Abstract

We introduce the concepts of set-valued homomorphism and strong set-valued homomorphism of a quantale which are the extended notions of congruence and complete congruence, respectively. The properties of generalized lower and upper approximations, constructed by a set-valued mapping, are discussed.

1. Introduction

The concept of Rough set was introduced by Pawlak [1] as a mathematical tool for dealing with vagueness or uncertainty. In Pawlak’s rough sets, the equivalence classes are the building blocks for the construction of the lower and upper approximations. It soon invoked a natural question concerning a possible connection between rough sets and algebraic systems. Biswas and Nanda [2] introduced the notion of rough subgroups. Kuroki [3] and Qimei [4] introduced the notions of a rough ideal and a rough prime ideal in a semigroup, respectively. Davvaz in [5] introduced the notion of rough subring with respect to an ideal of a ring. Rough modules have been investigated by Davvaz and Mahdavipour [6]. Rasouli and Davvaz studied the roughness in MV-algebra [7]. In [8–12], the roughness of various hyperstructures are discussed. Further, some authors consider the rough set in a fuzzy algebraic system, see [13–16]. The concept of quantale was introduced by Mulvey [17] in 1986 with the purpose of studying the spectrum of C*-algebra, as well as constructive foundations for quantum mechanics. There are abundant contents in the structure of quantales, because quantale can be regarded as the generalization of frame. Since quantale theory provides a powerful tool in studying noncommutative structures, it has wide applications, especially in studying noncommutative C*-algebra theory, the ideal theory of commutative ring, linear logic, and so on. The quantale theory has aroused great interests of many researchers, and a great deal of new ideas and applications of quantale have been proposed in twenty years [18–24].

The majority of studies on rough sets for algebraic structures such as semigroups, groups, rings, and modules have been concentrated on a congruence relation. An equivalence relation is sometimes difficult to be obtained in real-world problems due to the vagueness and incompleteness of human knowledge. From this point of view, Davvaz [25] introduced the concept of set-valued homomorphism for groups. And then, Yamak et al. [26, 27] introduced the concepts of set-valued homomorphism and strong set-valued homomorphism of a ring and a module. In this paper, the concepts of set-valued homomorphism and strong set-valued homomorphism in quantales are introduced. We discuss the properties of generalized lower and upper approximations in quantales.

2. Preliminaries

In this section, we give some basic notions and results about quantales and rough set theory (see [19, 22, 25, 28]), which will be necessary in the next sections.

Definition 2.1. A quantale is a complete lattice 𝑄 with an associative binary operation ∘ satisfying π‘Žβˆ˜βŽ›βŽœβŽξ˜π‘–βˆˆπΌπ‘π‘–βŽžβŽŸβŽ =ξ˜π‘–βˆˆπΌξ€·π‘Žβˆ˜π‘π‘–ξ€Έ,βŽ›βŽœβŽξ˜π‘–βˆˆπΌπ‘Žπ‘–βŽžβŽŸβŽ βˆ˜π‘=ξ˜π‘–βˆˆπΌξ€·π‘Žπ‘–βˆ˜π‘ξ€Έ,(2.1) for all π‘Ž,𝑏,π‘Žπ‘–,π‘π‘–βˆˆπ‘„(π‘–βˆˆπΌ).
An element π‘’βˆˆπ‘„ is called a left (right) unit if and only if π‘’βˆ˜π‘Ž=π‘Ž (π‘Žβˆ˜π‘’=π‘Ž), 𝑒 is called a unit if it is both a right and left unit.
A quantale 𝑄 is called a commutative quantale if π‘Žβˆ˜π‘=π‘βˆ˜π‘Ž for all π‘Ž,π‘βˆˆπ‘„.
A quantale 𝑄 is called an idempotent quantale if π‘Žβˆ˜π‘Ž=π‘Ž for all π‘Žβˆˆπ‘„.
A subset 𝑆 of 𝑄 is called a subquantale of 𝑄 if it is closed under ∘ and arbitrary sups.
In a quantale 𝑄, we denote the top element of 𝑄 by 1 and the bottom by 0. For 𝐴,π΅βŠ†π‘„, we write 𝐴∘𝐡 to denote the set {π‘Žβˆ˜π‘βˆ£π‘Žβˆˆπ΄,π‘βˆˆπ΅}, 𝐴∨𝐡 to denote {π‘Žβˆ¨π‘βˆ£π‘Žβˆˆπ΄,π‘βˆˆπ΅} and β‹π‘–βˆˆπΌπ΄π‘–={β‹π‘–βˆˆπΌπ‘Žπ‘–βˆ£π‘Žπ‘–βˆˆπ΄π‘–}.

Definition 2.2. Let 𝑄 be a quantale, a subset βˆ…β‰ πΌβŠ†π‘„ is called a left (right) ideal of 𝑄 if(1)π‘Ž,π‘βˆˆπΌ implies π‘Žβˆ¨π‘βˆˆπΌ,(2)π‘ŽβˆˆπΌ,π‘βˆˆπ‘„ and π‘β‰€π‘Ž imply π‘βˆˆπΌ for all,(3)π‘Žβˆˆπ‘„ and π‘₯∈𝐼 imply π‘Žβˆ˜π‘₯∈𝐼 (π‘₯βˆ˜π‘ŽβˆˆπΌ).
A subset πΌβŠ†π‘„ is called an ideal if it is both a left and a right ideal.
Let 𝑋 be a subset of 𝑄, we write ↓𝑋={π‘¦βˆˆπ‘„βˆ£π‘¦β‰€π‘₯ for some π‘₯βˆˆπ‘‹}, 𝑋 is a lower set if and only if 𝑋=↓𝑋. It is obvious that an ideal 𝐼 is a directed lower set. For every (left, right) ideal 𝐼 of 𝑄, it is easy to see that 0∈𝐼.
An ideal of 𝑄 is called a prime ideal if π‘Žβˆ˜π‘βˆˆπΌ implies π‘ŽβˆˆπΌ or π‘βˆˆπΌ for all π‘Ž,π‘βˆˆπ‘„.
An ideal 𝐼 of 𝑄 is called a semi-prime ideal if π‘Žβˆ˜π‘ŽβˆˆπΌ implies π‘ŽβˆˆπΌ for all π‘Žβˆˆπ‘„.
An ideal 𝐼 of 𝑄 (𝐼≠𝑄) is called a primary ideal if for all π‘Ž,π‘βˆˆπ‘„, π‘Žβˆ˜π‘βˆˆπΌ and π‘Žβˆ‰πΌ imply π‘π‘›βˆˆπΌ for some 𝑛>0. (𝑏𝑛=π‘βˆ˜π‘βˆ˜β‹―βˆ˜π‘ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘›).

