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.
Let be a SSV-Hom as defined by , , . Let be a -ideal of defined by . Then GLA and GUA of the -ideal of are as follows: and . It is easily verified that and are -ideals of .
Consider defined by and . Then is a SV-Hom.
Let be a fuzzy subset of defined by . Then is a -ideal of . Hence GLA and GUA of -ideal of are and . It is observed that is not a -ideal of and is a constant -ideal. Hence it is important to take SSV-Hom.
Definition 20. Let be a SV-Hom and let be a fuzzy subset of a quantale . Then is called an upper a lower GRF prime ideal of if is a -prime ideal of . A fuzzy subset of , which is both an upper and a lower GRF prime ideal, is called GRF prime ideal of .
Similarly, we can define upper lower GRF semiprime primary ideals of quantale. Thus the concept of generalized rough fuzzy ideals of quantales extends the notion of rough fuzzy ideals.
Proposition 21. Let be a SSV-Hom. If is a -prime ideal of , then is a -prime ideal of .
Proof. As is a -prime ideal of , therefore or and hence, is a -ideal of , so by Theorem 14, is a -ideal of .
ConsiderSince is a SSV-Hom, therefore for there exist and such that .
Hence,Thus, or . Hence is a -prime ideal of .
Proposition 22. Let be a SSV-Hom. If is a -prime ideal of , then is a -prime ideal of .
Proof. The proof is similar as reported in Proposition 21.
By the above two theorems, we have immediately the following corollary.
Corollary 23. Let be a SSV-Hom and let be a -prime ideal of . Then is a prime ideal of .
Theorem 24. Let be a SSV-Hom and let be a -ideal of . Then is a -prime ideal of if and only if .
Proof. Let be a -prime ideal of . Then or .
This implies thatAs is a -ideal of , hence by definition of -ideal, we haveBy (26) and (27), we obtain . Conversely, suppose that . We have to show that is a -prime ideal. As is totally ordered, so or . Hence or . This shows that is a -prime ideal of .
Theorem 25. Let be a SSV-Hom and let be a -prime ideal of . Then (respectively, ) is a -prime ideal of if and only if, for each , (respectively, ), where , is a prime ideal of .
Proof. As is a -prime ideal of , therefore or . Suppose is a -prime ideal of , then is a -ideal of . By Theorem 18, is an ideal of . In order to show that is a prime ideal for all , we have to show that for implies that or . Let . Then or . Thus, or . Hence is a prime ideal of .
Conversely, suppose that is a prime ideal of , then is an ideal of . By Theorem 18, is a -ideal of .
ConsiderSince is a SSV-Hom, we have , such that .
Hence,Therefore or . Hence is a -prime ideal of
Theorem 26. Let be a SSV-Hom and let be a -semiprime ideal of . Then is a -semiprime ideal of .
Proof. As is a -semiprime ideal of , therefore and is a -ideal of , so by Theorem 14, is a -ideal of .
Hence considerThus . Therefore is a -semiprime ideal of .
Theorem 27. Let be a SSV-Hom and let be a -semiprime ideal of . Then is a -semiprime ideal of .
Proof. Proof is similar as reported in Theorem 26.
Corollary 28. Let be a SSV-Hom and let be a -semiprime ideal of . Then is a semiprime ideal of .
Theorem 29. Let be a -semiprime ideal of and let be a SSV-Hom. Then (respectively, ) is a -semiprime ideal of if and only if, for each , (respectively, ), where , is a semiprime ideal of .
Proof. Suppose is a -semiprime ideal of , then is a -ideal of . By Theorem 18, is an ideal of . In order to show that is a semiprime ideal , we have to show that for implies . Let . Since is a -semiprime ideal, we have . Thus, we have . Hence is a semiprime ideal of .
Conversely, suppose that is a semiprime ideal of . Then is an ideal of . By Theorem 18, is a -ideal.
For to be a -semiprime ideal, we have to show that . As is a SSV-Hom and is a -semiprime ideal of , considerThus . Hence is a -semiprime ideal of .
Example 30. 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.
Let be a SSV-Hom as defined in Example 19.
Let be a fuzzy subset of defined by . Then one can verify that is a -prime ideal of .
