Abstract
This research article offers a study on a new relation of rough sets and soft sets with an algebraic structure quantale module by using soft reflexive and soft compatible relations. The lower approximation and upper approximation of subsets of quantale module are utilized by aftersets and foresets. As a sequel of this relation, different characterizations of rough soft substructures of quantale modules are obtained. To ensure the results, soft reflective and soft compatible relations are focused and these are interpreted by aftersets and foresets. Furthermore, the algebraic relations between upper (lower) approximation of substructures of quantale module and the upper (lower) approximations of their homomorphic images with the help of quantale module 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
The quantale module has piqued the interest of many researchers since it was first proposed by Abramsky and Vickers [1]. The concept of a quantale module was inspired by the concept of module over a ring. Rings are replaced by quantales, while abelian groups are replaced by complete lattices. For the first time, the concept of quantale module appeared out of nowhere as the central concept in Abramsky and Vickers' unified treatment of process semantics. Mulvey [2] proposed the Quantale theory. It is defined on the basis of a complete lattice as an algebraic structure.
Pawlak developed the famous rough set theory [3], which deals with inadequate knowledge. The rough set deals with the categorization and investigation of inadequate information and knowledge. After Pawlak’s work, some contributions and a new view on rough set theory were suggested by Zhu [4]. In [5], some properties and characterization of generalized rough sets were presented by Ali et al. Rough sets are now used in a variety of fields, including cognitive sciences, machine learning, pattern recognition, and process control.
Rough sets theory was brought to algebraic structures and soft algebraic structures by a number of authors. Iwinski explored rough set algebraic characteristics [6]. In Q-module [7], Qurashi and Shabir presented the concept of roughness. Xiao and Li [8] proposed the concept of generalized rough quantales (subquantales). Yang and Xu examined rough ideals (prime, semi prime) in quantales [9]. Luo and Wang [10] introduced fuzzy ideals and its type in quantales. Generalized roughness of fuzzy substructures in quantale based on soft relation was studied by Qurashi et al. [11]. Topological structures of lower and upper rough subsets in a hyperring were introduced by Abughazalah et al. [12]. In [13], criteria selection and decision making of hotels using dominance-based rough set theory were presented. Approximations of substructures in partially ordered LA-semihypergroups were presented by Yaqoob and Tang [14]. In [15], roughness of bipolar soft sets and their related applications are discussed. In [16], Feng et al. presented the relationship between soft and rough sets and proposed rough soft sets and soft rough sets. Integrated Best-Worst Method in terms of Green supplier selection based on the information system performance was suggested by Fazlollahtabar and Kazemitash [17].
Many issues emerge in different fields such as engineering, economics, and social sciences where data have some degree of ambiguity. Because well-known mathematical tools are designed for certain situations, they have numerous restrictions. Many theories exist to deal with uncertainty, such as fuzzy set theory, probability theory, rough sets, and ambiguous sets, but they are constrained by their design.
Molodtsov introduced the concept of soft set [18], which is a mathematical tool for overcoming the problems that plague the above theories. Soft set theory is a general mathematical technique for dealing with items that are unclear, imprecise, or not precisely defined. Many authors offer different set operations and attempt to unify the algebraic aspects of soft sets like Maji et al. [19]. A new and different idea of operations was presented by Ali et al. [20]. Soft sets and algebraic structures were combined in various ways by researchers like soft intersection semigroups [21]. Soft linear programming and applications of soft vector spaces were presented in [22]. Khan et al. applied uni-soft structures to ordered -Semihypergroups [23]. Complex intuitionistic fuzzy algebraic structures in groups were introduced by Gulzar et al. [24]. Development of a rough-MABAC-DoE-based metamodel for supplier selection in an iron and steel industry was introduced by Chattopadhyay et al. [25].
The central theme and objective of soft sets is to capture the essence of parametrization, which has been adapted to the creation of soft binary relations (SBRs), which are a parameterized collection of binary relations on a universe of interest. This mentioned the problem of complicated objects that can be interpreted differently from different perspectives.
By using aftersets and foresets and then notions associated to soft binary relation (SBR), a new method of 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 [26]. Motivated by the idea in [26], soft substructures in quantale module 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 approximation of soft subsets of quantale, further soft substructures are employed for further characterization.
