Discrete Dynamics in Nature and Society

Volume 2018, Article ID 1085201, 11 pages

https://doi.org/10.1155/2018/1085201

## Generalized Rough Fuzzy Ideals in Quantales

^{1}Department of Mathematics, Government College University, Faisalabad, Pakistan^{2}Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Correspondence should be addressed to Saqib Mazher Qurashi; kp.ude.fucg@rahzambiqas

Received 7 September 2017; Revised 19 November 2017; Accepted 10 December 2017; Published 16 January 2018

Academic Editor: Rigoberto Medina

Copyright © 2018 Saqib Mazher Qurashi and Muhammad Shabir. 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.

#### Abstract

The paper examines the generalized rough fuzzy ideals of quantales. There are some intrinsic relations between fuzzy prime (primary) ideals of quantales and generalized rough fuzzy prime (primary) ideals of quantales. Homomorphic images of “generalized rough ideals, generalized rough prime (primary) ideals, and generalized rough fuzzy prime (primary) ideals” which are incited by quantale homomorphism are examined.

#### 1. Introduction

The idea of “theory of rough sets” proposed by Pawlak [1, 2] to manage uncertainty and granularity in the information system has attracted the concern and attention of scientists and experts in different fields of science and technology. Late years have seen its wide applications in algebraic systems, knowledge discovery, data mining, expert systems, pattern recognition, granular computing, graph theory, machine learning, partially ordered sets, and so forth [3–15]. It is noted that the significant concepts in the classical theory of rough set are the lower and upper approximations obtained from equivalence relation on a universal set. In many cases, as pointed out by numerous researchers, the implementation of theory of rough set becomes restrictive if we use the condition of the equivalence relation in the model of Pawlak rough set. To get control of this issue, several authors generalized the classical rough set theory by using more general binary relations [16–20] or by employing coverings [21, 22]. Besides, theory of rough sets can also be generalized to the fuzzy environment by employing the notion of fuzzy sets of Zadeh [23], and the resulting notions are called fuzzy rough sets [24–27].

Recently, researchers have connected the ideas and techniques for rough set hypothesis to different algebraic structures. Biswas and Nanda [4] took a group as a ground set and presented the notions of rough groups and rough subgroups. Kuroki and Mordeson [28] discussed the structure of rough sets and rough groups. At that point in [29], Kuroki presented the thought of rough ideals in semigroup. Rough prime ideals and rough fuzzy prime ideals in semigroups were proposed by Xiao and Zhang [17]. Davvaz [30] gave the concepts of rough ideals in rings. He also wrote a short note on algebraic -rough sets [31]. Kazancı and Davvaz in [32] gave rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings. To overcome the confinement of equivalence relations in the process of establishing rough sets in a ring, Yamak et al. [19] introduced the concept of set-valued mappings as the basis of the generalized upper and lower approximations of a ring with the help of an ideal. Roughness in modules was researched by Davvaz and Mahdavipour [33]. Rasouli and Davvaz [14] presented the notion of rough ideals in MV-algebra. In BCK-algebras, rough ideals were defined by Jun [34]. Classical approximation theory has also been applied to some partially ordered structures. For instance, in [10], Estaji et al. investigated the concepts of upper and lower rough ideals in a lattice by introducing the relationships between lattice theory and rough sets. Zhan et al. discussed “a new rough set theory: rough soft hemirings” in [35]. Ma et al. gave the “applications of a kind of novel -soft fuzzy rough ideals to hemirings” and investigated “a survey of decision making methods based on certain hybrid soft set models” [36, 37].

The combination of rough set theory to soft set theory is very important. Feng et al. proposed rough soft sets by combining Pawlak rough sets and soft sets. In particular, Feng et al. put forth a novel concept of soft rough fuzzy sets by combining rough sets, soft sets, and fuzzy sets and we call it Feng-soft rough fuzzy set [38]. And in 2011, Meng et al. further discussed the Feng-soft rough fuzzy sets and put forward another kind of soft rough fuzzy sets, which is called Meng-soft rough fuzzy set [39]. These sets are limited and have a rigorous restrictive condition. Based on the above reason, Zhan and Zhu provided a novel concept of soft rough fuzzy sets, which is called a -soft rough fuzzy set [40]. As reported in [41, 42], characterizations of two kinds of hemirings based on probability spaces and reviews on decision making methods based on (fuzzy) soft sets and rough soft sets are discussed, respectively.

