.

The following theorem is the main result of this work. Without loss of generality, we can assume that𝑔₁ = 1 in

Definition 6andProposition 7. First, we prove the following specialization of [5, Proposition 1.10].

Proposition 8. Under the previous assumptions, if 𝑆 =

∑𝑛_𝑖=1𝛽_𝑔𝑖(𝑃(𝑋))with𝑔₁= 1, . . . , 𝑔_𝑛 ∈ 𝐺, then1_𝑆= ⋃𝑛_𝑖=1𝛽_𝑔𝑖(𝑋) is the identity element of𝑆.

Proof. By induction on𝑛, it is enough to consider the case

with two summands, that is,𝑆 = 𝑃(𝑋) + 𝛽_𝑔₂(𝑃(𝑋)). Then,

1_𝑆 = 1_𝑃(𝑋)+ 1_𝛽

𝑔2(𝑃(𝑋))− 1𝑃(𝑋)1𝛽𝑔2(𝑃(𝑋)). Since the addition is

the symmetric difference and the product is the intersection, we obtain that1_𝑆= 𝑋 Δ 𝛽_𝑔2(𝑋) Δ (𝑋∩𝛽_𝑔2(𝑋)) = 𝑋∪𝛽_𝑔2(𝑋).

Theorem 9. Let (𝑋, 𝛼) be a partial action of 𝐺 on the set

𝑋with enveloping action(𝑇, 𝛽). The following statements are

equivalent.

(1)(𝑋, 𝛼)is of finite type.

(2)𝐸 = 𝑃(𝑇).

(3)(𝑃(𝑋), 𝛼)is of finite type.

Proof. 1 ⇒ 2. Suppose that there exist𝑔₁ = 1, . . . , 𝑔_𝑛 ∈ 𝐺 such that𝑇 = ⋃𝑛_𝑖=1𝛽_𝑔𝑖(𝑋). ByProposition 8, the identity element of the ring 𝑆 = ∑𝑛_𝑖=1𝛽_𝑔𝑖(𝑃(𝑋)) ⊆ 𝐸 is 1_𝑆 =

⋃𝑛_𝑖=1𝛽𝑔𝑖(𝑋) = 𝑇. So,𝑇 ∈ 𝐸, and since𝐸is an ideal of𝑃(𝑇),

we conclude that𝐸 = 𝑃(𝑇).

2 ⇒ 3. If𝐸 = 𝑃(𝑇), then𝑇 ∈ 𝐸. So,𝐸is a ring with identity, and byProposition 7the result follows.

3 ⇒ 1. If(𝑃(𝑋), 𝛼) is of finite type, then there exist

𝑔₁ = 1, . . . , 𝑔_𝑛 ∈ 𝐺such that𝐸 = ∑𝑛_𝑖=1𝛽_𝑔𝑖(𝑃(𝑋)). Thus, for each𝑔 ∈ 𝐺, there exist𝐴𝑔₁, . . . , 𝐴𝑔_𝑛 ⊆ 𝑋such that𝛽_𝑔(𝑋) =

∑𝑛_𝑖=1𝛽_𝑔𝑖(𝐴𝑔_𝑖), which implies that𝛽_𝑔(𝑋) ⊆ ⋃𝑛_𝑖=1𝛽_𝑔𝑖(𝑋) for each 𝑔 ∈ 𝐺. Hence, 𝑇 = ⋃_𝑔∈𝐺𝛽_𝑔(𝑋) ⊆ ⋃𝑛_𝑖=1𝛽_𝑔𝑖(𝑋). In conclusion,(𝑋, 𝛼)is of finite type.

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

Example 10. Let𝑋be the set of even integers and𝐺the group

Z. We define a partial action𝛼of𝐺on𝑋as follows:𝑆_𝑛 = 𝑋 if𝑛is even and𝑆_𝑛 = 0if𝑛is odd; for𝑛, an even integer𝛼_𝑛 :

𝑆_−𝑛 → 𝑆_𝑛is defined by𝛼_𝑛(𝑘) = 𝑘 + 𝑛for all𝑘 ∈ 𝑆_−𝑛, and for

𝑛, an odd integer𝛼_𝑛is the empty function. The enveloping action of(𝑋, 𝛼)is (𝑇, 𝛽)where 𝑇 = Zand𝛽is the action ofZon𝑇, defined by𝛽_𝑛(𝑘) = 𝑘 + 𝑛for all𝑘 ∈ 𝑇. Since

𝑋 ∈ 𝛽₀(𝑃(𝑋))and the set of odd integers𝑌 ∈ 𝛽₁(𝑃(𝑋)), we have𝑋 + 𝑌 = 𝑇 ∈ 𝐸. Hence,𝐸 = 𝑃(𝑇), and thus(𝑃(𝑇), 𝛽)is the enveloping action of(𝑃(𝑋), 𝛼).

Example 11. Let𝑋 = {𝑛₀}where𝑛₀is a fixed integer and𝐺is the groupZ. We define a partial action𝛼of𝐺on𝑋as follows:

𝑆₀= 𝑋and𝑆_𝑛 = 0for𝑛 ̸= 0;𝛼₀: 𝑆₀ → 𝑆₀is the identity and

𝛼_𝑛is the empty function in other case. The enveloping action of(𝑋, 𝛼)is(𝑇, 𝛽), that of the previous example.

Note that each singleton of𝑇is an element of𝛽_𝑛(𝑃(𝑋)) for some integer𝑛. So,𝐸 = ∑_𝑛∈Z𝛽_𝑔(𝑃(𝑋))the enveloping action of(𝑃(𝑋), 𝛼)coincides with the collection of all finite subsets of𝑇. Hence,𝑇 ∉ 𝐸because𝑇is an infinite set, and we conclude that𝐸 ⫋ 𝑃(𝑇).

In [5] it was proved that if𝑅is a ring and𝛼is a partial action of a group𝐺on𝑅with enveloping action(𝑇, 𝛽), then

𝑇is right (left) Noetherian (Artinian), if and only if𝑅is right (left) Noetherian (Artinian) and𝛼is of finite type (Corollary 1.3). Under the same assumptions, they also proved that𝑇is semisimple if and only if𝑅is semisimple and𝛼is 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

follow-ing statements are equivalent.

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

(2)The ring 𝑃(𝑋)is Noetherian (Artinian, semisimple),

and(𝑃(𝑋), 𝛼)is of finite type.

(3)The ring𝐸is Noetherian (Artinian, semisimple).

Proof. It is enough to observe that𝑋 is finite if and only if

the ring𝑃(𝑋)is noetherian (Artinian, semisimple) and apply

Theorem 9.

Acknowledgments

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4 International Journal of Mathematics and Mathematical Sciences

References

[1] R. Exel, “Partial actions of groups and actions of inverse semigroups,”Proceedings of the American Mathematical Society, vol. 126, no. 12, pp. 3481–3494, 1998.

[2] F. Abadie, “Enveloping actions and Takai duality for partial actions,”Journal of Functional Analysis, vol. 197, no. 1, pp. 14– 67, 2003.

[3] M. Dokuchaev and R. Exel, “Associativity of crossed products by partial actions, enveloping actions and partial representations,”

Transactions of the American Mathematical Society, vol. 357, no.

5, pp. 1931–1952, 2005.

[4] J. ´Avila, M. Ferrero, and J. Lazzarin, “Partial actions and partial fixed rings,”Communications in Algebra, vol. 38, no. 6, pp. 2079– 2091, 2010.

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