A new generalized approximation space is commonly used these days by utilizing the aftersets and foresets notions related with soft binary relations. Kanawal and Shabir [26] approximated different soft substructures in semigroups using a generalized approximation space based on soft binary relations. Roughness of intuitionistic fuzzy sets by soft relations was discussed by Anwar et al. [27]. Roughness of Pythagorean fuzzy sets based on soft binary relations was proposed in [28] by Bilal and Shabir. Using soft relations, soft substructures were defined by Zhou et al. [29] and these were approximated by soft relations. Soft substructures in quantale modules are defined in this paper, and aftersets and foresets are used to construct the lower and upper approximation of substructures, respectively.
The following scheme is for the remainder of the paper. Section 2 connects some key explanations about quantale modules, their substructures, soft substructures, and their relevant sequels. Section 3 discusses the concept of crisp sub sets approximations over quantale module created by soft binary relations. In Section 4, generalized soft substructures are defined and further fundamental algebraic properties of these phenomena are investigated utilizing these ideas. In Section 5, we also extend this research by defining the relationship between homomorphic images of substructures in quantale module and their approximation by soft binary relations.
2. Preliminaries
In this section, we will review some fundamental concepts related to quantale module and its substructures, soft sets, and rough sets.
2.1. Definition (see [2])
A quantale is a complete lattice equipped with an associative, binary operation distributing over an arbitrary joins. That is for any It holds and .
Let . Then, the followings are defined:
Throughout the paper, quantales are denoted by .
2.2. Definition (see [1])
Let be a quantale and be a lattice equipped with a left action Then, M is called left module over the quantale if for any , , , we have
Right quantale modules can be defined in the same way. For the rest of the paper, -module M will stand for a left quantale module over the quantale . The symbol will denote the top element and will stand for the bottom one for quantale module, unless stated otherwise.
2.3. Example
The following are the examples of -modules M:(1)Let be a complete lattice where 0 is the bottom element and 1 is the top element of , as shown in Figure 1 and the operation on is shown in Table 1. Then, it is straightforward to verify that is a quantale. Let be a lattice. The order relation of M is given in Figure 2 and let be the left action on M as shown in Table 2. Then, it is straightforward that M is a -module.(2)Every quantale is certainly a -module over .
2.4. Definition (see [1])
Let be a -module. A subset is called a sub--module of if for any , , , it holds that and .
2.5. Definition (see [1])
Let be a -module and Then, I is a -module ideal of .(1)If , then (2) and implies (3) implies
2.6. Definition (see [1])
Let M be a -module. A binary relation on M is called congruence on M if it is an equivalence relation on M; for any given, , and , it satisfies the following conditions: for all impliesand implies.
2.7. Definition (see [1])
Let and be two -modules. A map is a - module homomorphism if it is a sup-lattice homomorphism which also preserves scalar multiplication. That is,for any , ,
A -module homomorphism is called an epimorphism if is onto and is called a monomorphism if is one-one. It is an isomorphism, if is bijective.
2.8. Theorem (see [1])
Let and be two -modules. If is a -module homomorphism, then is a congruence of -modules.
2.9. Definition (see [18])
A pair is called soft set over if where is a subset of (the set of parameters).
2.10. Definition (see [20])
Let and be two soft sets over . Then, soft subset if the following conditions are fulfilled:(1)(2),
2.11. Definition (see [30])
Let be a soft set over , i.e., . Then, is called a soft binary relation (SBYR) over
2.12. Definition
Let be a soft set over quantale module . Then,(1) is called a soft sub--module over iff is a sub--module of (2) is a called soft -module ideal over iff is -module ideal of ,
2.13. Definition (see [3])
Let be a finite set and be an equivalence relation on Let denote the equivalence class of the relation containing . If a subset of is expressed as a union of equivalence classes of then that is said to be definable set in Let a subset of cannot be expressed as a union of equivalence classes of Then, we say it is undefinable set. However, we can approximate that undefinable set by two definable sets in . The first one is called -lower approximation of , and the second is called -upper approximation of . They are defined as follows:
A rough set is the pair ; if , then is definable.
3. Approximation of Subsets of Quantale Module by Soft Binary Relation
In this section, applications of soft relation on quantale module are discussed. A subset of quantale module M can be approximated by soft relations in two ways. Aftersets and foresets are applied to approximate a subset of M. Two sets named as soft set corresponding to each subset are called the lower approximation () and the upper approximation () with respect to the aftersets and foresets, respectively.