The structure of quantale was proposed by Mulvey [43] to study the spectrum of -algebras. The idea of ideals (prime, primary) of quantale was given by Wang and Zhao [44, 45]. Xiao and Li [16] generalized the ideals of quantale by means of set-valued mappings. The start of theory of rough sets for applying in algebraic structures, for example, semigroups, rings, modules, and groups, has been focused on a congruence relation. However, we obtain the restricted applications by using the congruence relation. To take care of this issue, Davvaz [19, 31] introduced the idea of a set-valued homomorphism for rings and groups. In this paper, we intend to generalize the results which have been proved in [46].

The arrangement of the paper is as per the following. In Section 2, we review some principal properties of rough sets, rough fuzzy sets, and ideals of quantale. In Section 3, we have introduced “generalized rough fuzzy ideal” and “generalized rough fuzzy prime (semiprime, primary) ideals” of quantales and give a few properties of such ideals. In Section 4, we will describe the images of generalized rough ideals and discuss how they are related. We will explain the relation between lower (upper) generalized rough and lower (upper) generalized approximations of their homomorphic images by using quantale homomorphism and set-valued homomorphism of quantale. In Section 5, we will discuss generalized rough fuzzy prime (primary) ideals based on quantale homomorphism. At last, the conclusion is given in Section 6.

#### 2. Preliminaries

Here, we review a few ideas and results which will be vital in the following.

*Definition 1 (see [2]). *Let be an approximation space, where is a nonempty set, and let be an equivalence relation on . For , the equivalence class of , containing , is denoted by . For , the upper and lower approximations of are, respectively, defined as , . It is easy to verify that for all .

For more details on rough sets, rough fuzzy sets, and fuzzy rough sets, we refer to [2, 24, 26, 27]. Throughout this paper, we shall use and for quantales, unless stated otherwise.

*Definition 2 (see [47]). *A complete lattice having associative binary operation is called a quantale if it satisfies ; , for all

We will represent the top element of by and the bottom element by throughout the paper. Let , and we define by the set , by and .

*Definition 3 (see [44]). *Let be a quantale. A nonempty subset of is said to be an ideal of if the following conditions hold:(1)For all , is implied.(2)If , and imply .(3)For all and , then and .

An ideal is said to be a prime ideal if implies or for all .

An ideal is said to be a semiprime ideal if implies for all .

Primary ideal is an ideal of if for all , and imply for some positive integer , where .

As it is well known in the fuzzy theory established by Zadeh [23], a fuzzy subset of is defined as a map from to the unit interval . The symbols and will denote the respective infimum and supremum.

*Definition 4 (see [24]). *Let be an approximation space. A fuzzy subset is a mapping from to , then for , one definesThey are called the lower and upper approximations of , respectively. If , then is called a rough fuzzy set with respect to . For , the setsare called -cut and strong -cut of the fuzzy set , respectively.

*Definition 5 (see [46]). *A nonempty fuzzy subset of is called a fuzzy ideal of , if the following conditions are satisfied:(1)If , then .(2).(3).

From and in Definition 5, it is observed that for all . Thus, a fuzzy set is a fuzzy ideal of if and only if and for all .

*Definition 6 (see [46]). *A nonconstant fuzzy ideal of a quantale is called a fuzzy prime ideal of if for all , or .

Note that we require a fuzzy prime ideal of a quantale to be a nonconstant in order to keep consistent with the definition of prime ideals of quantales [45]. Therefore, throughout this paper, a fuzzy ideal of a quantale is always assumed to be nonconstant. For fuzzy semiprime and fuzzy primary ideals, see [46].

Proposition 7 (see [46]). *Let be a fuzzy subset of a quantale . Then is a fuzzy (prime, semiprime, primary) ideal of if and only if for each , (resp., ) is either empty or (prime, semiprime, primary) ideal of .*

Throughout this paper, -ideal, -prime, -semiprime, and -primary ideals will denote fuzzy ideal, fuzzy prime, fuzzy semiprime, and fuzzy primary ideals of quantales, unless stated otherwise. We use to denote the set of all fuzzy subsets of .

The concept of generalized rough sets is a generalization of Pawlak’s rough set. In rough set theory, an equivalence relation is the basic requirement for lower and upper approximations. Sometimes it is difficult to find such an equivalence relation among the elements of the set under investigation. In such situations, generalized rough set approach can be useful.

