#### 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 Pawlakβs 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 [8β12], the roughness of various hyperstructures are discussed. Further, some authors consider the rough set in a fuzzy algebraic system, see [13β16]. 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 [18β24].

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 with an associative binary operation satisfying
for all .

An element is called a *left (right) unit* if and only if (), is called a *unit* if it is both a right and left unit.

A quantale is called a *commutative quantale* if for all .

A quantale is called an *idempotent quantale* if for all .

A subset of is called a *subquantale* of if it is closed under and arbitrary sups.

In a quantale , we denote the top element of by 1 and the bottom by 0. For , we write to denote the set , to denote and .

*Definition 2.2. *Let be a quantale, a subset is called a *left (right) ideal* of if(1) implies ,(2) and imply for all,(3) and imply ().

A subset is called an *ideal* if it is both a left and a right ideal.

Let be a subset of , we write for some , is a *lower set* if and only if . It is obvious that an ideal is a directed lower set. For every (left, right) ideal of , it is easy to see that .

An ideal of is called a *prime ideal* if implies or for all .

An ideal of is called a *semi-prime ideal* if implies for all .

An ideal of () is called a *primary ideal* if for all , and imply for some . ().

*Definition 2.3. *A nonempty subset is called an -*system* of , if for all , .

A nonempty subset is called a *multiplicative set* of , if for all .

Every ideal of is both an -system and a multiplicative set.

*Definition 2.4. *Let be a quantale, an equivalence relation on is called a *congruence* on if for all , we have(1),(2).It is obvious that , for all .

*Definition 2.5. *Let be a quantale, a congruence on is called a *complete congruence*, if(1) for all ,(2) for all .

*Definition 2.6. *Let and be two quantales. A map is said to be a *homomorphism* if(1) for all ;(2) for all .

*Definition 2.7. *Let and be two nonempty universes. Let be a set-valued mapping given by , where denotes the set of all subsets of . Then the triple is referred to as a generalized approximation space. For any set , the generalized lower and upper approximations, and , are defined by

The pair is referred to as a *generalized rough set. *

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

Theorem 2.8. *Let be nonempty universes and be a set-valued mapping, where denotes the set of all nonempty subsets of . If , then .*

If and is an equivalence relation on , then the pair is the Pawlak approximation space. Therefore, a generalized rough set is an extended notion of Pawlakβs rough set.

Theorem 2.9. *Let be a generalized approximation space, its lower and upper approximation operators satisfy the following properties. For all , *(1)*, ,*(2)*, ,*(3)*, ,*(4)*, ,*(5)*, ,**where is the complement of the set .*

#### 3. Generalized Rough Subsets in Quantales

In this paper, and are two quantales.

Theorem 3.1. *Let be a set-valued mapping and . Then*(1)*, if ,*(2)*, if ,*(3)*,*(4)*, if is an idempotent quantale,*(5)*, if ,*(6)*, if .*

*Proof. *(1) Suppose that , we have for . So . Similarly, . So . By Theorem 2.9, we have .

(2) Suppose that , we have for . So . Similarly, . So . By Theorem 2.9, we have .

(3) It is obvious that . By Theorem 2.9, we have .

(4) Since is an idempotent quantale, we have . By Theorem 2.9, we have .

(5) and (6) The proofs are similar to (1) and (2), respectively.

*Definition 3.2. *A set-valued mapping is called a *set-valued homomorphism* if(1) for all ,(2) for all .

is called a *strong set-valued homomorphism* if the equalities in (1), (2) hold.

*Example 3.3. *(1) Let be a congruence on . Then the set-valued mapping defined by is a set-valued homomorphism but not necessarily a strong set-valued homomorphism. If is complete, then is a strong set-valued homomorphism.

(2) Let be a quantale homomorphism from to . Then the set-valued mapping defined by is a strong set-valued homomorphism.

Theorem 3.4. *Let be a set-valued homomorphism and . Then*(1)*,*(2)*,*(3)*,*(4)*, if is an idempotent quantale.*

*Proof. *(1) Suppose that , there exist such that . So there exist and . Hence and . Since is a set-valued homomorphism, we have . Therefore, which implies that .