3.1. Definition (see [11])
Let . Then, is a soft binary relation on a set M, where (set of parameters). For and of with respect to aftersets are basically the two soft sets over defined as follows:
Further, and of S with respect to foreset are basically the two soft sets over M, defined as follows: and .
For all , is called afterset of k and is called foreset of k. Moreover, and are defined as for aftersets. for foresets for each . Generally, , , and . However, they are equal if is a symmetric relation. This is justified in the next example.
3.2. Example
Let and . Define by and
Thus, the aftersets of elements M are as follows.
, , , and , , , and , and the foresets of elements of M are as follows: , , , and , , , . Let . Then, and of with respect to aftersets and foresets are as follows: , and , , , and , . This shows that and .
3.3. Definition
A SBR on a quantale module M is called soft compatible relation (SCRE), if it satisfies the following conditions: if and for any and .
3.4. Example
Let be a complete lattice as shown in Figure 3, and the operation on is . Then, it is easy to verify that is a quantale. Let be the left action of on as shown in Table 3. In this case, . Then, it is easy to check that is a -module over and represented by M. Let . Define by
Then, is a soft compatible relation (SCRE) and soft reflexive relation (SRRE) on M.
3.5. Remark
Let be a SCRE on a -module M. Then, it is easily verified that and for all and .
3.6. Example
Let be a -module M as given in example 3.4 and let . Define by and . Then, is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows:
Thus, we have
Hence, Also, similarly we can check that .
3.7. Remark
If is a SCRE on a -module M, then and with respect to foresets.
3.8. Definition
A SCRE on a -module M is called soft join-complete with respect to to aftersets if and is called soft complete with respect to if
A SCRE which is both join-complete and -complete with respect to aftersets is called soft complete relation (SCTR) with respect to aftersets.
3.9. Example
Let be a -module M as given in example 3.4 and let . Define by
Then, is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows: . It is easily checked that That is, etc. Further, we can check that .
So, is a SCTR with respect to aftersets.
3.10. Definition
A SCRE on a -module M is called soft join-complete with respect to foresets if and is called soft complete with respect to if .
A SCRE which is both join-complete and -complete is called SCTR with respect to foresets.
3.11. Remark
It has been observed that if we have SCTR for aftersets, not need it is SCTR for foresets. This is demonstrated in the following example.
3.12. Example
Let be a -module M as given in example 3.4 and let . Define by and . Then, is a SCRE and SRRE on M. The aftersets and foresets calculated by the elements of M are as follows: , . It is observed that . Likewise, we can check that So, is a SCTR with respect to aftersets. But, So, is not a SCTR with respect to foresets.
In ref [31], the following theorems are helpful for our further study.
3.13. Theorem (see [31])
Let be the subsets of -module M and and be SRRE on M. Then, the following hold for all (1)(2)(3)(4)(5)(6)(7)(8)(9)
3.14. Theorem (see [31])
Let be the subset of -module M and and be SRRE on M. Then, the following hold for all (10)(11)(12)(13)(14)(15)(16)(17)(18)
3.15. Theorem (see [31])
Let be the subsets of -module M and and be SRRE on M. Then, the following hold for all (1)(2)(3)(4)
3.16. Theorem
Let be SRRE and SCRE with respect to the aftersets on a -module M. Then, for non-empty subset S and R of M, we have and .
Proof. Let Then, where such that . Thus, there exist such that That is, . Since is SCRE, we have and , , . So, . Thus,
Let . Then, where and such that and . That is, and So, , and , , and Since is SCRE, we have and , So, . Thus, .
3.17. Example
Let be a -module M as given in example 3.4 and let . Define by and . Then, is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows: , Let and . Then, , and . Now, and . So, . Similarly, we can prove that .
3.18. Theorem
Let be SRRE and SCRE with respect to the foresets on a -module M. Then, for non-empty subset S and R of M, we have and .
Proof. The proof is similar to the proof of Theorem 3.16.
4. Approximation of Substructures in Quantale Module
In this section, foresets and aftersets are applied to different type of substructures in quantale module through soft relations to discuss their lower and upper approximations. These are then characterized by soft reflexive and soft compatible relations to present different characteristics of them.
4.1. Definition
Let and be a SBR on a -module M. Then, S is said to be generalized upper soft sub -module of M with respect to aftersets if is a soft sub--module of M.
4.2. Theorem
Let be a SRRE and SCRE on a -module M. Then, S is said to be generalized upper soft sub -module of M with respect to aftersets if S is a soft sub--module of M.
