Partial Actions and Power Sets

Abstract

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 [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).

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 𝑆_𝑔, 𝑔∈𝐺, of 𝑋 and bijections 𝛼_𝑔∶𝑆_𝑔−1→𝑆_𝑔 such that for all 𝑔,ℎ∈𝐺, the following statements hold.(1)𝑆₁=𝑋 and 𝛼₁ is the identity of 𝑋. (2)𝑆_(𝑔ℎ)⁻¹⊇𝛼_ℎ⁻¹(𝑆_ℎ∩𝑆_𝑔−1). (3)𝛼_𝑔∘𝛼_ℎ(𝑥)=𝛼_𝑔ℎ(𝑥), for all 𝑥∈𝛼_ℎ⁻¹(𝑆_ℎ∩𝑆_𝑔−1).

The partial action 𝛼 will be denoted by (𝑋,𝛼) or 𝛼={𝑆_𝑔,𝛼_𝑔}_𝑔∈𝐺. 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 𝛾_𝑔∶𝑌→𝑌 and let 𝑋 be a subset of 𝑌. Set 𝑆_𝑔=𝑋∩𝛾_𝑔(𝑋) and let 𝛼_𝑔 be the restriction of 𝛾_𝑔 to 𝑆_𝑔−1, for each 𝑔∈𝐺. Then, it is easy to see that 𝛼={𝑆_𝑔,𝛼_𝑔}_𝑔∈𝐺 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 𝑆_𝑔, 𝑔∈𝐺, is an ideal of 𝑅 and that every map 𝛼_𝑔∶𝑆_𝑔−1→𝑆_𝑔 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 𝑔∈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 all 𝑥∈𝑆_𝑔−1. (3)𝐸 is 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. 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 𝑆_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 set 𝑋 and a partial action 𝛼={𝑋_𝑔,𝛼_𝑔}_𝑔∈𝐺 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={𝛽_𝑔∣𝑔∈𝐺} of 𝐺 on 𝑇, where each 𝛽_𝑔 is a bijection of 𝑇, such that the partial action 𝛼 is given by restriction. Thus, we can assume that 𝑋⊆𝑇, 𝑇 is the orbit of 𝑋, 𝑆_𝑔=𝑋∩𝛽_𝑔(𝑋) for each 𝑔∈𝐺 and 𝛼_𝑔(𝑥)=𝛽_𝑔(𝑥) for all 𝑔∈𝐺 and all 𝑥∈𝑆_𝑔−1.

