# Soft Relations Applied to the Substructures of Quantale Module and Their Approximation

Authors: Saqib Mazher Qurashi; Khushboo Zahra Gilani; Muhammad Shabir; Muhammad Gulzar; Ashraful Alam
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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 Γ-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. Section2 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 quantaleKd is a complete lattice equipped with an associative, binary operation ⊛ distributing over an arbitrary joins. That is for any r∈Kd,ri,si∈Kd,i∈I. It holdsr⊛∨i∈Isi=∨i∈Ir⊛si and viɛIri⊗s=viɛiri⊗s.LetXi,X,Y⊆Kdi∈I. Then, the followings are defined:(1)X⊛Y=x⊛y|x∈X,y∈Y;X∨Y=x∨y|x∈X,y∈Y;∨i∈IXi=∨i∈Ixi|xi∈Xi.Throughout the paper, quantales are denoted byKd.


### 2.2. Definition (see [1])

LetKd be a quantale and M be a Sup− lattice equipped with a left action ⊕:Kd×M⟶M. Then, M is called left Kd−module over the quantale Kd if for any ai,a,b∈Kd,x∈M,xj⊆M, i∈I, j∈J, we have(2)∨i∈Iai⊕x=∨i∈Iai⊕x;Right quantale modules can be defined in the same way. For the rest of the paper,Kd-module M will stand for a left quantale module over the quantale Kd. The symbol T 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 ofKd-modules M:(1)
LetKd=0,r,s,1 be a complete lattice where 0 is the bottom element and 1 is the top element of Kd, as shown in Figure 1 and the operation ⊛ on Kd is shown in Table 1. Then, it is straightforward to verify that Kd,⊛ is a quantale. Let M=⊥,x,T be a Sup− lattice. The order relation of M is given in Figure 2 and let ⊕:Kd×M⟶M be the left action on M as shown in Table 2. Then, it is straightforward that M is a Kd-module.(2)
Every quantaleKd is certainly a Kd-module over Kd.Figure 1
Illustration ofkd.Table 1
Binary operation subject to ⊗.
⊗0rsI00000r0r0rs00ssI0rsIFigure 2
Illustration of M.Table 2
Binary opreation subject to⊕.
⊕⊥xT0⊥⊥⊥r⊥xTs⊥xxI⊥xT


### 2.4. Definition (see [1])

LetM be a Kd-module. A subset M1⊆M is called a sub-Kd-module of M if for any r∈M1, si⊆M1, k∈Kd, it holds that ∨i∈Isi∈M1 and k⊕r∈M1.


### 2.5. Definition (see [1])

LetM be a Kd-module and ∅≠I⊆M. Then, I is a Kd-module ideal of M.(1)
Ifri∈Ii∈I, then ∨i∈Iri∈I(2)
r∈I and c≤r implies c∈I(3)
r∈I implies a⊕r∈I,∀a∈Kd


### 2.6. Definition (see [1])

LetM be a Kd-module. A binary relation Γ on M is called congruence on M if it is an equivalence relation on M; for any givenmi, ni⊆M,m,nɛM, and q∈Kd, it satisfies the following conditions: for alliɛI,miΓniimpliesΔi∈ImiΓΔi∈IniandmΓn impliesqΔmΓqΔn.


### 2.7. Definition (see [1])

LetM1 and M2 be two Kd-modules. A map Ω:M1⟶M2 is a Kd- module homomorphism if it is a sup-lattice homomorphism which also preserves scalar multiplication. That is,(3)Ω∨i∈Iri=∨i∈IΩri,Ωb⊕r=b⊕Ωr,for any b∈Kd, r∈M,, ri⊆Mi∈IAKd-module homomorphism Ω:M1⟶M2 is called an epimorphism if Ω is onto M2 and Ω is called a monomorphism if Ω is one-one. It is an isomorphism, if Ω is bijective.