Proof. (1)Suppose that for and for some . Then, . There are such that and . That is, and , Since S is a sub--module of M and is a SCRE Thus, we have and . That is, and . So, . Hence, . This shows that .(2)Let such that . This shows . Then, there is such that and . That is, and . Since S is a sub--module of M and is a SRRE and SCRE. Thus, we have and , and , . Hence, . This shows that . Therefore, is a sub -module of M. That is, S is a generalized upper soft sub--module of M with respect to aftersets.It is mentioned in the next example that the converse is not true.
4.3. Example
Suppose be a complete lattice as shown in Figure 4 and the operation on is that . Then, it is easily checked that is a quantale. Suppose be the left action of on as shown in Table 4. In this case, ; then, it is easy to check that is a quantale module M. Let . Then, defined by and . Then, is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows:
Let . Then, S is not a sub--module of M. But, and are sub--module of M. Hence, S is a generalized upper soft sub--module of M with respect to aftersets.
4.4. Definition
Let and be a SBR on a -module M. Then, S is said to be generalized upper soft sub -module of M with respect to foresets if is a soft sub--module of M.
4.5. Theorem
Let be a SRRE and SCRE on a -module M. Then, S is said to be generalized upper soft sub -module of M with respect to foresets if S is a soft sub--module of M.
Proof. The proof is similar to the proof of Theorem 4.2.
4.6. Definition
Let and be a SRRE and SCRE on a -module M. Then, S is said to be generalized lower soft sub--module of M with respect to aftersets if is a soft sub--module of M.
4.7. Theorem
Let be a SRRE and SCTE on a -module M. Then, S is said to be generalized lower soft sub -module of M with respect to aftersets if S is a soft sub--module of M.
Proof. (1)Suppose that for and for . This shows that . Hence, S is a sub--module of M and is SCTR Thus, and . Hence, . This shows that .(2)Let be such that . This shows for . Hence, S is a sub--module of M and is a SRRE and SCTE. Thus, we have . So, we can write . Hence, . This shows that . Therefore, is a sub -module of M. That is, S is a generalized lower soft sub--module of M with respect to aftersets.Observe in the next example that the converse is not true.
4.8. Example
Let be a -module M as given in example 4.3 and let . Define by and . The aftersets calculated by the elements of M are as follows: , . Then, is a SCTR and SRRE with respect to aftersets. Let . Then, S is not a sub--module of M. However, are sub--module of M. Hence, S is generalized lower soft sub--module of M.
4.9. Definition
Let and be a SRRE and SCRE on a -module M. Then, S is said to be generalized lower soft sub--module of M with respect to foresets if is a soft sub--module of M.
4.10. Theorem
Let be a SCTR and SRRE on a -module M. Then, S is said to be generalized lower soft sub -module of M with respect to foresets if S is a soft sub--module of M.
Proof. The proof is obvious.
4.11. Remark
The results in this section related to foresets are similar to the results with respect to aftersets.
4.12. Definition
Let and be a SRRE and SCRE on a -module M if is a soft -module ideal M. Then, S is said to be generalized upper soft -module ideal of M with respect to aftersets.
4.13. Theorem
Let and be a SRRE, SCRE and soft join-complete relation on a -module M. Then, S is said to be generalized upper soft -module ideal of M with respect to aftersets if S is an -module ideal of M.
Proof. (1)Suppose that such that for . Therefore, and . Then ,there are and such that , , and . This implies , , and . S is a -module ideal of M and is a SCRE Hence, we have and . That is, and . So, . Hence, . This shows that .(2)Let and Therefore, . Then, there is such that and . Since is a SCRE and soft join-complete relation, we have . Then, there is and such that S is a -module ideal of M and We have so (3)Let be such that . This shows . Then, there are such that and . That is, and . S is a module ideal of M and is a SRRE and SCRE. We have and and , . Hence, . This shows that . Therefore, is a -module ideal of M. That is, S is a generalized upper soft -module ideal of M with respect to aftersets.It is observed in the next example that the converse is not true.
4.14. Example
Let be the quantale module as given in example 4.3 and . Then, and .
The aftersets calculated by the elements of M are as follows:then is a SCRE and SRRE and soft join-complete relation with respect to aftersets on M. Let . Then, S is not a -module ideal of M. However, and are -module ideal of M. Hence, S is a generalized upper soft -module ideal of M with respect to aftersets.