Definition 2.3. A nonempty subset π‘€βŠ†π‘„ is called an π‘š-system of 𝑄, if for all π‘Ž,π‘βˆˆπ‘€, ↓(π‘Žβˆ˜1βˆ˜π‘)βˆ©π‘€β‰ βˆ….
A nonempty subset π‘†βŠ†π‘„ is called a multiplicative set of 𝑄, if π‘Žβˆ˜π‘βˆˆπ‘† for all π‘Ž,π‘βˆˆπ‘†.
Every ideal of 𝑄 is both an π‘š-system and a multiplicative set.

Definition 2.4. Let 𝑄 be a quantale, an equivalence relation πœƒ on 𝐿 is called a congruence on 𝑄 if for all π‘Ž,𝑏,𝑐,𝑑,π‘Žπ‘–,π‘π‘–βˆˆπ‘„(π‘–βˆˆπΌ), we have(1)π‘Žπœƒπ‘,π‘πœƒπ‘‘β‡’(π‘Žβˆ˜π‘)πœƒ(π‘βˆ˜π‘‘),(2)π‘Žπ‘–πœƒπ‘π‘–(π‘–βˆˆπΌ)β‡’(β‹π‘–βˆˆπΌπ‘Žπ‘–)πœƒ(β‹π‘–βˆˆπΌπ‘π‘–).It is obvious that [π‘Ž]πœƒβˆ˜[𝑏]πœƒβŠ†[π‘Žβˆ˜π‘]πœƒ, β‹π‘–βˆˆπΌ[π‘Žπ‘–]πœƒβŠ†[β‹π‘–βˆˆπΌπ‘Žπ‘–]πœƒ for all π‘Ž,𝑏,π‘Žπ‘–βˆˆπ‘„(π‘–βˆˆπΌ).

Definition 2.5. Let 𝑄 be a quantale, a congruence πœƒ on 𝑄 is called a complete congruence, if(1)[π‘Ž]πœƒβˆ˜[𝑏]πœƒ=[π‘Žβˆ˜π‘]πœƒ for all π‘Ž,π‘βˆˆπ‘„,(2)β‹π‘–βˆˆπΌ[π‘Žπ‘–]πœƒ=[β‹π‘–βˆˆπΌπ‘Žπ‘–]πœƒ for all π‘Žπ‘–βˆˆπ‘„(π‘–βˆˆπΌ).

Definition 2.6. Let (𝑄1,∘1) and (𝑄2,∘2) be two quantales. A map π‘“βˆΆπ‘„1→𝑄2 is said to be a homomorphism if(1)𝑓(π‘Žβˆ˜1𝑏)=𝑓(π‘Ž)∘2𝑓(𝑏) for all π‘Ž,π‘βˆˆπ‘„1;(2)𝑓(β‹π‘–βˆˆπΌπ‘Žπ‘–)=β‹π‘–βˆˆπΌπ‘“(π‘Žπ‘–) for all π‘Žπ‘–βˆˆπ‘„1(π‘–βˆˆπΌ).

Definition 2.7. Let π‘ˆ and π‘Š be two nonempty universes. Let 𝑇 be a set-valued mapping given by π‘‡βˆΆπ‘ˆβ†’π‘ƒ(π‘Š), where 𝑃(π‘Š) denotes the set of all subsets of π‘Š. Then the triple (π‘ˆ,π‘Š,𝑇) is referred to as a generalized approximation space. For any set π΄βŠ†π‘Š, the generalized lower and upper approximations, π‘‡βˆ’(𝐴) and π‘‡βˆ’(𝐴), are defined by π‘‡βˆ’(𝐴)={π‘₯βˆˆπ‘ˆβˆ£π‘‡(π‘₯)βŠ†π΄},π‘‡βˆ’(𝐴)={π‘₯βˆˆπ‘ˆβˆ£π‘‡(π‘₯)βˆ©π΄β‰ βˆ…}.(2.2)
The pair (π‘‡βˆ’(𝐴),π‘‡βˆ’(𝐴)) is referred to as a generalized rough set.

From the definition, the following theorems can be easily derived.

Theorem 2.8. Let π‘ˆ,π‘Š be nonempty universes and π‘‡βˆΆπ‘ˆβ†’π‘ƒβˆ—(π‘Š) be a set-valued mapping, where π‘ƒβˆ—(π‘Š) denotes the set of all nonempty subsets of π‘Š. If π΄βŠ†π‘Š, then π‘‡βˆ’(𝐴)βŠ†π‘‡βˆ’(𝐴).

If π‘Š=π‘ˆ and 𝑅𝑇={(π‘₯,𝑦)βˆ£π‘¦βˆˆπ‘‡(π‘₯)} is an equivalence relation on π‘ˆ, then the pair (π‘ˆ,𝑅𝑇) is the Pawlak approximation space. Therefore, a generalized rough set is an extended notion of Pawlak’s rough set.

Theorem 2.9. Let (π‘ˆ,π‘Š,𝑇) be a generalized approximation space, its lower and upper approximation operators satisfy the following properties. For all 𝐴,π΅βˆˆπ‘ƒ(π‘Š), (1)π‘‡βˆ’(𝐴)=(π‘‡βˆ’(𝐴𝑐))𝑐, π‘‡βˆ’(𝐴)=(π‘‡βˆ’(𝐴𝑐))𝑐,(2)π‘‡βˆ’(π‘Š)=π‘ˆ, π‘‡βˆ’(βˆ…)=βˆ…,(3)π‘‡βˆ’(𝐴∩𝐡)=π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡), π‘‡βˆ’(𝐴βˆͺ𝐡)=π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡),(4)π΄βŠ†π΅β‡’π‘‡βˆ’(𝐴)βŠ†π‘‡βˆ’(𝐡), π‘‡βˆ’(𝐴)βŠ†π‘‡βˆ’(𝐡),(5)π‘‡βˆ’(𝐴βˆͺ𝐡)βŠ‡π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡), π‘‡βˆ’(𝐴∩𝐡)βŠ†π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡),where 𝐴𝑐 is the complement of the set 𝐴.

