Abstract
The aim of this research article is to derive a new relation between rough sets and soft sets with an algebraic structure quantale by using soft binary relations. The aftersets and foresets are utilized to define lower approximation and upper approximation of soft subsets of quantales. As a consequence of this new relation, different characterization of rough soft substructures of quantales is obtained. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Moreover, this concept generalizes the concept of rough soft substructures of other structures. Furthermore, the algebraic relations between the upper (lower) approximation of soft substructures of quantales and the upper (lower) approximation of their homomorphic images with the help of soft quantales homomorphism are examined. In comparison with the different type of approximations in different type of algebraic structures, it is concluded that this new study is much better.
1. Introduction
Quantale theory was proposed by Mulvey[1]. It is based on defining an algebraic structure on a complete lattice. Since quantale was defined on a complete lattice, there must be a correlation between linear logic and quantale theory which was studied by Yetter, in his study. He presented a new class of models for linear intuitionistic logic [2]. In recent years, quantale is applied in vast research areas, such as algebraic theory [3], rough set theory [4–7], topological theory [8], theoretical computer science [9], and linear logic [10].
In 1982, Pawlak developed the famous rough set theory [11], which is a mathematization of inadequate knowledge. The rough set deals with the categorization and investigation of inadequate information and knowledge. After Pawlak’s work, Zhu [12] provided some new views on the rough set theory. In [13], Ali et al. studied some properties of generalized rough sets. Nowadays, rough sets are applied in many different areas, such as cognitive sciences, machine learning, pattern recognition, and process control.
There are many problems that arise in different fields such as engineering, economics, and social sciences in which data have some sort of uncertainty. Well-known mathematical tools have so many limitations because these tools are introduced for particular circumstances. There are many theories to overcome uncertainty such as fuzzy set theory, probability theory, rough sets, and vague sets, but these are limited due to its design.
In 1998, Molodtsov present the idea of soft set theory, which is a mathematical tool to overcome the adversities affecting the above theories [14]. Many authors like Maji et al. present different operations on soft sets and try to consolidate the algebraic aspects of soft sets [15]. A new and different idea of operations was presented by Ali et al. [16]. Many soft algebraic structures such as soft modules [17], soft groups [18], soft rings [19], and soft ordered semigroups [20] were studied. The basic theme and purpose of soft sets are to create the idea of parametrization, and this idea has been utilized to find soft binary relation (SBR) which is a parameterized collection of binary relations on a universe under consideration. This puts forward the consideration for complicated objects that may be perceived from different points of view. In [21–23], Feng et al. presented the relationship between soft, rough, and fuzzy sets and produced rough soft sets, soft rough sets, and soft-rough fuzzy sets.
By using aftersets and foresets notions associated with SBR, a new approximation space is widely utilized these days. By using generalized approximation space based on SBR, different soft substructures in semigroups were approximated by Kanawal and Shabir [24]. Motivated by the idea in [24], soft substructures in quantales are defined, and the aftersets and foresets are employed to construct the lower approximation and upper approximation of soft substructures. Since we are dealing with the approximation of soft subsets of quantale, further soft substructures are employed for further characterization.
There are several authors who introduced rough sets theory in algebraic structures and soft algebraic structures. Iwinski analyzes algebraic properties of rough sets [25]. Qurashi and Shabir present the idea of roughness in - module [5]. Idea of the generalized rough quantales (subquantales) was presented by Xiao and Li [6]. Rough prime (semiprime and primary) ideals in quantales were investigated by Yang and Xu [7]. Fuzzy ideals (prime, semiprime, and primary) in quantales were introduced by Luo and Wang [4]. Generalized roughness of fuzzy substructures in quantale is studied by Qurashi et al. [26]. In [27], Yamak et al. proposed the idea of set-valued mappings as the basis of the generalized upper (lower) approximations of a ring with the help of ideals. Rough prime bi -hyper ideals of -semihypergroups were proposed by Yaqoob et al. [28, 29]. Rough substructures of semigroups were studied by Kuroki [30].
The following scheme is designed for the rest of the paper. Some essential explanations related to quantales, its substructures, soft substructures, and their corresponding sequels are connected in Section 2. Notion of approximations of soft sets over quantale generated by soft binary relations is discussed in Section 3. In Section 4, by using these ideas, generalized soft substructures are defined and investigated further fundamental algebraic characteristics of these phenomena. Additionally, we extend this study to define the relationship between homomorphic images and their approximation by soft binary relation in Section 5.