### 2.8. Theorem (see [1])

LetM1 and M2 be two Kd-modules. If Ω:M1⟶M2 is a Kd-module homomorphism, then KerΩ=a,b∈M1×M1:Ωa=Ωb is a congruence of Kd-modules.


### 2.9. Definition (see [18])

A pairF,C is called soft set over M if F:C⟶PM where C is a subset of E (the set of parameters).


### 2.10. Definition (see [20])

LetF,C1 and H,C2 be two soft sets over M. Then, F,C1 soft subset H,C2 if the following conditions are fulfilled:(1)
C1⊆C2(2)
Fc⊆Hc, ∀c∈C1


### 2.11. Definition (see [30])

LetΓ,C be a soft set over M×M, i.e., Γ:C⟶PM×M. Then, Γ,C is called a soft binary relation (SBYR) over M.


### 2.12. Definition

LetΓ,C be a soft set over quantale module M. Then,(1)
Γ,C is called a soft sub-Kd-module over M iff Γc is a sub-Kd-module of M∀c∈C(2)
Γ,C is a called soft Kd-module ideal over M iff Γc is Kd-module ideal of M, ∀c∈C


### 2.13. Definition (see [3])

LetM≠∅ be a finite set and Γ be an equivalence relation on M. Let tΓ denote the equivalence class of the relation Γ containing t∈M. If a subset of M is expressed as a union of equivalence classes of M, then that is said to be definable set in M. Let a subset R of M cannot be expressed as a union of equivalence classes of M. Then, we say it is undefinable set. However, we can approximate that undefinable set by two definable sets in M. The first one is called Γ-lower approximation Γ−LAP of R, and the second is called Γ-upper approximation Γ−UAP of R. They are defined as follows:(4)Γ¯R=t∈M:tΓ⊆MandΓ¯R=t∈M:tΓ∩M≠∅.A rough set is the pairΓ¯R,Γ¯R; if Γ¯R=Γ¯R, then R is definable.


## 2.1. Definition (see [2])

A quantaleKd is a complete lattice equipped with an associative, binary operation ⊛ distributing over an arbitrary joins. That is for any r∈Kd,ri,si∈Kd,i∈I. It holdsr⊛∨i∈Isi=∨i∈Ir⊛si and viɛIri⊗s=viɛiri⊗s.LetXi,X,Y⊆Kdi∈I. Then, the followings are defined:(1)X⊛Y=x⊛y|x∈X,y∈Y;X∨Y=x∨y|x∈X,y∈Y;∨i∈IXi=∨i∈Ixi|xi∈Xi.Throughout the paper, quantales are denoted byKd.


## 2.2. Definition (see [1])

LetKd be a quantale and M be a Sup− lattice equipped with a left action ⊕:Kd×M⟶M. Then, M is called left Kd−module over the quantale Kd if for any ai,a,b∈Kd,x∈M,xj⊆M, i∈I, j∈J, we have(2)∨i∈Iai⊕x=∨i∈Iai⊕x;Right quantale modules can be defined in the same way. For the rest of the paper,Kd-module M will stand for a left quantale module over the quantale Kd. The symbol T 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 ofKd-modules M:(1)
LetKd=0,r,s,1 be a complete lattice where 0 is the bottom element and 1 is the top element of Kd, as shown in Figure 1 and the operation ⊛ on Kd is shown in Table 1. Then, it is straightforward to verify that Kd,⊛ is a quantale. Let M=⊥,x,T be a Sup− lattice. The order relation of M is given in Figure 2 and let ⊕:Kd×M⟶M be the left action on M as shown in Table 2. Then, it is straightforward that M is a Kd-module.(2)
Every quantaleKd is certainly a Kd-module over Kd.Figure 1
Illustration ofkd.Table 1
Binary operation subject to ⊗.
⊗0rsI00000r0r0rs00ssI0rsIFigure 2
Illustration of M.Table 2
Binary opreation subject to⊕.
⊕⊥xT0⊥⊥⊥r⊥xTs⊥xxI⊥xT


## 2.4. Definition (see [1])

LetM be a Kd-module. A subset M1⊆M is called a sub-Kd-module of M if for any r∈M1, si⊆M1, k∈Kd, it holds that ∨i∈Isi∈M1 and k⊕r∈M1.