3. Generalized Rough Subsets in Quantales

In this paper, (𝑄1,∘1) and (𝑄2,∘2) are two quantales.

Theorem 3.1. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued mapping and βˆ…β‰ π΄,π΅βŠ†π‘„2. Then(1)π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡), if 0∈𝐴∩𝐡,(2)π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡), if π‘’βˆˆπ΄βˆ©π΅,(3)π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡),(4)π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡), if 𝑄2 is an idempotent quantale,(5)π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡), if 0∈𝐴∩𝐡,(6)π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡), if π‘’βˆˆπ΄βˆ©π΅.

Proof. (1) Suppose that π‘Žβˆˆπ΄, we have π‘Ž=π‘Žβˆ¨0∈𝐴∨𝐡 for 0∈𝐡. So π΄βŠ†π΄βˆ¨π΅. Similarly, π΅βŠ†π΄βˆ¨π΅. So 𝐴βˆͺπ΅βŠ†π΄βˆ¨π΅. By Theorem 2.9, we have π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡)=π‘‡βˆ’(𝐴βˆͺ𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡).
(2) Suppose that π‘Žβˆˆπ΄, we have π‘Ž=π‘Žβˆ˜2π‘’βˆˆπ΄βˆ˜2𝐡 for π‘’βˆˆπ΅. So π΄βŠ†π΄βˆ˜2𝐡. Similarly, π΅βŠ†π΄βˆ˜2𝐡. So 𝐴βˆͺπ΅βŠ†π΄βˆ˜2𝐡. By Theorem 2.9, we have π‘‡βˆ’(𝐴)βˆͺπ‘‡βˆ’(𝐡)=π‘‡βˆ’(𝐴βˆͺ𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡).
(3) It is obvious that π΄βˆ©π΅βŠ†π΄βˆ¨π΅. By Theorem 2.9, we have π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡)=π‘‡βˆ’(𝐴∩𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡).
(4) Since 𝑄2 is an idempotent quantale, we have π΄βˆ©π΅βŠ†π΄βˆ˜π΅. By Theorem 2.9, we have π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡)=π‘‡βˆ’(𝐴∩𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡).
(5) and (6) The proofs are similar to (1) and (2), respectively.

Definition 3.2. A set-valued mapping π‘‡βˆΆπ‘„1→𝑃(𝑄2) is called a set-valued homomorphism if(1)𝑇(π‘Ž)∘2𝑇(𝑏)βŠ†π‘‡(π‘Žβˆ˜1𝑏) for all π‘Ž,π‘βˆˆπ‘„1,(2)β‹π‘–βˆˆπΌπ‘‡(π‘Žπ‘–)βŠ†π‘‡(β‹π‘–βˆˆπΌπ‘Žπ‘–) for all π‘Žπ‘–βˆˆπ‘„1(π‘–βˆˆπΌ).
𝑇 is called a strong set-valued homomorphism if the equalities in (1), (2) hold.

Example 3.3. (1) Let πœƒ be a congruence on 𝑄2. Then the set-valued mapping π‘‡βˆΆπ‘„1→𝑃(𝑄2) defined by 𝑇(π‘₯)=[π‘₯]πœƒ is a set-valued homomorphism but not necessarily a strong set-valued homomorphism. If πœƒ is complete, then 𝑇 is a strong set-valued homomorphism.
(2) Let 𝑓 be a quantale homomorphism from 𝑄1 to 𝑄2. Then the set-valued mapping π‘‡βˆΆπ‘„1→𝑃(𝑄2) defined by 𝑇(π‘Ž)={𝑓(π‘Ž)} is a strong set-valued homomorphism.

Theorem 3.4. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued homomorphism and βˆ…β‰ π΄,π΅βŠ†π‘„2. Then(1)π‘‡βˆ’(𝐴)βˆ¨π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡),(2)π‘‡βˆ’(𝐴)∘1π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡),(3)π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡),(4)π‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡), if 𝑄1 is an idempotent quantale.

Proof. (1) Suppose that π‘βˆˆπ‘‡βˆ’(𝐴)βˆ¨π‘‡βˆ’(𝐡), there exist π‘Žβˆˆπ‘‡βˆ’(𝐴),π‘βˆˆπ‘‡βˆ’(𝐡) such that 𝑐=π‘Žβˆ¨π‘. So there exist π‘₯βˆˆπ΄βˆ©π‘‡(π‘Ž) and π‘¦βˆˆπ΅βˆ©π‘‡(𝑏). Hence π‘₯βˆ¨π‘¦βˆˆπ΄βˆ¨π΅ and π‘₯βˆ¨π‘¦βˆˆπ‘‡(π‘Ž)βˆ¨π‘‡(𝑏). Since 𝑇 is a set-valued homomorphism, we have π‘₯βˆ¨π‘¦βˆˆπ‘‡(π‘Žβˆ¨π‘). Therefore, 𝑇(π‘Žβˆ¨π‘)∩(𝐴∨𝐡)β‰ βˆ… which implies that 𝑐=π‘Žβˆ¨π‘βˆˆπ‘‡βˆ’(𝐴∨𝐡).
(2) The proof is similar to (1).
(3) Suppose that π‘₯βˆˆπ‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡), there exist π‘Žβˆˆπ΄βˆ©π‘‡(π‘₯) and π‘βˆˆπ΅βˆ©π‘‡(π‘₯). Since 𝑇 is a set-valued homomorphism, we have π‘Žβˆ¨π‘βˆˆπ‘‡(π‘₯)βˆ¨π‘‡(π‘₯)βŠ†π‘‡(π‘₯). So π‘Žβˆ¨π‘βˆˆπ‘‡(π‘₯)∩(𝐴∨𝐡) which implies that π‘₯βˆˆπ‘‡βˆ’(𝐴∨𝐡).
(4) Suppose that π‘₯βˆˆπ‘‡βˆ’(𝐴)βˆ©π‘‡βˆ’(𝐡), there exist π‘Žβˆˆπ΄βˆ©π‘‡(π‘₯) and π‘βˆˆπ΅βˆ©π‘‡(π‘₯). Since 𝑇 is a set-valued homomorphism and 𝑄1 is idempotent, we have π‘Žβˆ¨π‘βˆˆπ‘‡(π‘₯)∘2𝑇(π‘₯)βŠ†π‘‡(π‘₯∘1π‘₯)=𝑇(π‘₯). So π‘Žβˆ¨π‘βˆˆπ‘‡(π‘₯)∩(𝐴∨𝐡) which implies that π‘₯βˆˆπ‘‡βˆ’(𝐴∘2𝐡).