Hence GUA and GLA of the -prime ideal are and . It is observed that and are nonconstant -prime ideals of .
Let be a fuzzy subset of defined by . Then is a -semiprime ideal of . Hence GLA and GUA of -semiprime ideal are as follows: and . It is clear that and are -semiprime ideals of .
Theorem 31. Let be a -primary ideal of and let be a SSV-Hom. Then is a -primary ideal of .
Proof. As is a -primary ideal of , therefore or and hence, is a -ideal of , so by Theorem 14, is -ideal of . Since is given as SSV-Hom, considerHere , up to times for some positive integer . Thus or . Therefore is a -primary ideal of .
Theorem 32. Let be a -primary ideal of and let be a SSV-Hom. Then is a -primary ideal of .
Proof. The proof is similar to the proof of Theorem 31.
Theorem 33. Let be a SSV-Hom and let be a nonconstant -primary ideal of . Then (respectively, ) is a -primary ideal of if and only if for each , (respectively, ), where , is a primary ideal of .
4. Homomorphic Images of Generalized Rough Ideals Based on Quantale Homomorphism
In this section, we will describe the images of generalized lower and upper approximations by using quantale homomorphism and set-valued homomorphism of quantales.
Definition 34 (see [47]). Let and be two quantales. A map is called a quantale homomorphism if(1);(2).A quantale homomorphism is called an epimorphism if is onto and is called a monomorphism if is one-one. If is bijective, then it is called an isomorphism.
It is clear that if , then ; that is, is order-preserving.
Proposition 35. Let and be two quantales, let be an epimorphism, and let be a SV-Hom. Then one has the following:
(1) If is one to one and , then is a SV-Hom from to
(2) If is a SSV-Hom, then is a SSV-Hom.
Proof. First of all, we show that is a well-defined mapping. Suppose , then we have . Thus we have . Now we show that is SV-Hom. Suppose , then there exist and such that . Since is a SV-Hom and is a quantale homomorphism, then . Therefore, . Hence . Thus, we have . Now we show that . Let , then there exists such that . Hence . Finally, . Hence . So, is a SV-Hom from to .
It is similar to part.
Theorem 36. Let be a quantale isomorphism and let be a SV-Hom. Set and , then(1);(2);(3) .
Proof. Let . Then there exists such that . Since , . Suppose , then , and by the definition of , we obtain . Thus, , and hence . Thus, we obtain .
Now we take , then there is such that . Hence . So there is such that and . By the definition of , we have . Thus . This gives . Hence, . Thus . Finally, we obtain .
Suppose , then there exists such that and . Suppose , then there is such that ; hence . Thus , so . Hence, . Thus , so we have .
Now let . Then there exists such that and . Suppose , then and hence . Thus and we obtain . Hence and thus, . Hence finally, we have .
Let . Then . Conversely, suppose that , then there is such that . Since is ono-one, hence .
Remark 37. From Theorem 36, it is easily obtained that .
Theorem 38. Let be a surjective quantale homomorphism and let be a SV-Hom. Let and . Then,(1) is an ideal of iff is an ideal of ;(2) is a prime ideal of iff is a prime ideal of ;(3) is a semiprime ideal of iff is a semiprime ideal of ;(4) is a primary ideal of iff is a primary ideal of .
Proof. Suppose is an ideal of . We show that is an ideal of , where by Theorem 36.
(i) Let . Then there exists such that and . Since is a surjective quantale homomorphism and is an ideal of , we have . Therefore .
(ii) Let . Then we obtain and such that and . Since , we have . But is a lower set and , and we have . Thus .
(iii) Let and . Then there exist and such that and . Since is an ideal and is a quantale homomorphism, we have . Hence . In the same way, we have . Hence, is an ideal of . But . So is an ideal of .
Conversely, suppose is an ideal of .
(i) Let . Then . Since is directed, . So by Theorem 36, we have . Hence is directed.
(ii) Let . Then . Since is a lower set, then . By Theorem 36, we obtain . So is a lower set.
(iii) Suppose and , then and . But is an ideal of , and we have . Thus by Theorem 36, we have . Similarly, . So, is an ideal of .