(2) The proof is similar to (1).

(3) Suppose that , there exist and . Since is a set-valued homomorphism, we have . So which implies that .

(4) Suppose that , there exist and . Since is a set-valued homomorphism and is idempotent, we have . So which implies that .

Theorem 3.5. *Let be a strong set-valued homomorphism and . Then*(1)*,*(2)*.*

*Proof. * (1) Suppose that , there exist such that . Hence and . Since is a strong set-valued homomorphism, we have which implies that .

(2) The proof is similar to (1).

#### 4. Generalized Rough Ideals in Quantales

Theorem 4.1. *Let be a set-valued mapping. If and are, respectively, a right and a left ideal of , then*(1)*,*(2)*,*(3)*,*(4)*,*(5)*,*(6)*.**If is an idempotent quantale and is a set-valued homomorphism, then the equalities in (1)β(3) hold.*

*Proof. *Since and are, respectively, a right and a left ideal of , we have , and . By Theorem 2.9, we get the conclusion (1)β(4). By Theorem 3.1, we get (5) and (6).

If is idempotent, we first show that . Suppose that , there exist such that . So, and . Since T is a set-valued homomorphism and is idempotent, we have . Therefore, . So the equality in (1) holds. Since is idempotent, we have . So . By Theorem 2.9, we get . Since the equality in (1) holds, we have .

Lemma 4.2. *Let be a set-valued homomorphism. If is a lower set of , then is a lower set of .*

*Proof. *Suppose , then . Let , we have . Since is a lower set, we have . Therefore, which implies that .

Lemma 4.3. *Let be a strong set-valued homomorphism. If is a lower set of , then is a lower set of .*

*Proof. *Suppose that , there exists . Since is a strong set-valued homomorphism, we have . So there exist such that . Since is a lower set, we have . Thus which follows that .

Lemma 4.4. *Let be a set-valued homomorphism and a nonempty subset of . If is closed under arbitrary (resp., finite) sups, then is closed under arbitrary (resp., finite) sups.*

*Proof. *Let . For each , we have , then there exist . Since is a set-valued homomorphism, we have . And we have for A is closed under arbitrary sups. So which implies that .

Lemma 4.5. *Let be a strong set-valued homomorphism and a nonempty subset of . If is closed under arbitrary (resp., finite) sups, then is closed under arbitrary (resp., finite) sups.*

*Proof. *Let . For each , we have . Since is a strong set-valued homomorphism, we have . Suppose , there exist () such that . Since is closed under arbitrary sups, we have . So which implies that .

Theorem 4.6. *Let be a set-valued homomorphism. If is a subquantale of , then is a subquantale of .*

*Proof. *Since is a set-valued homomorphism and is a subquantale, by Theorems 3.4 and 2.9, we have . So, is closed under .

Since is closed under arbitrary sups, by Lemma 4.4, we get is closed under arbitrary sups.

Theorem 4.7. *Let be a strong set-valued homomorphism. If is a subquantale of , then is a subquantale of .*

*Proof. *The proof is similar to Theorem 4.6.

Theorem 4.8. *Let be a strong set-valued homomorphism and a right (left) ideal of . Then is, if it is nonempty, a right (left) ideal of .*

*Proof. *Suppose that . Since is closed under finite sups, by Lemma 4.4, we have is closed under finite sups. So .

Since is a lower set, by Lemma 4.3, we have is a lower set.

Suppose , there exists . Since is a right ideal of , we have for each . So . Therefore, which implies that .

Theorem 4.9. *Let be a strong set-valued homomorphism and a right (left) ideal of . Then is, if it is nonempty, a right (left) ideal of .*

*Proof. *Suppose . Since is closed under finite sups, by Lemma 4.5, we have is closed under finite sups. So, .

Since is a lower set, by Lemma 4.2, we have is a lower set.

Suppose , we have . Let . Since is a strong set-valued homomorphism, we have , then there exist , such that . Since is a right ideal, we have . Therefore, which implies that .

*Definition 4.10. *A subset is called a *generalized rough ideal* (subquantale) of if and are ideals (subquantales) of .