4.15. Definition
Let and be a SRRE and SCRE on a -module M if is a soft -module ideal M. Then, S is said to be generalized upper soft -module of M with respect to foresets if S is a -module ideal of M.
4.16. Theorem
Let and be a SRRE, SCRE, and soft join-complete relation on a -module M. Then, S is said to be generalized upper soft -module ideal of M with respect to foresets if S is an -module ideal of M.
Proof. The proof is obvious.
4.17. Definition
Let and be a SRRE and SCRE on a -module M if is a soft -module ideal M. Then, S is said to be generalized lower soft -module of M with respect to aftersets.
4.18. Theorem
Let and be a SRRE and SCTR on a -module M. Then, S is said to be generalized lower soft -module ideal of M with respect to aftersets if S is an -module ideal of M.
Proof. (1)Suppose that such that for . This shows and . Hence, S is a -module ideal of M and is SCTR . So, we have and . Hence, . This shows that .(2)Let and Therefore, . For , we have . Since is a SRRE and SCRE, we have . That is, . Thus, . Since S is a -module ideal of M, we have . Thus, and .(3)Let be such that . This shows for . Hence, S is a -module ideal of M. Then, for all , we have . Also given that is a SRRE and SCTE. Thus, we can write . Hence, . This shows that . Therefore, is a -module ideal of M. That is, S is a generalized lower soft -module ideal of M with respect to aftersets.Observe in the next example that the converse is not true.
4.19. Example
Let be a -module M as given in example 3.4 and let . Define by and . The aftersets calculated by the elements of M are as follows: , . Then, is a SCTR and SRRE with respect to aftersets. Let . Then, S is not an ideal of -module of M. However, are -module ideal of M. Hence, S is generalized lower soft -module ideal of M with respect to aftersets.
4.20. Definition
Let and be a SBR on a -module M if is a soft -module ideal M. Then, S is said to be generalized lower soft -module of M with respect to foresets.
4.21. Theorem
Let and be a SRRE and SCTR on a -module M. Then, S is said to be generalized lower soft -module ideal of M with respect to foresets if S is an -module ideal of M.
Proof. The proof is simple.
5. Homomorphic Images of Generalized Rough Soft Substructures
The relationship between the upper and lower generalized soft substructures of the - module, as well as the images of upper (lower) approximations under -module homomorphism, is being discussed in this section. Further, we study some properties of these approximations.
5.1. Lemma
Let and be -modules and be a SBR on . Let be a surjective -module homomorphism. Set . Then, the following holds:(1) is a SBR on (2) is SRRE if is SRRE(3) is SCRE if is SCRE
Proof. The proof is obvious.
5.2. Lemma
Let and be a -modules and be a SBR on . Let be a surjective -module homomorphism. Set Then, is SCTR with respect to aftersets if is SCTR with respect to aftersets and is one-one.
Proof. Clearly .
Conversely, suppose that . Then, by definition of aftersets . Thus, , that is, . This implies that . Since is SCTR with respect to aftersets, . Then, there is and such that . As is onto, we have , and . Thus, . This implies, . As is one-one, we have . Now, , , , , , . Also now, , , , , , . Now, . That is, . This implies that . Hence, . Similarly, we can prove that . This show that is SCTR with respect to aftersets.
5.3. Lemma
Let and be a -modules and be a SBR on . Let be a surjective -module homomorphism. Set . Then, is SCTR with respect to foresets if is SCTR with respect to foresets and is one-one.
Proof. The proof is obvious.
5.4. Lemma
Let and be -modules and be a SBR on . Let be a surjective -module homomorphism. Set Then, for and
Proof. Let . Then, . Then, there is such that and . So, and . Since is onto, there exist and such that and . Thus, , that is, (. Hence, , that is, and . This implies . So, we have , , . This show that .
Conversely, let . Then, there is such that and . Thus, such that and . So, and , and (, and . Hence, , that is, , , . This shows that . Hence, .
5.5. Lemma
Let and be -modules and be a SBR on . Let be a surjective -module homomorphism. Set . Then, the following holds:(1); if is one-one, then (2)If is one-one, then if and only if
Proof. (1)Let for . Then, there is such that and . Suppose that . Then, there is such that and , that is, (. Hence, by definition of aftersets . Hence, and . So, using above , . Thus, . Hence, which shows that . Conversely, let Then, . Suppose that is one-one. Then, there is such that and . Let by definition of aftersets, we have . Then, , that is, . Since , , . Thus, , that is, , since . Hence, . So,. Thus, .(2)Let . Then, there is such that . Since is one-one, we get . Thus, . Conversely, let . Then, .