Theorem 3.5. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and βˆ…β‰ π΄,π΅βŠ†π‘„2. Then(1)π‘‡βˆ’(𝐴)βˆ¨π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∨𝐡),(2)π‘‡βˆ’(𝐴)∘1π‘‡βˆ’(𝐡)βŠ†π‘‡βˆ’(𝐴∘2𝐡).

Proof. (1) Suppose that π‘βˆˆπ‘‡βˆ’(𝐴)βˆ¨π‘‡βˆ’(𝐡), there exist π‘Žβˆˆπ‘‡βˆ’(𝐴),π‘βˆˆπ‘‡βˆ’(𝐡) such that 𝑐=π‘Žβˆ¨π‘. Hence 𝑇(π‘Ž)βŠ†π΄ and 𝑇(𝑏)βŠ†π΅. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(π‘Žβˆ¨π‘)=𝑇(π‘Ž)βˆ¨π‘‡(𝑏)βŠ†π΄βˆ¨π΅ which implies that 𝑐=π‘Žβˆ¨π‘βˆˆπ‘‡βˆ’(𝐴∨𝐡).
(2) The proof is similar to (1).

4. Generalized Rough Ideals in Quantales

Theorem 4.1. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued mapping. If 𝐼 and 𝐽 are, respectively, a right and a left ideal of 𝑄2, then(1)π‘‡βˆ’(𝐼∘2𝐽)βŠ†π‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽),(2)π‘‡βˆ’(𝐼∘2𝐽)βŠ†π‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽),(3)π‘‡βˆ’(𝐼∧𝐽)βŠ†π‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽),(4)π‘‡βˆ’(𝐼)βˆ§π‘‡βˆ’(𝐽)=π‘‡βˆ’(𝐼∧𝐽),(5)π‘‡βˆ’(𝐼)βˆͺπ‘‡βˆ’(𝐽)βŠ†π‘‡βˆ’(𝐼∨𝐽),(6)π‘‡βˆ’(𝐼)βˆͺπ‘‡βˆ’(𝐽)βŠ†π‘‡βˆ’(𝐼∨𝐽).
If 𝑄1 is an idempotent quantale and 𝑇 is a set-valued homomorphism, then the equalities in (1)–(3) hold.

Proof. Since 𝐼 and 𝐽 are, respectively, a right and a left ideal of 𝑄2, we have 𝐼∘2π½βŠ†πΌβˆ©π½, 𝐼∧𝐽=𝐼∩𝐽 and π‘œβˆˆπΌβˆ©π½. By Theorem 2.9, we get the conclusion (1)–(4). By Theorem 3.1, we get (5) and (6).
If 𝑄1 is idempotent, we first show that π‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽)βŠ†π‘‡βˆ’(𝐼∘2𝐽). Suppose that π‘₯βˆˆπ‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽), there exist π‘¦βˆˆπΌ,π‘§βˆˆπ½ such that 𝑦,π‘§βˆˆπ‘‡(π‘₯). So, π‘¦βˆ˜2π‘§βˆˆπΌβˆ˜2𝐽 and π‘¦βˆ˜2π‘§βˆˆπ‘‡(π‘₯)∘2𝑇(π‘₯). Since T is a set-valued homomorphism and 𝑄1 is idempotent, we have π‘¦βˆ˜2π‘§βˆˆπ‘‡(π‘₯∘1π‘₯)=𝑇(π‘₯). Therefore, π‘₯βˆˆπ‘‡βˆ’(𝐼∘2𝐽). So the equality in (1) holds. Since 𝑄1 is idempotent, we have πΌβˆ©π½βŠ†πΌβˆ˜2𝐽. So 𝐼∩𝐽=𝐼∘2𝐽=𝐼∩𝐽. By Theorem 2.9, we get π‘‡βˆ’(𝐼∘2𝐽)=π‘‡βˆ’(𝐼∩𝐽)=π‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽). Since the equality in (1) holds, we have π‘‡βˆ’(𝐼∧𝐽)=π‘‡βˆ’(𝐼∘2𝐽)=π‘‡βˆ’(𝐼)βˆ©π‘‡βˆ’(𝐽).

Lemma 4.2. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued homomorphism. If 𝐴 is a lower set of 𝑄2, then π‘‡βˆ’(𝐴) is a lower set of 𝑄1.

Proof. Suppose π‘₯β‰€π‘¦βˆˆπ‘‡βˆ’(𝐴), then 𝑇(𝑦)βŠ†π΄. Let π‘§βˆˆπ‘‡(π‘₯),π‘Žβˆˆπ‘‡(𝑦), we have π‘§βˆ¨π‘Žβˆˆπ‘‡(π‘₯)βˆ¨π‘‡(𝑦)βŠ†π‘‡(π‘₯βˆ¨π‘¦)=𝑇(𝑦)βŠ†π΄. Since 𝐴 is a lower set, we have π‘§βˆˆπ΄. Therefore, 𝑇(π‘₯)βŠ†π΄ which implies that π‘₯βˆˆπ‘‡βˆ’(𝐴).

Lemma 4.3. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism. If 𝐴 is a lower set of 𝑄2, then π‘‡βˆ’(𝐴) is a lower set of 𝑄1.