## 2.5. Definition (see [1])

LetM be a Kd-module and ∅≠I⊆M. Then, I is a Kd-module ideal of M.(1)
Ifri∈Ii∈I, then ∨i∈Iri∈I(2)
r∈I and c≤r implies c∈I(3)
r∈I implies a⊕r∈I,∀a∈Kd


## 2.6. Definition (see [1])

LetM be a Kd-module. A binary relation Γ on M is called congruence on M if it is an equivalence relation on M; for any givenmi, ni⊆M,m,nɛM, and q∈Kd, it satisfies the following conditions: for alliɛI,miΓniimpliesΔi∈ImiΓΔi∈IniandmΓn impliesqΔmΓqΔn.


## 2.7. Definition (see [1])

LetM1 and M2 be two Kd-modules. A map Ω:M1⟶M2 is a Kd- module homomorphism if it is a sup-lattice homomorphism which also preserves scalar multiplication. That is,(3)Ω∨i∈Iri=∨i∈IΩri,Ωb⊕r=b⊕Ωr,for any b∈Kd, r∈M,, ri⊆Mi∈IAKd-module homomorphism Ω:M1⟶M2 is called an epimorphism if Ω is onto M2 and Ω is called a monomorphism if Ω is one-one. It is an isomorphism, if Ω is bijective.


## 2.8. Theorem (see [1])

LetM1 and M2 be two Kd-modules. If Ω:M1⟶M2 is a Kd-module homomorphism, then KerΩ=a,b∈M1×M1:Ωa=Ωb is a congruence of Kd-modules.


## 2.9. Definition (see [18])

A pairF,C is called soft set over M if F:C⟶PM where C is a subset of E (the set of parameters).


## 2.10. Definition (see [20])

LetF,C1 and H,C2 be two soft sets over M. Then, F,C1 soft subset H,C2 if the following conditions are fulfilled:(1)
C1⊆C2(2)
Fc⊆Hc, ∀c∈C1


## 2.11. Definition (see [30])

LetΓ,C be a soft set over M×M, i.e., Γ:C⟶PM×M. Then, Γ,C is called a soft binary relation (SBYR) over M.


## 2.12. Definition

LetΓ,C be a soft set over quantale module M. Then,(1)
Γ,C is called a soft sub-Kd-module over M iff Γc is a sub-Kd-module of M∀c∈C(2)
Γ,C is a called soft Kd-module ideal over M iff Γc is Kd-module ideal of M, ∀c∈C


## 2.13. Definition (see [3])

LetM≠∅ be a finite set and Γ be an equivalence relation on M. Let tΓ denote the equivalence class of the relation Γ containing t∈M. If a subset of M is expressed as a union of equivalence classes of M, then that is said to be definable set in M. Let a subset R of M cannot be expressed as a union of equivalence classes of M. Then, we say it is undefinable set. However, we can approximate that undefinable set by two definable sets in M. The first one is called Γ-lower approximation Γ−LAP of R, and the second is called Γ-upper approximation Γ−UAP of R. They are defined as follows:(4)Γ¯R=t∈M:tΓ⊆MandΓ¯R=t∈M:tΓ∩M≠∅.A rough set is the pairΓ¯R,Γ¯R; if Γ¯R=Γ¯R, then R 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 moduleM 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 (LAP) and the upper approximation (UAP) with respect to the aftersets and foresets, respectively.