The action 𝛽 on 𝑇 can be extended to an action on 𝑃(𝑇). Moreover, since 𝛽_𝑔, 𝑔∈𝐺, is a bijective function, we have that 𝛽_𝑔(𝐴Δ𝐵)=𝛽_𝑔(𝐴)Δ𝛽_𝑔(𝐵) and 𝛽_𝑔(𝐴∩𝐵)=𝛽_𝑔(𝐴)∩𝛽_𝑔(𝐵) for all 𝐴,𝐵∈𝑃(𝑇) and all 𝑔∈𝐺. 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={𝑋_𝑔,𝛼_𝑔}_𝑔∈𝐺 a partial action of 𝐺 on 𝑋 and consider the collection 𝛼^′={𝑆_𝑔,𝛼^′_𝑔}_𝑔∈𝐺, where 𝑆_𝑔=𝑃(𝑋_𝑔), 𝑔∈𝐺, and 𝛼^′_𝑔∶𝑆_𝑔−1→𝑆_𝑔 is defined by 𝛼^′_𝑔(𝐴)={𝛼_𝑔(𝑎)∶𝑎∈𝐴} for all 𝑔∈𝐺 and all 𝐴∈𝑃(𝑋_𝑔). It is clear that 𝛼^′_𝑔, 𝑔∈𝐺, 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 𝐴∈𝛼_ℎ^′−1(𝑆_ℎ∩𝑆_𝑔−1), then 𝛼^′_ℎ(𝐴)∈𝑃(𝑋_ℎ∩𝑋_𝑔−1). Thus, 𝑎∈𝛼_ℎ⁻¹(𝑋_ℎ∩𝑋_𝑔−1)⊆𝑋_(𝑔ℎ)⁻¹ for each 𝑎∈𝐴. Hence, 𝐴∈𝑃(𝑋_(𝑔ℎ)⁻¹)=𝑆_(𝑔ℎ)⁻¹, and we conclude that 𝛼_ℎ^′−1(𝑆_ℎ∩𝑆_𝑔−1)⊆𝑆_(𝑔ℎ)⁻¹.
(3) For all 𝐴∈𝛼_ℎ^′−1(𝑆_ℎ∩𝑆_𝑔−1), we have that (𝛼^′_𝑔∘𝛼^′_ℎ)(𝐴)={(𝛼_𝑔∘𝛼_ℎ)(𝑎)∶𝑎∈𝐴}. Since 𝐴∈𝑆_(𝑔ℎ)⁻¹ (item 2), then (𝛼^′_𝑔∘𝛼^′_ℎ)(𝐴)={𝛼_𝑔ℎ(𝑎)∶𝑎∈𝐴}=𝛼^′_𝑔ℎ(𝐴). In conclusion, (𝛼^′_𝑔∘𝛼^′_ℎ)(𝐴)=𝛼^′_𝑔ℎ(𝐴) for all 𝐴∈𝛼_ℎ^′−1(𝑆_ℎ∩𝑆_𝑔−1).
Finally, for all 𝐴,𝐵∈𝑃(𝑋_𝑔−1), we have that 𝛼^′_𝑔(𝐴Δ𝐵)=𝛼^′_𝑔(𝐴)Δ𝛼^′_𝑔(𝐵) and 𝛼^′_𝑔(𝐴∩𝐵)=𝛼^′_𝑔(𝐴)∩𝛼^′_𝑔(𝐵), because each 𝛼^′_𝑔, 𝑔∈𝐺, 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 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. Let 𝛼 be a partial action of 𝐺 on the nonempty set 𝑋. The following statements hold. (1)𝑃(𝑋)  is an ideal of 𝑃(𝑇). (2)𝐸=∑_𝑔∈𝐺𝛽_𝑔(𝑃(𝑋))  is a 𝛽-invariant ideal of 𝑃(𝑇). (3)The enveloping action of  (𝑃(𝑋),𝛼)  is  (𝐸,𝛽), where each  𝛽_𝑔, 𝑔∈𝐺,acts on  𝐸  by restriction.

Proof. (1) It is a direct consequence of the inclusion 𝑋⊆𝑇.
(2) Since 𝑃(𝑋) is an ideal of 𝑃(𝑇), we have that 𝐸=∑_𝑔∈𝐺𝛽_𝑔(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 𝐴∈𝑃(𝑆_𝑔−1). Then, 𝛼^′_𝑔(𝐴)={𝛼_𝑔(𝑥)∣𝑥∈𝐴}. Since 𝐴⊆𝑆_𝑔−1⊆𝑋 and (𝑇,𝛽) is the enveloping action of (𝑋,𝛼), we have that 𝛼_𝑔(𝑥)=𝛽_𝑔(𝑥) for all 𝑥∈𝐴 and all 𝑔∈𝐺. Thus, 𝛼^′_𝑔(𝐴)={𝛽_𝑔(𝑥)𝑥∈𝐴}=𝛽_𝑔(𝐴), and we conclude that 𝛼^′_𝑔(𝐴)=𝛽_𝑔(𝐴) for all 𝐴∈𝑃(𝑆_𝑔−1).
To prove 1, let 𝐴∈𝑃(𝑆_𝑔). Then, 𝛽_𝑔−1(𝐴)=𝛼^′_𝑔−1(𝐴) and thus 𝐴=𝛽_𝑔(𝛼^′_𝑔−1(𝐴))∈𝛽_𝑔(𝑃(𝑋)). Hence, 𝑃(𝑆_𝑔)⊆𝑃(𝑋)∩𝛽_𝑔(𝑃(𝑋)) for all 𝑔∈𝐺.
For the other inclusion, let 𝐴,𝐵∈𝑃(𝑋) such that 𝐴=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 have 𝑆_𝑔=𝑋∩𝛽_𝑔(𝑋) for all 𝑔∈𝐺. Hence, 𝐴=𝑆_𝑔∩𝛽_𝑔(𝐵)⊆𝑆_𝑔 and thus 𝐴∈𝑃(𝑆_𝑔).
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,…,𝑔_𝑛∈𝐺 such that 𝑇=⋃^𝑛_𝑖=1𝛽_𝑔𝑖(𝑋).