Proof. Suppose that π‘₯β‰€π‘¦βˆˆπ‘‡βˆ’(𝐴), there exists π‘§βˆˆπ‘‡(𝑦)∩𝐴. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(π‘₯)βˆ¨π‘‡(𝑦)=𝑇(π‘₯βˆ¨π‘¦)=𝑇(𝑦). So there exist π‘Žβˆˆπ‘‡(π‘₯),π‘βˆˆπ‘‡(𝑦) such that 𝑧=π‘Žβˆ¨π‘. Since 𝐴 is a lower set, we have π‘Žβˆˆπ΄. Thus 𝑇(π‘₯)βˆ©π΄β‰ βˆ… which follows that π‘₯βˆˆπ‘‡βˆ’(𝐴).

Lemma 4.4. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued homomorphism and 𝐴 a nonempty subset of 𝑄2. If 𝐴 is closed under arbitrary (resp., finite) sups, then π‘‡βˆ’(𝐴) is closed under arbitrary (resp., finite) sups.

Proof. Let π΅βŠ†π‘‡βˆ’(𝐴). For each π‘βˆˆπ΅, we have π‘βˆˆπ‘‡βˆ’(𝐴), then there exist π‘₯π‘βˆˆπ‘‡(𝑏)∩𝐴. Since 𝑇 is a set-valued homomorphism, we have β‹π‘βˆˆπ΅π‘₯π‘βˆˆβ‹π‘βˆˆπ΅π‘‡(𝑏)βŠ†π‘‡(⋁𝐡). And we have β‹π‘βˆˆπ΅π‘₯π‘βˆˆπ΄ for A is closed under arbitrary sups. So 𝑇(⋁𝐡)βˆ©π΄β‰ βˆ… which implies that β‹π΅βˆˆπ‘‡βˆ’(𝐴).

Lemma 4.5. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐴 a nonempty subset of 𝑄2. If 𝐴 is closed under arbitrary (resp., finite) sups, then π‘‡βˆ’(𝐴) is closed under arbitrary (resp., finite) sups.

Proof. Let π΅βŠ†π‘‡βˆ’(𝐴). For each π‘βˆˆπ΅, we have 𝑇(𝑏)βŠ†π΄. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(⋁𝐡)=⋁𝑇(𝐡). Suppose π‘§βˆˆπ‘‡(⋁𝐡)=⋁𝑇(𝐡), there exist π‘₯π‘βˆˆπ‘‡(𝑏)βŠ†π΄ (π‘βˆˆπ΅) such that 𝑧=β‹π‘βˆˆπ΅π‘₯𝑏. Since 𝐴 is closed under arbitrary sups, we have 𝑧=β‹π‘βˆˆπ΅π‘₯π‘βˆˆπ΄. So 𝑇(⋁𝐡)βŠ†π΄ which implies that β‹π΅βˆˆπ‘‡βˆ’(𝐴).

Theorem 4.6. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued homomorphism. If 𝐴 is a subquantale of 𝑄2, then π‘‡βˆ’(𝐴) is a subquantale of 𝑄1.

Proof. Since 𝑇 is a set-valued homomorphism and 𝐴 is a subquantale, by Theorems 3.4 and 2.9, we have π‘‡βˆ’(𝐴)∘1π‘‡βˆ’(𝐴)βŠ†π‘‡βˆ’(𝐴∘2𝐴)βŠ†π‘‡βˆ’(𝐴). So, π‘‡βˆ’(𝐴) is closed under ∘1.
Since 𝐴 is closed under arbitrary sups, by Lemma 4.4, we get π‘‡βˆ’(𝐴) is closed under arbitrary sups.

Theorem 4.7. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism. If 𝐴 is a subquantale of 𝑄2, then π‘‡βˆ’(𝐴) is a subquantale of 𝑄1.

Proof. The proof is similar to Theorem 4.6.

Theorem 4.8. Let π‘‡βˆΆπ‘„1β†’π‘ƒβˆ—(𝑄2) be a strong set-valued homomorphism and 𝐼 a right (left) ideal of 𝑄2. Then π‘‡βˆ’(𝐼) is, if it is nonempty, a right (left) ideal of 𝑄1.

Proof. Suppose that π‘Ž,π‘βˆˆπ‘‡βˆ’(𝐼). Since 𝐼 is closed under finite sups, by Lemma 4.4, we have π‘‡βˆ’(𝐼) is closed under finite sups. So π‘Žβˆ¨π‘βˆˆπ‘‡βˆ’(𝐼).
Since 𝐼 is a lower set, by Lemma 4.3, we have π‘‡βˆ’(𝐼) is a lower set.
Suppose π‘Žβˆˆπ‘„1,π‘₯βˆˆπ‘‡βˆ’(𝐼), there exists π‘¦βˆˆπΌβˆ©π‘‡(π‘₯). Since 𝐼 is a right ideal of 𝑄2, we have π‘¦βˆ˜2π‘βˆˆπΌ for each π‘βˆˆπ‘‡(π‘Ž)βŠ†π‘„2. So π‘¦βˆ˜2π‘βˆˆπ‘‡(π‘₯)∘2𝑇(π‘Ž)βŠ†π‘‡(π‘₯∘1π‘Ž). Therefore, 𝑇(π‘₯∘1π‘Ž)βˆ©πΌβ‰ βˆ… which implies that π‘₯∘1π‘Žβˆˆπ‘‡βˆ’(𝐼).

Theorem 4.9. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a right (left) ideal of 𝑄2. Then π‘‡βˆ’(𝐼) is, if it is nonempty, a right (left) ideal of 𝑄1.