Let be a prime ideal of . Then is obviously an ideal of and . By part , is an ideal of . We also have that . Now suppose and . Since is surjective, there are such that , . Then . By Theorem 36, we obtain . But is prime, and we have or . Thus or . So is a prime ideal of .
Conversely, let be a prime ideal of . Then is an ideal of . Since , thus . By part , is an ideal of . Now suppose and . So . Since is prime, we have or . So by Theorem 36, we have or . Thus is a prime ideal of .
The remaining parts and are similar to the proof .
Theorem 39. Let be a surjective quantale homomorphism and let be a SV-Hom. Set and . Then the following hold:(1) is an ideal of iff is an ideal of ;(2) is a prime ideal of iff is a prime ideal of ;(3) is a semiprime ideal of iff is a semiprime ideal of ;(4) is a primary ideal of iff is a primary ideal of .
Proof. The proofs of all the parts can be obtained by Theorem 38.
5. Generalized Rough Fuzzy Prime (Primary) Ideals Induced by Quantale Homomorphism
Theorem 40. Let be a surjective quantale homomorphism, let be a SV-Hom, and let be a fuzzy subset of . If , then(1) is a -ideal of iff is a -ideal of ;(2) is a -prime ideal of iff is a -prime ideal of ;(3) is a -semiprime ideal of iff is a -semiprime ideal of ;(4) is a -primary ideal of iff is a -primary ideal of .
In the above, , ; that is, is the standard Zadeh image of the fuzzy subset under the mapping .
Proof. We first point out that, for each , and if and only if .
Let be a -ideal of . Then for all , if , then . By Theorem 18, we have is an ideal of . Also by using Proposition 12, we obtain that is an ideal of . Now, by Theorem 38, we have that is an ideal of . Thus, by Theorem 18, we have that is a -ideal of .
Conversely, suppose is a -ideal of . We have that is an ideal of by utilizing Theorem 18. It is obtained from Theorem 38 that is an ideal of . Hence by Theorem 18, is a -ideal of .
Let be a -prime ideal of . Now for , then for any . Since is a -prime ideal of , then by Theorem 25, we have that is a prime ideal of . It is also obtained from Proposition 12 that is a prime ideal of . Hence is a prime ideal of , by Theorem 38. Thus, by Theorem 25, we have that is a -prime ideal of .
Conversely, suppose is a -prime ideal of . By Theorem 25, we obtain that is a prime ideal of . Thus it is obtained, from Theorem 38, that is a prime ideal of . Hence is a -prime ideal of by Theorem 25.
Axioms and can be obtained in a similar way.
Theorem 41. Let be a surjective quantale homomorphism from a quantale onto a quantale . Let be a SV-Hom and let be a fuzzy subset of . If , then(1) is a -ideal of iff is a -ideal of ;(2) is a -prime ideal of iff is a -prime ideal of ;(3) is a -semiprime ideal of iff is a -semiprime ideal of ;(4) is a -primary ideal of iff is a -primary ideal of .
Proof. The proof is similar as reported in Theorem 40.
6. Conclusion
Pure and applied mathematics are two important branches of mathematics and rough set theory has its own importance in both the branches. When we combine rough set theory with algebraic structures, we obtain new interesting results and research topics. These research topics are attracted by computer scientists and mathematicians. Researchers apply roughness into the algebraic system and find interesting algebraic properties of them. The combination of fuzzy set and rough set theory leads to various models. The relations between fuzzy sets, rough sets, and quantale theory have been already considered in [46]. We have examined the generalized rough fuzzy set theory and its properties in quantale.
In the present paper, we substituted a universe set by a quantale and introduced the notions of generalized rough fuzzy prime (semiprime, primary) ideals in quantale. We see that the lower and upper approximations of fuzzy ideals, using SSV-Hom, are fuzzy ideals, respectively. It is also seen that the approximations of fuzzy prime (semiprime, primary) ideals using SSV-Hom are fuzzy prime (semiprime, primary) ideals, respectively. We have discussed the relation between upper (lower) generalized rough fuzzy (prime, semiprime, primary) ideals and upper (lower) generalized rough fuzzy approximations of their homomorphic images.
We believe that in the near future the idea of generalized roughness will be extended to other algebraic structures.
Conflicts of Interest
There are no conflicts of interest related to this paper.