### 3.1. Definition (see [11])

LetΓ: C⟶PM×M. Then, Γ,C is a soft binary relation on a set M, where ϕ≠C⊆E (set of parameters). For ϕ≠S⊆M,Γ¯S,C and Γ¯S,C of S with respect to aftersets are basically the two soft sets over M, defined as follows: (5)Γ¯Su=k∈M|kΓu⊆SandΓ¯Su=k∈M|kΓu∩S≠∅Further,Γ¯S,C and Γ¯S,C of S with respect to foreset are basically the two soft sets over M, defined as follows: Γ¯Su=k∈M|Γuk⊆S and Γ¯Su=k∈M|Γuk∩S≠∅.For allu∈C, kΓu=r∈M:k,r∈Γu is called afterset of k and Γuk=r∈M:r,k∈Γu is called foreset of k. Moreover, LAP and UAP are defined as Γ¯S:C⟶PMandΓ¯S:C⟶PM for aftersets. Γ¯S:C⟶PMandΓ¯S:C⟶PM for foresets for each S∈PM. Generally, kΓu≠Γuk, Γ¯Su≠Γ¯Su, and Γ¯Su≠Γ¯Su. However, they are equal if Γu is a symmetric relation. This is justified in the next example.


### 3.2. Example

LetKd=⊥,r,s,T and C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,T,T,⊥,r and Γu2=⊥,⊥,r,r,s,s,T,T,r,TThus, the aftersets of elementsM are as follows.⊥Γu1=⊥,r, rΓu1=r, sΓu1=s, TΓu1=T and ⊥Γu2=⊥, rΓu2=r,T, sΓu2=s, and TΓu2=T, and the foresets of elements of M are as follows: Γu1⊥=⊥, Γu1r=⊥,r, Γu1s=s, Γu1T=T and Γu2⊥=⊥, Γu2r=r, Γu2s=s, Γu2T=r,T. Let S=⊥,T. Then, LAP and UAP of S⊆M with respect to aftersets and foresets are as follows: Γ¯Su1=T, Γ¯Su2=⊥,T and Γ¯Su1=⊥,T, Γ¯Su2=⊥,r,T, Γ¯Su1=⊥,T, Γ¯Su2=⊥ and Γ¯Su1=⊥,r,T, Γ¯Su2=⊥,T. This shows that Γ¯S,C≠Γ¯S,C and Γ¯S,C≠Γ¯S,C.


### 3.3. Definition

A SBRΓ on a quantale module M is called soft compatible relation (SCRE), if it satisfies the following conditions: ∀i∈I if riΓsi⇒∨i∈IriΓ∨i∈Isi and rΓs⇒k⊕rΓk⊕s for any r,s∈M,ri,si⊆M,i∈I and k∈Kd.


### 3.4. Example

LetKd=⊥,r,s,T be a complete lattice as shown in Figure 3, and the operation ⊛ on Kd is a⊛b=⊥. Then, it is easy to verify that Kd,⊛ is a quantale. Let ⊕:Kd×M⟶M be the left action of Kd on M as shown in Table 3. In this case, M=Kd. Then, it is easy to check that Kd is a Kd-module over Kd and represented by M. Let C=u1,u2. Define Γ:C⟶PM×M by(6)Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,Γu2=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ.Figure 3
Illustration of M.Table 3
Binary operation subject to⊕.
⊕⊥rsT⊥⊥rsTr⊥rsTs⊥rsTT⊥rsTThen,Γ,C is a soft compatible relation (SCRE) and soft reflexive relation (SRRE) on M.


### 3.5. Remark

LetΓ,C be a SCRE on a Kd-module M. Then, it is easily verified that k⊕sΓu⊆k⊕sΓu and rΓu∨sΓu⊆r∨sΓu for all r,s∈M and k∈Kd.


### 3.6. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,Τ,s,r,Τ,Τ,r and Γu2=⊥,⊥,r,r,s,s,Τ,Τ. Then, Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows:(7)⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,Τ.ΤΓu1=r,s,Τ. Thus, we haverΓu1∨sΓu1=r,Τ∨s,Τ=Τ⊆r,s,Τ=r∨sΓu1. Hence, rΓu1∨sΓu1⊆r∨sΓu1 Also, similarly we can check that k⊕xΓu⊆k⊕xΓu∀k∈Kd,x∈M∀u∈C.