Proof. Suppose π‘Ž,π‘βˆˆπ‘‡βˆ’(𝐼). Since 𝐼 is closed under finite sups, by Lemma 4.5, we have π‘‡βˆ’(𝐼) is closed under finite sups. So, π‘Žβˆ¨π‘βˆˆπ‘‡βˆ’(𝐼).
Since 𝐼 is a lower set, by Lemma 4.2, we have π‘‡βˆ’(𝐼) is a lower set.
Suppose π‘Žβˆˆπ‘„1,π‘₯βˆˆπ‘‡βˆ’(𝐼), we have 𝑇(π‘₯)βŠ†πΌ. Let π‘¦βˆˆπ‘‡(π‘₯∘1π‘Ž). Since 𝑇 is a strong set-valued homomorphism, we have π‘¦βˆˆπ‘‡(π‘₯)∘2𝑇(π‘Ž), then there exist 𝑦1βˆˆπ‘‡(π‘₯)βŠ†πΌ, 𝑦2βˆˆπ‘‡(π‘Ž) such that 𝑦=𝑦1∘2𝑦2. Since 𝐼 is a right ideal, we have 𝑦=𝑦1∘2𝑦2∈𝐼. Therefore, 𝑇(π‘₯∘1π‘Ž)βŠ†πΌ which implies that π‘₯∘1π‘Žβˆˆπ‘‡βˆ’(𝐼).

Definition 4.10. A subset π΄βŠ†π‘„2 is called a generalized rough ideal (subquantale) of 𝑄1 if π‘‡βˆ’(𝐴) and π‘‡βˆ’(𝐴) are ideals (subquantales) of 𝑄1.
The following corollary follows from Theorems 4.6–4.9.

Corollary 4.11. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 an ideal (a subquantale) of 𝑄2. If π‘‡βˆ’(𝐼) and π‘‡βˆ’(𝐼) are nonempty, then 𝐼 is a generalized rough ideal (subquantale) of 𝑄1.

From the above, we know that an ideal is a generalized rough ideal with respect to a strong set-valued homomorphism. The following example shows that the converse does not hold in general.

Example 4.12. Let 𝑄1={0,π‘Ž,1} and 𝑄2={0ξ…ž,𝑏,𝑐,1ξ…ž} be quantales shown in Figures 1 and 2 and Tables 1 and 2.
Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism as defined by 𝑇(0)={0β€²},𝑇(π‘Ž)={𝑏,𝑐},𝑇(1)={1β€²}. Let 𝐴={0β€²,𝑏,𝑐}βŠ†π‘„2, 𝐡={𝑏,𝑐}βŠ†π‘„2, then π‘‡βˆ’(𝐴)={0,π‘Ž}=π‘‡βˆ’(𝐴) and π‘‡βˆ’(𝐡)={π‘Ž}=π‘‡βˆ’(𝐡). It is obvious that 𝐴 is a generalized rough ideal of 𝑄1 but 𝐴 is not an ideal of 𝑄2 and 𝐡 is a generalized rough subquantale of 𝑄1 but 𝐡 is not a subquantale of 𝑄2.

Table 1 
Table 2 

Figure 1 

Figure 2 

Theorem 4.13. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a prime ideal of 𝑄2. Then π‘‡βˆ’(𝐼) is, if it is nonempty, a prime ideal of 𝑄1.

Proof. By Theorem 4.8, we get π‘‡βˆ’(𝐼) is an ideal of 𝑄1.
Let π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝐼), there exists π‘₯βˆˆπΌβˆ©π‘‡(π‘Žβˆ˜1𝑏). Since 𝑇 is a strong set-valued homomorphism, we have π‘₯βˆˆπ‘‡(π‘Ž)∘2𝑇(𝑏), then there exist π‘¦βˆˆπ‘‡(π‘Ž),π‘§βˆˆπ‘‡(𝑏) such that π‘₯=π‘¦βˆ˜2π‘§βˆˆπΌ. Since 𝐼 is a prime ideal of 𝑄2, we have π‘¦βˆˆπΌ or π‘§βˆˆπΌ. So 𝑇(π‘Ž)βˆ©πΌβ‰ βˆ… or 𝑇(𝑏)βˆ©πΌβ‰ βˆ… which implies that π‘Žβˆˆπ‘‡βˆ’(𝐼) or π‘βˆˆπ‘‡βˆ’(𝐼).

Theorem 4.14. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a prime ideal of 𝑄2. Then π‘‡βˆ’(𝐼) is, if it is nonempty, a prime ideal of 𝑄1.

Proof. By Theorem 4.9, we get π‘‡βˆ’(𝐼) is an ideal of 𝑄1.
Let π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝐼), we have 𝑇(π‘Žβˆ˜1𝑏)βŠ†πΌ. Since 𝑇 is a strong set-valued homomorphism, we have 𝑇(π‘Ž)∘2𝑇(𝑏)βŠ†πΌ. We assume that π‘Žβˆ‰π‘‡βˆ’(𝐼), then 𝑇(π‘Ž)ξ„€βŠ†πΌ, there exists π‘₯βˆˆπ‘‡(π‘Ž) but π‘₯βˆ‰πΌ. If π‘¦βˆˆπ‘‡(𝑏), then π‘₯∘2π‘¦βˆˆπ‘‡(π‘Ž)∘2𝑇(𝑏)βŠ†πΌ. Since 𝐼 is a prime ideal of 𝑄2, we have π‘¦βˆˆπΌ. Therefore, 𝑇(𝑏)βŠ†πΌ which implies that π‘βˆˆπ‘‡βˆ’(𝐼).

We call πΌβŠ†π‘„2 is a generalized rough prime ideal of 𝑄1 if π‘‡βˆ’(𝐼) and π‘‡βˆ’(𝐼) are ideals of 𝑄1.

Theorem 4.15. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a semiprime ideal of 𝑄2. Then π‘‡βˆ’(𝐼) is, if it is nonempty, a semiprime ideal of 𝑄1.

Proof. By Theorem 4.9, we get π‘‡βˆ’(𝐼) is an ideal of 𝑄1.
Suppose that π‘Žβˆ˜1π‘Žβˆˆπ‘‡βˆ’(𝐼), we have 𝑇(π‘Žβˆ˜1π‘Ž)βŠ†πΌ. Let π‘₯βˆˆπ‘‡(π‘Ž), we have π‘₯∘2π‘₯βˆˆπ‘‡(π‘Ž)∘2𝑇(π‘Ž)βŠ†π‘‡(π‘Žβˆ˜1π‘Ž)βŠ†πΌ. Since 𝐼 is a semi-prime ideal, we have π‘₯∈𝐼. So 𝑇(π‘Ž)βŠ†πΌ which implies that π‘Žβˆˆπ‘‡βˆ’(𝐼).