2. Preliminaries
Let be a nonempty finite set called the universe set and be an E.R (equivalence relation) over . Let denotes the equivalence class of the relation containing . Any definable set in would be written as finite union of equivalence classes of . Let in general is not a definable set in . However, the set can be approximated by two definable sets in . The first one is called -lower approximation () of , and the second is called -upper approximation (). They are defined as follows:
The of in is the greatest definable in contained in . The of in is the least definable set in containing . For any nonempty subset in , is called rough set with respect to or simply a -rough subset of if , where denotes the set of all subsets of .
Definition 1 (see [31]). Let be a complete lattice. Define an associative binary relation on satisfying. Then, is called quantale.
Let , . We define some notions as follows:Throughout the paper, quantales are denoted by and .
Let . Then, W is called a subquantale of if the following holds:(1), .(2), .That is, closed under and arbitrary supremum.
Definition 2 (see [32]). Let be a quantale, is called left (right) ideal if the following satisfied:(1) implies (2), such that implies (3) and implies A nonempty subset is called ideal of if it is left as well as right ideal.
Example 1. Let complete lattices are shown in Figure 1. We define be the associative binary operation on as shown in Table 1.
Then, is a quantale. Then, , , , , and are all of quantale .
Definition 3 (see [32]). Let be an ideal. is called prime ideal if, , or . is called semiprime if, , is called primary if, , and implies for some .
Definition 4 (see [14]). A pair is called a soft set over if where is a subset of (the set of parameters).
Definition 5 (see [16]). Let and be two soft sets over . Then, soft subset if the following conditions are fulfilled:(1)(2),
Definition 6 (see [33]). Let be a soft set over , that is, . Then, is called a soft binary relation (SBR) over . A SBR over is a soft set over . That is, .
Definition 7. Let be a soft set over quantale . Then,(1) is called soft subquantale over iff is a subquantale of , (2) is called soft ideal over iff is an ideal of , (3) is called soft prime ideal over iff is a prime ideal of , (4) is called soft semiprime ideal over iff is a semiprime ideal of , (5) is called soft primary ideal over iff is a primary ideal of ,
3. Approximation of Soft Sets over Quantale by Soft Binary Relation
In this section, we present some important aspects regarding to the approximation of soft sets in quantale by SBR. We utilized aftersets and foresets to approximate soft sets.
Definition 8 (see [34]). Let be a SBR over , where (parametric set). Then, . For a soft set over , the and of w.r.t the afterset are essentially two soft sets over , which is defined asAnd for a soft set over , the and of w.r.t the foreset are actually two soft sets over , which is defined asFor all , where is called the afterset of and is called the foreset of .
Remark 1. (1)For each soft set over , and (2)For each soft set over , and
Definition 9. Let be a SBR over , that is, . Then, is called soft compatible relation (SCPR) if for all and , we have(1) (2) for every .
Definition 10. A SCPR over is called soft complete relation (SCTR) with respect to the afterset if, for all , we have(1)(2)for all .
A SCPR is called -complete w.r.t the aftersets if it satisfies only condition (1). A SCPR is called -complete w.r.t the aftersets if it satisfies only condition (2).
A SCPR over is called soft complete relation (SCTR) with respect to the foreset if for all , and we have(1)(2)for all .
A SCPR is called -complete w.r.t the foresets if it satisfies only condition (1).
A SCPR is called -complete w.r.t the foresets if it satisfies only condition (2).
Theorem 1. Let be a SCPR with respect to the afterset over . Then, for any two soft sets and over , we have(1)(2)
Proof. For arbitrary , let Then, for some and . This implies that and , so there exist elements such that and . Thus, and So and imply ; that is, . Also, ; therefore, This shows that
Now, for arbitrary , let Then, for some and . This implies that and , so there exist elements such that and . Thus, , and So and imply ; that is, . Also, ; therefore, This shows that
Theorem 2. Let be a SCPR with respect to the foreset over . Then, for any two soft sets and over , we have(1)(2)
Proof. The proof is simple.
Theorem 3. Let be a SCTR w.r.t the afterset over . Then, for any two soft sets and over , we have(1)(2)
Proof. For arbitrary , if at least one of and is empty, then (1) is obvious. Now, for arbitrary , consider that and . Then, . So, let . Then, for some and . This implies that and . As . This shows that . Hence, (1) is proved.
For arbitrary , if at least one of and is empty, then (2) is obvious. Now, for arbitrary , consider that and . Then, . So, let . Then, for some and . This implies that and . As . This shows that . Hence, (2) is proved.
Theorem 4. Let be a SCTR with respect to the foreset over . Then, for any two soft sets and over , we have(1)(2)
Proof. The proof is obvious.