### 3.7. Remark

IfΓ,C is a SCRE on a Kd-module M, then k⊕Γus⊆Γuk⊕s and Γur∨Γus⊆Γur∨s with respect to foresets.


### 3.8. Definition

A SCREΓ,C on a Kd-module M is called soft join-complete with respect to to aftersets if pΓu∨qΓu=p∨qΓu and is called soft complete with respect to ⊕ if k⊕pΓu=k⊕pΓu∀k∈Kdp,q∈M∀u∈C.A SCREΓ,C which is both join-complete and ⊕-complete with respect to aftersets is called soft complete relation (SCTR) with respect to aftersets.


### 3.9. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by(8)Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,r,ΤandΓu2=⊥,⊥,r,r,s,s,Τ,Τ.Then,Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,Τ,ΤΓu1=Τ. It is easily checked that pΓu1∨qΓu1=p∨qΓu1​∀p,q∈M That is, rΓu1∨sΓu1=r,Τ∨s,Τ=Τ=r∨sΓu1 etc. Further, we can check that k⊕pΓu1=k⊕pΓu1∀k∈Kd,p∈M∀u∈C.So,Γ,C is a SCTR with respect to aftersets.


### 3.10. Definition

A SCREΓ,C on a Kd-module M is called soft join-complete with respect to foresets if Γup∨Γuq=Γup∨q and is called soft complete with respect to ⊕ if k⊕Γup=Γuk⊕p∀k∈Kdp,q∈M∀u∈C.A SCREΓ,C 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

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,r,Τ and Γu2=⊥,⊥,r,r,s,s,Τ,Τ. Then, Γ,C is a SCRE and SRRE on M. The aftersets and foresets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,Τ,ΤΓu1=Τ, Γu1⊥=⊥,Γu1r=r,Γu1s=s,Γu1Τ=r,s,Τ. It is observed that rΓu1∨sΓu1=r,Τ∨s,Τ=Τ=r∨sΓu1. Likewise, we can check that k⊕pΓu1=k⊕pΓu1∀k∈Kd,p∈M,∀u∈C So, Γ,C is a SCTR with respect to aftersets. But, Γu1r∨Γu1s=r∨s=Τ≠r,s,Τ=Γu1r∨s So, Γ,C 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∅≠S,∅≠R be the subsets of Kd-module M and Γ,C and γ,C be SRRE on M. Then, the following hold for all u∈C:(1)
Γ¯Su⊆S⊆Γ¯Su(2)
S⊆R⇒Γ¯Su⊆Γ¯Ru(3)
S⊆R⇒Γ¯Su⊆Γ¯Ru(4)
Γ¯S,C∩Γ¯R,C=Γ¯S∩R,C(5)
Γ¯S,C∩Γ¯R,C⊇Γ¯S∩R,C(6)
Γ¯S,C∪Γ¯R,C⊆Γ¯S∪R,C(7)
Γ¯S,C∪Γ¯R,C=Γ¯S∪R,C(8)
Γ,C⊆γ,C⇒Γ¯S,C⊇γ¯S,C(9)
Γ,C⊆γ,C⇒Γ¯S,C⊆γ¯S,C


### 3.14. Theorem (see [31])

Let∅≠S,∅≠R be the subset of Kd-module M and Γ,C and γ,C be SRRE on M. Then, the following hold for all u∈C:(10)
Γ¯Su⊆S⊆Γ¯Su(11)
S⊆R⇒Γ¯Su⊆Γ¯Ru(12)
S⊆R⇒Γ¯Su⊆Γ¯Ru(13)
Γ¯S,C∩Γ¯R,C=Γ¯S∩R,C(14)
Γ¯S,C∩Γ¯R,C⊇Γ¯S∩R,C(15)
Γ¯S,C∪Γ¯R,C⊆Γ¯S∩R,C(16)
Γ¯S,C∪Γ¯R,C=Γ¯S∪R,C(17)
Γ,C⊆γ,C⇒Γ¯S,C⊇γ¯S,C(18)
Γ,C⊆γ,C⇒Γ¯S,C⊆γ¯S,C