Theorem 4.16. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a primary ideal of 𝑄2. Then π‘‡βˆ’(𝐼) is, if π‘‡βˆ’(𝐼)β‰ βˆ… and π‘‡βˆ’(𝐼)≠𝑄1, a primary ideal of 𝑄1.

Proof. By Theorem 4.8, we get π‘‡βˆ’(𝐼) is an ideal of 𝑄1.
Suppose that π‘Ž,π‘βˆˆπ‘„,π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝐼) and π‘Žβˆ‰π‘‡βˆ’(𝐼), there exists π‘₯βˆˆπΌβˆ©π‘‡(π‘Žβˆ˜1𝑏). Since 𝑇 is a strong set-valued homomorphism, we have π‘₯βˆˆπ‘‡(π‘Ž)∘2𝑇(𝑏), there exist π‘¦βˆˆπ‘‡(π‘Ž), π‘§βˆˆπ‘‡(𝑏) such that π‘₯=π‘¦βˆ˜2π‘§βˆˆπΌ. Since π‘Žβˆ‰π‘‡βˆ’(𝐼), we get π‘¦βˆ‰πΌ. Since 𝐼 is a primary ideal, we have π‘§π‘›βˆˆπΌ for some 𝑛>0 Since 𝑇 is a strong set-valued homomorphism, we have π‘§π‘›βˆˆπ‘‡(𝑏𝑛). So 𝑇(𝑏𝑛)βˆ©πΌβ‰ βˆ… which implies that π‘π‘›βˆˆπ‘‡βˆ’(𝐼).

Theorem 4.17. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a strong set-valued homomorphism and 𝐼 a primary ideal of 𝑄2. Let 𝑄2 be a commutative quantale and 𝑇(π‘₯) a finite set for each π‘₯βˆˆπ‘„1. Then π‘‡βˆ’(𝐼) is, if π‘‡βˆ’(𝐼)β‰ βˆ… and π‘‡βˆ’(𝐼)≠𝑄1, a primary ideal of 𝑄1.

Proof. By Theorem 4.9, we get π‘‡βˆ’(𝐼) is an ideal of 𝑄1.
Suppose π‘Ž,π‘βˆˆπ‘„,π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝐼) and π‘Žβˆ‰π‘‡βˆ’(𝐼), then 𝑇(π‘Žβˆ˜1𝑏)βŠ†πΌ and 𝑇(π‘Ž)ξ„€βŠ†πΌ. So there exists π‘₯βˆˆπ‘‡(π‘Ž) with π‘₯βˆ‰πΌ. We assume that 𝑇(𝑏)={𝑦1,𝑦2,…,π‘¦π‘š}, then π‘₯∘2π‘¦π‘–βˆˆπ‘‡(π‘Ž)∘2𝑇(𝑏)βŠ†π‘‡(π‘Žβˆ˜1𝑏)βŠ†πΌ(𝑖=1,2,…,π‘š). Since 𝐼 is a primary ideal, there exists π‘¦π‘›π‘–π‘–βˆˆπΌ for some 𝑛𝑖>0(𝑖=1,2,…,π‘š). Let 𝑛=max{π‘›π‘–βˆ£π‘–=1,2,…,π‘š}. Since 𝐼 is an ideal, we have π‘¦π‘™π‘–βˆˆπΌ(𝑙β‰₯𝑛). Let π‘§βˆˆπ‘‡(π‘π‘šπ‘π‘›)=𝑇(𝑏)π‘šπ‘›. Since 𝑄2 is commutative, there exists {𝑖1,𝑖2,…,𝑖𝑠}βŠ†{1,2,…,π‘š} such that 𝑧=𝑦𝑑1𝑖1∘2𝑦𝑑2𝑖2∘2β‹―βˆ˜2𝑦𝑑𝑠𝑖𝑠 with 𝑑1+𝑑2+β‹―+𝑑𝑠=π‘šπ‘›, where 𝑑𝑗>0(𝑗=1,2,…,𝑠). Assume that 𝑑𝑗<𝑛,(𝑗=1,2,…,𝑠), then 𝑑1+𝑑2+β‹―+𝑑𝑠<π‘ π‘›β‰€π‘šπ‘›. It contradicts with 𝑑1+𝑑2+β‹―+𝑑𝑠=π‘šπ‘›. Therefore, there is 1β‰€π‘˜β‰€π‘  such that π‘‘π‘˜β‰₯𝑛,(𝑗=1,2,…,𝑠), we have π‘¦π‘‘π‘˜π‘–π‘˜βˆˆπΌ. Since 𝐼 is an ideal, we get π‘§βˆˆπΌ. Hence 𝑇(π‘π‘šπ‘›)βŠ†πΌ which implies that π‘π‘šπ‘›βˆˆπ‘‡βˆ’(𝐼).

Theorem 4.18. Let π‘‡βˆΆπ‘„1→𝑃(𝑄2) be a set-valued homomorphism. If 𝑆 is a multiplicative set of 𝑄2, then π‘‡βˆ’(𝑆) is, if it is nonempty, a multiplicative of 𝑄1. If 𝑇 is a strong set-valued homomorphism, then π‘‡βˆ’(𝑆) is, if it is nonempty, a multiplicative of 𝑄1.