*Definition 8 (see [19]). *Let and be two nonempty universes. Let be a set-valued mapping given by , where is the power set of . Then the triple is referred to as a generalized approximation space or generalized rough set. Any set-valued function from to defines a binary relation from to by setting . Obviously, if is an arbitrary relation from to , then a set-valued mapping can be defined by , where . For any set , the lower and upper approximations represented by and , respectively, are defined asWe call the pair generalized rough set, and , are termed as lower and upper generalized approximation operators, respectively.

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 Pawlak’s rough set [16].

*Definition 9 (see [16]). *Let and be two quantales. A set-valued mapping , where represents the collection of all nonempty subsets of , is called a set-valued homomorphism if, for all ,(1),(2).

A set-valued mapping is called a strong set-valued homomorphism if we replace inclusion by equality in and .

From here onwards by SV-Hom, we will mean the set-valued homomorphism. For strong set-valued homomorphism, we will use SSV-Hom. Besides will mean the map , unless stated otherwise.

#### 3. Generalized Rough Fuzzy Prime (Primary) Ideals in Quantale

In this section, we will introduce the generalized rough fuzzy ideal in quantales and resulting properties of such ideals are presented. Now we use the concept from Definition 4 and generalized it in the following.

*Definition 10. *Let and be two quantales and let be a SV-Hom. Let be any fuzzy subset of . Then for every , one definesHere is the generalized lower approximation and is the generalized upper approximation of the fuzzy subset . The pair is called generalized rough fuzzy set of if .

From here onward by GLA, GUA, and GRF, we will mean generalized lower approximation, generalized upper approximation, and generalized rough fuzzy set, respectively.

Lemma 11. *Let be a SV-Hom. Then for every collection ,*(1)*;*(2)*.*

*Proof. * For , we haveThe other item has the similar proof.

*Proposition 12. Let and be two quantales and let be a SV-Hom. Let be a fuzzy subset of . Then for each , one has the following:(1);(2);(3);(4).*

*Proof. * LetAxioms , , and are similar to the proof of .

*Definition 13. *Let be a SV-Hom. A fuzzy subset of the quantale is called a lower GRF ideal of if is a -ideal of . A fuzzy subset of , which is both an upper and a lower GRF ideal of , is called GRF ideal of .

*Now, lower approximations and upper approximations of -ideals of quantales are being studied in the following.*

*Theorem 14. Let be a SSV-Hom and let be a -ideal of . Then is a -ideal of .*

*Proof. *Since is a -ideal of , by Definition 5, we have and . As is a SSV-Hom, so , .

Therefore,Since , there exist and such that .

Hence,Hence,Again since is a SSV-Hom, hence .

Thus we haveNow since , there exist , such that .

Thus,Hence,Thus, by (9) and (12), is a -ideal of .

*Theorem 15. Let be a SSV-Hom and let be a -ideal of . Then is a -ideal of .*

*Proof. *Since is a SSV-Hom, therefore . Also is -ideal of ; hence .

ConsiderFor , we have and such that .

Hence,Thus, Now,For , there exist and such that .

Hence,Thus, Hence by (15) and (18), we have is a -ideal of .

*By the above two theorems, we have immediately the following corollary.*

*Corollary 16. Let be a SSV-Hom and let be a -ideal of . Then is a ideal of .*

*Proposition 17. Let be a SSV-Hom. Let be a family of -ideals of . Then is a -ideal of .*

*Proof. *Since every is a -ideal for , therefore ,Hence,Hence,Therefore, is a -ideal of .

*Theorem 18. Let be a SSV-Hom and let be a -ideal of . Then (respectively, ) is a -ideal of if and only if for each , (respectively, ), where , is an ideal of .*

*Proof. *Suppose is a -ideal of . We need to show that is an ideal of . Let . Then , . But since is a -ideal, so . Hence . Let , , and . Then . Thus . Suppose and , then , and we get . Similarly, . Hence, is an ideal of .

Conversely, assume is an ideal of . We will show that is a -ideal of . For any , let rang. Then and ; that is, and . Hence, .

ConsiderSince is a SSV-Hom, for , there exist and such that .

Hence we obtainSo .

Now for and , we obtain and . Hence, and . If either or , in both the cases, . We suppose . So . Hence, is a -ideal of .

*Example 19. *Let and be two quantales, where and are depicted in Figures 1 and 2 and the binary operations and on both the quantales are the same as the meet operation in the lattices and as shown in Tables 1 and 2.