4. Approximation of Soft Substructures in Quantales
In this section, we consider two quantales and and approximate different soft substructures of quantales by using different SBR over . We will show that of a soft substructure of quantales by using SCPR is again a soft substructure of quantales and provide counter examples to support the argument that the converse is not true. Also, we will show that of a soft substructure of quantales by using SCTR is again a soft substructure of quantales and provide a counter example to support the argument that the converse is not true.
Throughout this section, we consider to be the SBR over and for all , , and for all , unless otherwise specified.
Definition 11. Let be a SBR over and be a soft set over . If . is a soft subquantale of , then is called generalized upper soft () subquantale of w.r.t the aftersets. If is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of , then is called ideal (prime ideal, semiprime ideal, and primary ideal) of w.r.t the aftersets.
Definition 12. Let be a SBR over and be a soft set over . If is a soft subquantale of , then is called generalized upper soft () subquantale of w.r.t the foresets. If is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of , then is called ideal (prime ideal, semiprime ideal, and primary ideal) of w.r.t the foresets.
Theorem 5. Let be a SCPR over . If is a soft subquantale of , then is a subquantale of w.r.t the aftersets.
Proof. Suppose that is a soft subquantale, then for any . Let , . Then, . So, there exists . Thus, and since is a SCPR. Therefore, , implies . This implies that . Also, (as is a soft subquantale). So, . Hence, .
Let . Then, and . So, there exists and . Thus, , , , and since is a SCPR. Therefore, implies . This implies that . Also, (as is a soft subquantale). So, . Hence, . This completes the proof.
With the same arguments, the next Theorem 6 can be achieved.
Theorem 6. Let be a SCPR over . If is a soft subquantale of , then is a subquantale of w.r.t the foresets.
Theorem 7. Let be a soft -complete relation over w.r.t the aftersets. If is a soft left (right) ideal of , then is a left (right) ideal of w.r.t the aftersets.
Proof. Suppose that is a soft left ideal of , then for any . Let . Then, and . So, there exists and . Thus, , , , and since is a SCPR. Therefore, ; that is, . Also, (as is a soft left ideal). So, . Hence, .
Now, let such that and . So, . Since , so there exist . Thus, and . Since is a soft -complete relation, therefore, . This implies that , for some and . Thus, and imply (as is ideal). So, . Hence, .
Let and . Then, . So, there exist . Thus, and . Since is a soft left ideal so, for any . This implies that . So, ; that is, . So, . Hence, . Similarly, we can show that .
Theorem 8. Let be a SCTR over w.r.t the aftersets. If is a soft prime ideal of , then is a prime ideal of w.r.t the aftersets.
Proof. Assume that is a soft prime ideal of , then for any . Then, by Theorem 5, is generalized upper soft ideal of . Let such that . Then, So, there exist . This implies that and . Since is a SCTR, . Thus, for some and . Thus, and is a soft prime ideal of so, or . Thus, or . Hence, or .
Theorem 9. Let be a SCTR over w.r.t the aftersets. If is a soft semiprime ideal of , then is a semiprime ideal of w.r.t the aftersets.
Proof. Assume that is a soft semiprime ideal of , then for any . Then, by Theorem 5, is generalized upper soft ideal of . Let such that . Then, . So, there exist . This implies that and . Since is a SCTR, . Thus, for some . Thus, and is a soft semiprime ideal of so, . Thus, . Hence, .
Theorem 10. Let be a SCTR over w.r.t the aftersets. If is a soft primary ideal of , then is a primary ideal of w.r.t the aftersets.
Proof. Assume that is a soft primary ideal of , then for any . Then, by Theorem 5, is generalized upper soft ideal of . Let such that and . Then, . So, there exist . This implies that and . Since is a SCTR, . Thus, for some and . Thus, and is a soft primary ideal of so for some . Also, for . Thus, . Hence, .
Remark 2. In general, the converse of the above theorem is not true. We will present examples to justify our claim as follows.
Example 2. Let and be two complete lattices described in Figures 2 and 3, respectively.
We define and the associative binary operation on and , respectively, as shown in Tables 2 and 3. Then, and are quantales.(1)Let and define SBR over by the rule Then, is SCPR. The aftersets with respect to and are given as follows: Define soft set over by the rule Then, is not a soft subquantale of . But and are subquantale of . So is a of w.r.t the aftersets. Foresets with respect to and are given as follows: Define soft set over by the rule Then, is not a soft subquantale of . But and are subquantale of . So, is a subquantale of w.r.t the foresets.(2)Now, let and define SBR over by the ruleAftersets with respect to and are given as follows:Then, is -complete relation over w.r.t the aftersets. Define soft set over by the ruleThen, is not a soft ideal of . But and are ideal of . So, is a ideal of w.r.t the aftersets.