Proof. Suppose that π‘Ž,π‘βˆˆπ‘‡βˆ’(𝑆), there exist π‘₯βˆˆπ‘‡(π‘Ž)βˆ©π‘†, π‘¦βˆˆπ‘‡(𝑏)βˆ©π‘†. Hence π‘₯βˆ˜π‘¦βˆˆπ‘‡(π‘Ž)∘2𝑇(𝑏)βŠ†π‘‡(π‘Žβˆ˜1𝑏). Since 𝑆 is a multiplicative set, we have π‘₯∘2π‘¦βˆˆπ‘†. Therefore, π‘₯∘2π‘¦βˆˆπ‘‡(π‘Žβˆ˜1𝑏)βˆ©π‘† which implies that π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝑆).
Suppose π‘Ž,π‘βˆˆπ‘‡βˆ’(𝑆), then 𝑇(π‘Ž)βŠ†π‘†,𝑇(𝑏)βŠ†π‘†. Let π‘₯βˆˆπ‘‡(π‘Žβˆ˜1𝑏). Since 𝑇 is a strong set-valued homomorphism, we have π‘₯βˆˆπ‘‡(π‘Ž)∘2𝑇(𝑏). There exist π‘¦βˆˆπ‘‡(π‘Ž)βŠ†π‘†,π‘§βˆˆπ‘‡(𝑏)βŠ†π‘† such that π‘₯=π‘¦βˆ˜2𝑧. Since 𝑆 is a multiplicative set, we have π‘₯=π‘¦βˆ˜2π‘§βˆˆπ‘†. So 𝑇(π‘Žβˆ˜1𝑏)βŠ†π‘† which implies that π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝑆).

We call π΄βŠ†π‘„2 a generalized rough multiplicative set (π‘š-system) of 𝑄1, if π‘‡βˆ’(𝐴),π‘‡βˆ’(𝐴) are multiplicative sets (π‘š-systems) of 𝑄1.

Theorem 4.19. Let π‘‡βˆΆπ‘„1β†’π‘ƒβˆ—(𝑄2) be a set-valued homomorphism. If 𝑄1 is commutative and π΄βŠ†π‘„2 with 1βˆ‰π‘‡βˆ’(𝐴), then the following statements are equivalent:(1)𝐴 is a generalized rough prime ideal of 𝑄1,(2)𝐴 is a generalized rough ideal and 𝐴𝑐 is a generalized rough multiplicative set of 𝑄1,(3)𝐴 is a generalized rough ideal and 𝐴𝑐 is a generalized rough π‘š-system of 𝑄1.

Proof. (1)β‡’(2): Since 𝐴 is a generalized rough prime ideal of 𝑄1, we get 𝐴 is a generalized rough ideal of 𝑄1 and π‘‡βˆ’(𝐴), π‘‡βˆ’(𝐴) are prime ideals of 𝑄1. Now we show that π‘‡βˆ’(𝐴𝑐) is a multiplicative set. Let π‘Ž, π‘βˆˆπ‘‡βˆ’(𝐴𝑐). By Theorem 2.9(1), we have π‘Ž,π‘βˆˆ(π‘‡βˆ’(𝐴))𝑐. Since π‘‡βˆ’(𝐴) is a prime ideal, we have π‘Žβˆ˜1π‘βˆ‰π‘‡βˆ’(𝐴) which implies that π‘Žβˆ˜1π‘βˆˆ(π‘‡βˆ’(𝐴))𝑐=π‘‡βˆ’(𝐴𝑐). So π‘‡βˆ’(𝐴𝑐) is a multiplicative set. Similarly, π‘‡βˆ’(𝐴𝑐) is a multiplicative set.
(2)β‡’(3): Let π‘Ž, π‘βˆˆπ‘‡βˆ’(𝐴𝑐). Since 1βˆ‰π‘‡βˆ’(𝐴) and 𝑇(1)β‰ βˆ…, we have 1βˆˆπ‘‡βˆ’(𝐴𝑐). Since π‘‡βˆ’(𝐴𝑐) is a multiplicative set, we have π‘Žβˆ˜11∘1π‘βˆˆπ‘‡βˆ’(𝐴𝑐). So ↓(π‘Žβˆ˜11∘1𝑏)βˆ©π‘‡βˆ’(𝐴𝑐)β‰ βˆ…. Therefore, π‘‡βˆ’(𝐴𝑐) is an π‘š-system. Similarly, we have π‘‡βˆ’(𝐴𝑐) is a π‘š-system.
(3)β‡’(1): Let π‘Žβˆ˜1π‘βˆˆπ‘‡βˆ’(𝐴). Suppose π‘Žβˆ‰π‘‡βˆ’(𝐴) and π‘βˆ‰π‘‡βˆ’(𝐴), then π‘Ž, π‘βˆˆ(π‘‡βˆ’(𝐴))𝑐=π‘‡βˆ’(𝐴𝑐). Since π‘‡βˆ’(𝐴𝑐) is an π‘š-system, we have ↓(π‘Žβˆ˜11∘1𝑏)βˆ©π‘‡βˆ’(𝐴𝑐)β‰ βˆ…. Thus ↓(π‘Žβˆ˜11∘1𝑏)∩(π‘‡βˆ’(𝐴))π‘β‰ βˆ…. Since π‘‡βˆ’(𝐴) is an ideal and 𝑄1 is commutative, we have π‘Žβˆ˜11∘1π‘βˆˆπ‘‡βˆ’(𝐴) and ↓(π‘Žβˆ˜11∘1𝑏)βŠ†π‘‡βˆ’(𝐴). It contradicts with ↓(π‘Žβˆ˜11∘1𝑏)∩(π‘‡βˆ’(𝐴))π‘β‰ βˆ…. So, π‘Žβˆˆπ‘‡βˆ’(𝐴) or π‘βˆˆπ‘‡βˆ’(𝐴). Therefore, π‘‡βˆ’(𝐴) is a prime ideal. Similarly, we have π‘‡βˆ’(𝐴) as a prime ideal.

5. Conclusion

The Pawlak rough sets on the algebraic sets such as semigroups, groups, rings, modules, and lattices were mainly studied by a congruence relation. However, the generalized Pawlak rough set was defined for two universes and proposed on generalized binary relations. Can we extended congruence relations to two universes for algebraic sets? Therefore, Davvaz [25] introduced the concept of set-valued homomorphism for groups which is a generalization of the concept of congruence. In this paper, the concepts of set-valued homomorphism and strong set-valued homomorphism of quantales are introduced. We construct generalized lower and upper approximations by means of a set-valued mapping and discuss the properties of them. We obtain that the concept of generalized rough ideal (subquantale) is the extended notion of ideal (subquantale). Many results in [29] are the special case of the results in this paper. It is an interesting research topic of rough set, we will further study it in the future.

Acknowledgments

The authors are grateful to the reviewers for their valuable suggestions to improve the paper. This work was supported by the National Science Foundation of China (no. 11071061).