A partial action 𝛼^′={𝐷_𝑔,𝛼^′_𝑔}_𝑔∈𝐺 of 𝐺 on the ring 𝑅 is called of finite type [5, Definition 1.1] if there exists a finite subset {𝑔₁,…,𝑔_𝑛} of 𝐺, such that, ∑^𝑛_𝑖=1𝐷_𝑔𝑔𝑖=𝑅 for any 𝑔∈𝐺. 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. (1)𝛼^′is of finite type. (2)There exist 𝑔₁,…,𝑔_𝑛∈𝐺 such that 𝐸^′=∑^𝑛_𝑖=1𝛽^′_𝑔𝑖(𝑅). (3)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=∑^𝑛_𝑖=1𝛽_𝑔𝑖(𝑃(𝑋)) with 𝑔₁=1,…,𝑔_𝑛∈𝐺, then 1_𝑆=⋃^𝑛_𝑖=1𝛽_𝑔𝑖(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)+𝛽_𝑔2(𝑃(𝑋)). Then, 1_𝑆=1_𝑃(𝑋)+1_{𝛽𝑔2(𝑃(𝑋))}−1_𝑃(𝑋)1_{𝛽𝑔2(𝑃(𝑋))}. Since the addition is the symmetric difference and the product is the intersection, we obtain that 1_𝑆=𝑋Δ𝛽_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 𝑋 with 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 exist 𝑔₁=1,…,𝑔_𝑛∈𝐺 such that 𝑇=⋃^𝑛_𝑖=1𝛽_𝑔𝑖(𝑋). By Proposition 8, the identity element of the ring 𝑆=∑^𝑛_𝑖=1𝛽_𝑔𝑖(𝑃(𝑋))⊆𝐸 is 1_𝑆=⋃^𝑛_𝑖=1𝛽_𝑔𝑖(𝑋)=𝑇. So, 𝑇∈𝐸, and since 𝐸 is an ideal of 𝑃(𝑇), we conclude that 𝐸=𝑃(𝑇).
2⇒3. If 𝐸=𝑃(𝑇), then 𝑇∈𝐸. 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,…,𝑔_𝑛∈𝐺 such that 𝐸=∑^𝑛_𝑖=1𝛽_𝑔𝑖(𝑃(𝑋)). Thus, for each 𝑔∈𝐺, there exist 𝐴^𝑔₁,…,𝐴^𝑔_𝑛⊆𝑋 such that 𝛽_𝑔(𝑋)=∑^𝑛_𝑖=1𝛽_𝑔𝑖(𝐴^𝑔_𝑖), which implies that 𝛽_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. Let 𝑋 be the set of even integers and 𝐺 the group ℤ. We define a partial action 𝛼 of 𝐺 on 𝑋 as follows: 𝑆_𝑛=𝑋 if 𝑛 is even and 𝑆_𝑛=∅ if 𝑛 is odd; for 𝑛, an even integer 𝛼_𝑛∶𝑆_−𝑛→𝑆_𝑛 is defined by 𝛼_𝑛(𝑘)=𝑘+𝑛 for all 𝑘∈𝑆_−𝑛, and for 𝑛, an odd integer 𝛼_𝑛 is the empty function. The enveloping action of (𝑋,𝛼) is (𝑇,𝛽) where 𝑇=ℤ and 𝛽 is the action of ℤ on 𝑇, defined by 𝛽_𝑛(𝑘)=𝑘+𝑛 for all 𝑘∈𝑇. Since 𝑋∈𝛽₀(𝑃(𝑋)) and the set of odd integers 𝑌∈𝛽₁(𝑃(𝑋)), we have 𝑋+𝑌=𝑇∈𝐸. 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 ℤ. We define a partial action 𝛼 of 𝐺 on 𝑋 as follows: 𝑆₀=𝑋 and 𝑆_𝑛=∅ for 𝑛≠0; 𝛼₀∶𝑆₀→𝑆₀ is the identity and 𝛼_𝑛 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 𝛽_𝑛(𝑃(𝑋)) for some integer 𝑛. So, 𝐸=∑_𝑛∈ℤ𝛽_𝑔(𝑃(𝑋)) the enveloping action of (𝑃(𝑋),𝛼) coincides with the collection of all finite subsets of 𝑇. Hence, 𝑇∉𝐸 because 𝑇 is an infinite set, and we conclude that 𝐸⫋𝑃(𝑇).
In [5] 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. (1)𝑋 is 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 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.”