Now, define SBR over by the ruleForesets with respect to and are given as follows:Then, is soft -complete relation over w.r.t the foresets. Define soft set over by the ruleThen, is not a soft ideal of . But and are ideal of . So, is a ideal of w.r.t the foresets.
Similar examples can be presented to justify that converse of Theorems 11 to 13 is not true.
Definition 13. Let be a SBR over . Consider the soft set over , if is a soft subquantale of , then is called generalized lower soft () subquantale of w.r.t the aftersets. If is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of , then is called ideal (prime ideal, semiprime ideal, and primary ideal) of w.r.t the aftersets.
Definition 14. Let be a SBR over . Consider the soft set over , if is a soft subquantale of , then is called subquantale of w.r.t the foresets. If is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of , then is called ideal (prime ideal, semiprime ideal, and primary ideal) of w.r.t the foresets.
Theorem 11. Let be a SCTR over w.r.t the aftersets. If is a soft subquantale of , then is a subquantale of w.r.t the aftersets.
Proof. Suppose that is a soft subquantale of and for any . Let . Then, . Since is a SCTR, therefore, . Hence, .
Now, let . Then, and . Since is a SCTR and is a soft subquantale, therefore, implies . Hence, .
With the same arguments, next Theorem 12 can be achieved.
Theorem 12. Let be a SCTR over w.r.t the foresets. If is a soft subquantale of , then is a subquantale of w.r.t the foresets.
Theorem 13. Let be a SCTR over w.r.t the aftersets. If is a soft ideal of , then is a ideal of w.r.t the aftersets.
Proof. Suppose that is a soft ideal of and for any . Let . Then, and . Since is a SCTR and is a soft ideal of so ; that is, Hence, .
Now, let such that and . So, . Let, and . So, , that is, . Since is ideal so implies . Thus, . Hence, .
Now, let and . Then, . Consider since is a SCTR so . Thus, for some and . But so and is a soft ideal of ; therefore, , that is, . Thus, . Hence, . Similarly, we can show that .
Theorem 14. Let be a SCTR over w.r.t the aftersets. If is a soft prime ideal of , then is a prime ideal of w.r.t the aftersets.
Proof. Assume that is a soft prime ideal of and for any . Then, by Theorem 4.19, is ideal of . Let such that . Then, . Consider . Since is a SCTR, . Thus, for some and . This implies that . As is a soft prime ideal so, or . Thus, or . Hence, or .
Theorem 15. Let be a SCTR over w.r.t the aftersets. If is a soft semiprime ideal of , then is a semiprime ideal of w.r.t the aftersets.
Proof. Assume that is a soft semiprime ideal of and for any . Then, by Theorem 14, is ideal of . Let such that . Then, . Let . As is a SCTR so . Since is a soft semiprime ideal, implies . Thus, . Hence, .
Theorem 16. Let be a SCTR over w.r.t the aftersets. If is a soft primary ideal of , then is a primary ideal of w.r.t the aftersets.
Proof. Suppose that is a soft primary ideal of and for any . Then, by Theorem 4.19., is a ideal of . Let such that and . Then, . Let . Since is a SCTR, . Thus, for some and . Thus, for some . Also, . As is a soft primary ideal, and . Thus, . Hence, for some .
Remark 3. One can find examples like Example 2 to show that converse of Theorems 11 to 16 is not true.
5. Relationship between Soft Quantale Homomorphism and Their Approximation
In this section, we define soft weak quantale homomorphism (SWQH), and then, we established the relationship between homomorphic images and their approximation by SBR.
Definition 15 (see [4]). A function is called weak quantale homomorphism (WQH) if and , where and are quantales. If is one-one, then is monomorphism. If is onto, then is called epimorphism, and if is bijective, then is called isomorphism between and .
Definition 16. Let be a soft quantale over and be a soft quantale over . Then, is said to soft weak homomorphic to if there exist ordered pair of functions satisfies the following(1) is onto WQH, that is, and (2) is surjective(3), The ordered pair of functions is SWQH. If in ordered pair both and are one-to-one functions, then is said to soft weak isomorphic to and is called SWQI.