The following corollary follows from Theorems 4.6β4.9.

Corollary 4.11. *Let be a strong set-valued homomorphism and an ideal (a subquantale) of . If and are nonempty, then is a generalized rough ideal (subquantale) of .*

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 and be quantales shown in Figures 1 and 2 and Tables 1 and 2.

Let be a strong set-valued homomorphism as defined by . Let , , then and . It is obvious that is a generalized rough ideal of but is not an ideal of and is a generalized rough subquantale of but is not a subquantale of .

Theorem 4.13. *Let be a strong set-valued homomorphism and a prime ideal of . Then is, if it is nonempty, a prime ideal of .*

*Proof. *By Theorem 4.8, we get is an ideal of .

Let , there exists . Since is a strong set-valued homomorphism, we have , then there exist such that . Since is a prime ideal of , we have or . So or which implies that or .

Theorem 4.14. *Let be a strong set-valued homomorphism and a prime ideal of . Then is, if it is nonempty, a prime ideal of .*

*Proof. *By Theorem 4.9, we get is an ideal of .

Let , we have . Since is a strong set-valued homomorphism, we have . We assume that , then , there exists but . If , then . Since is a prime ideal of , we have . Therefore, which implies that .

We call is a generalized rough prime ideal of if and are ideals of .

Theorem 4.15. *Let be a strong set-valued homomorphism and a semiprime ideal of . Then is, if it is nonempty, a semiprime ideal of .*

*Proof. *By Theorem 4.9, we get is an ideal of .

Suppose that , we have . Let , we have . Since is a semi-prime ideal, we have . So which implies that .

Theorem 4.16. *Let be a strong set-valued homomorphism and a primary ideal of . Then is, if and , a primary ideal of .*

*Proof. *By Theorem 4.8, we get is an ideal of .

Suppose that and , there exists . Since is a strong set-valued homomorphism, we have , there exist , such that . Since , we get . Since is a primary ideal, we have for some Since is a strong set-valued homomorphism, we have . So which implies that .

Theorem 4.17. *Let be a strong set-valued homomorphism and a primary ideal of . Let be a commutative quantale and a finite set for each . Then is, if and , a primary ideal of .*

*Proof. *By Theorem 4.9, we get is an ideal of .

Suppose and , then and . So there exists with . We assume that , then . Since is a primary ideal, there exists for some . Let . Since is an ideal, we have . Let . Since is commutative, there exists such that with , where . Assume that , then . It contradicts with . Therefore, there is such that , we have . Since is an ideal, we get . Hence which implies that .

Theorem 4.18. *Let be a set-valued homomorphism. If is a multiplicative set of , then is, if it is nonempty, a multiplicative of . If is a strong set-valued homomorphism, then is, if it is nonempty, a multiplicative of .*

*Proof. *Suppose that , there exist , . Hence . Since is a multiplicative set, we have . Therefore, which implies that .

Suppose , then . Let . Since is a strong set-valued homomorphism, we have . There exist such that . Since is a multiplicative set, we have . So which implies that .

We call a generalized rough multiplicative set (-system) of , if are multiplicative sets (-systems) of .

Theorem 4.19. *Let be a set-valued homomorphism. If is commutative and with , then the following statements are equivalent:*(1)* is a generalized rough prime ideal of ,*(2)* is a generalized rough ideal and is a generalized rough multiplicative set of ,*(3)* is a generalized rough ideal and is a generalized rough -system of .*

*Proof. *(1)*β*(2): Since is a generalized rough prime ideal of , we get is a generalized rough ideal of and , are prime ideals of . Now we show that is a multiplicative set. Let , . By Theorem 2.9(1), we have . Since is a prime ideal, we have which implies that . So is a multiplicative set. Similarly, is a multiplicative set.

(2)*β*(3): Let , . Since and , we have . Since is a multiplicative set, we have . So . Therefore, is an -system. Similarly, we have is a -system.

(3)*β*(1): Let . Suppose and , then , . Since is an -system, we have . Thus . Since is an ideal and is commutative, we have and . It contradicts with . So, or . Therefore, is a prime ideal. Similarly, we have 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).