5.6. Theorem
Let and let be a surjective -module homomorphism on a -modules and be a SBR and SCTR with respect to aftersets on . Set . Then, the following holds:(1) is a -module ideal of if and only if is a -module ideal of (2) is a sub--module of if and only if is a sub--module of
Proof. (1) Let be a -module ideal of . Then, we have to show that is a -module ideal of By Lemma 5.4, we have for all and .
Let Then, there is such that . Hence, is a -module homomorphism and is a -module ideal of So, we have , . Thus, . Hence, .
Let and . Then, there is and such that and . Hence, , that is, and . Further, is a -module homomorphism. Thus, we have , , . By Lemma 5.5(2), we have . Hence, is a lower set and Thus, and so . Hence, is a lower set.
Now, we show that for all and . Let and . So, . Then, by Lemma 5.5(2), we have . Hence, is an -module ideal of . Thus, . Thus, . Hence, is an -module homomorphism. Thus, , that is, . Hence, . Thus, is a -module ideal of
Conversely, suppose is a -module ideal of
Suppose for some Then, . Hence, is a -module ideal and is a -module homomorphism. So, , Then, by Lemma 5.5(2), we have and is directed.
Let and . Then, and . Since is a lower set, we have . Thus, by Lemma 5.5(2), we have . So, is a lower set. Suppose and Then, and . Since is an -module ideal of , we have , and then by Lemma 5.5(2), we have . By the above discussion, we have is -module ideal of (2)Let is a sub--module of Then, we have to show that is sub--module of By Lemma 5.4, we have for all and Let Then, there is such that . Hence, is a -module homomorphism and is a sub--module of , So, , . So, we have . Hence, .
Now, we show that for all and . Let and So, . Then, by Lemma 5.5(2), we have . Hence, is a sub--module of . This shows that . Thus, . Hence, is an -module homomorphism. Thus, , that is, . Hence, . Thus, we have that is a sub--module of
Conversely, suppose is a sub--module of . Suppose, for some Then, . Hence, is a sub--module and is a -module homomorphism. So, , Then, by Lemma 5.5(2), we have .
Suppose and Then, and Since is an sub--module ideal of , we have and then by Lemma 5.5(2) we have . From the above discussion, we get is sub--module of
5.7. Remark
With similar arguments, Theorem 5.6 can be similarly proved but for the foresets.
5.8. Theorem
Let and let be a surjective -module homomorphism on a -module M and be a SBR and SCTR and with respect to aftersets on . Set . Then, the following holds:(1) is a -module ideal of if and only if is a -module ideal of (2) is a sub--module of if and only if is a sub--module of
Proof. The proof is similar in view of Theorem 5.6.
6. Comparison
Qurashi and Shabir presented the roughness in quantale modules with the help of congruence relation [7]. Furthermore, generalized roughness of fuzzy substructures in quantale with respect to soft relations in quantale was defined in [11]. It is clear that equivalence relation is a hurdle while evaluating roughness. In order to avoid this this hurdle, soft binary relations are presented in this paper. Since suitable soft binary relations are easy to find out, it is an easy approach to observe soft rough properties to discuss different characterizations of soft rough substructures in quantale modules with the help of aftersets and foresets. Different characterization of soft substructures in semigroups and their approximation based on soft relation was discussed by Kanwal and Shabir [26]. We are actually motivated from the paper roughness in quantale module and taken help from [11] to develop the idea of this paper.
7. Conclusion
In this paper, we have suggested a new relation of substructures of quantale module with rough set and soft sets. The properties of rough soft substructures in quantale module are discussed for the first time. On the one hand, we have presented different characterizations for soft relations to approach quantale module subsets, as well as the use of aftersets and foresets in this regard. Structural features of soft relations under aftersets and foresets are discussed. Furthermore, in the quantale module, foresets and aftersets are applied to various types of substructures using soft relations to explore their lower and upper approximations. The following work can be done in future:(1)Soft relations applied to the fuzzy substructures of quantale module and their approximations(2)Some studies of soft substructures of quantale module and their approximations
Data Availability
The paper includes the information used to verify the study’s findings.
Conflicts of Interest
The authors state that they have no conflicts of interest.
Authors’ Contributions
All authors have contributed equally to this paper in all aspects. All authors have read and agreed to the published version of the manuscript.