Lemma 1. Let be soft weak homomorphic to with SWQH . Let be a SBR over and . Define be a SBR over . Then, the following holds:(1) is SCPR if is SCPR(2)If is SWQI and is SCPR w.r.t the aftersets (w.r.t the foresets), then is SCPR w.r.t the aftersets (w.r.t the foresets)(3)(4) and if is SWQI, then (5)Let be a SWQI. Then, and
Proof. (1)and (2) are obvious(3)Suppose and for any , for some . Then, there exist such that and . Thus, . So, and . Thus, , that is, . Also, . So, . This implies that . Hence, . Now, let . Then, . This implies that . Thus, and . This implies that there exists and such that and . So, . This implies that . So, . Thus, . So, . Hence, . Consequently, .(4)Suppose and for any , for some . Then, there exist such that and . Thus, . Let . Then, there exist such that . So, , that is, . So, , that is, . Thus, . So, . Thus, . Hence, . Now, let . Then, there exist unique such that and . Let , that is, . Then, . Then, . So, . This implies that . So, . Then, . So, . Hence, . Consequently, .(6)Let for any . Then, . Conversely, suppose that . As is bijection so . Similarly, we can show that .
Remark 4. With a similar technique, Lemma 1 can be proved but for the foresets.
Theorem 17. Let be soft weak isomorphic to with SWQI . Let be a SCPR over and . Define for any . Then, the following holds:(1) is an ideal of iff is an ideal of for all (2) is a subquantale of iff is a subquantale of for all (3) is a prime ideal of iff is a prime ideal of for all (4) is a semiprime ideal of iff is a semiprime ideal of for all (5) is a primary ideal of iff is a primary ideal of for all
Proof. (1)Let be an ideal of for any . We will show that is an ideal of . By Lemma 1 (3), we have Let . Then, there exist such that and . Since is ideal and is SWQI so .
Now, let such that and . Then, there exist and such that and . So, implies . This implies that . This implies that and are ideal so . Thus, .
Finally, let and . Then, there exist and such that and . Since ideal, . Thus, . Similarly, . Hence, is ideal of .
Conversely, suppose that = be an ideal of for any . We will show that is ideal of .
Let . Then, . Since is ideal so . Then, by Lemma 5.2(5), .
Now, let such that and . Then, . Thus, . This implies that . Since is ideal . Then, by Lemma 5.2(5), . Finally, let and . Then, and . Since is ideal, , that is, . Thus, . In a similar way, we can show that . This completes the proof.
The proof of (2)–(5) is similar to the proof of (1).
Remark 5. Theorem 17with a similar technique can be proved but for the foresets.
With the same arguments, the next Theorem 18 can be achieved.
Theorem 18. Let be soft weak isomorphic to with SWQI . Let be a SCTR over and . Define for any . Then, the following holds:(1) is an ideal of iff is an ideal of for all (2) is a subquantale of iff is a subquantale of for all (3) is a prime ideal of iff is a prime ideal of for all (4) is a semiprime ideal of iff is a semiprime ideal of for all (5) is a primary ideal of iff is a primary ideal of for all
6. Comparison
Yang and Xu [7] introduced rough approximations in quantale which is a kind of partially ordered algebraic structure with an associative binary operation. The main idea of work in [7] is based on equivalence relation equipped with congruence relation in quantale. In fact, the generalization of Pawlak’s space is discussed in [7]. Further approximation of fuzzy substructures of quantale in crisp atmospheric space was discussed in [4]. Sometimes, it is difficult to find out an equivalence relation and then congruence while finding rough substructures in quantale. To remove this hurdle, soft binary relations are utilized in this paper. Since suitable soft binary relations are easy to find out, it is an easy approach to apply soft rough properties to approach different characterizations of soft rough structures in quantale with the help of aftersets and foresets.
7. Conclusion
The new combined effect of an algebraic structure quantale with rough and soft sets is presented by using soft binary relation, in this paper. The soft substructures of quantales like soft subquantale and soft ideal are discussed. The approximation w.r.t aftersets and foresets of these substructures by SBR which is an extended notion of Pawlak’s rough approximation space are presented. The more generalized version of approximation space implied from SBR over is employed. This new relation over enables us to use the concept of aftersets and foresets to express the lower and upper approximation. Important results regarding to the approximation of soft substructures of quantales under SBR with some essential algebraic conditions such as compatible and complete relations are introduced. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Crux of these results is that whenever we approximate a soft algebraic structure of quantale, corresponding upper and lower approximations, are again the same kind of soft algebraic structure. Furthermore, we presented the soft quantale homomorphism and established the relationship of soft homomorphic images with their approximation under SBR.
In future, one can use this work and generalize it to different soft algebraic structures such as soft quantale modules, soft hypergroups, soft hyperquantales, and soft hyperrings. One can take motivation from our generalized approximation space and define new approximation spaces.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.