### 3.15. Theorem (see [31])

Let∅≠S,∅≠R be the subsets of Kd-module M and Γ,C and γ,C be SRRE on M. Then, the following hold for all u∈C:(1)
Γ∩γ¯S,C⊆Γ¯S,C∩γ¯S,C(2)
Γ∩γ¯S,C⊇Γ¯S,C∪γ¯S,C(3)
Γ∩γ¯S,C⊆Γ¯S,C∩γ¯S,C(4)
Γ∩γ¯S,C⊇Γ¯S,C∪γ¯S,C


### 3.16. Theorem

LetΓ,C be SRRE and SCRE with respect to the aftersets on a Kd-module M. Then, for non-empty subset S and R of M, we have k⊕Γ¯Su⊆Γ¯k⊕Su and Γ¯Su∨Γ¯Ru⊆Γ¯S∨Ru∀​u∈C.Proof.
Letx∈k⊕Γ¯Su Then, x=k⊕p where p∈Γ¯Su such that pΓu∩S≠∅. Thus, there exist a∈pΓu∩S such that a∈S. That is, p,a∈Γu. Since Γ,C is SCRE, we have k⊕p,k⊕a∈Γu and k⊕a∈k⊕S, ⇒k⊕a∈k⊕pΓu∩k⊕S, ⇒∅≠k⊕pΓu∩k⊕S⇒k⊕p∈Γ¯k⊕Su. So, x∈Γ¯k⊕Su. Thus, k⊕Γ¯Su⊆Γ¯k⊕Su
Lety∈Γ¯Su∨Γ¯Ru. Then, y=p∨q where p∈Γ¯Su and q∈Γ¯Ru such that pΓu∩S≠∅ and qΓu∩R≠∅. That is, a∈pΓu∩S and b∈qΓu∩R. So, a∈pΓu, a∈S and b∈qΓu, b∈R, ⇒p,a∈Γu,a∈S and q,b∈Γu,b∈R. Since Γ,C is SCRE, we have p∨q,a∨b∈Γuand a∨b∈S∨R⇒a∨b∈p∨qΓu∩S∨R, ⇒∅≠p∨qΓu∩S∨R⇒p∨q∈Γ¯S∨Ru So, y∈Γ¯S∨Ru. Thus, Γ¯Su∨Γ¯Ru⊆Γ¯S∨Ru.


### 3.17. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ and Γu2=⊥,⊥,r,r,s,s,Τ,Τ. Then, Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,ΤΓu1=Τ, ⊥Γu2=⊥,rΓu2=r,sΓu2=s,ΤΓu2=Τ. Let R=s and S=T. Then, Γ¯Ru1=s,Γ¯Su1=r,T, and Γ¯Ru1∨Γ¯Su1=T. Now, S∨R=Tand Γ¯S∨Ru1=r,T. So, Γ¯Su1∨Γ¯Ru1⊆Γ¯S∨Ru1. Similarly, we can prove that k⊕Γ¯Su⊆Γ¯k⊕Su.


### 3.18. Theorem

LetΓ,C be SRRE and SCRE with respect to the foresets on a Kd-module M. Then, for non-empty subset S and R of M, we have k⊕Γ¯Su⊆Γ¯k⊕Su and Γ¯Su∨Γ¯Ru⊆Γ¯S∨Ru∀u∈C.Proof.
The proof is similar to the proof of Theorem 3.16.


## 3.1. Definition (see [11])

LetΓ: C⟶PM×M. Then, Γ,C is a soft binary relation on a set M, where ϕ≠C⊆E (set of parameters). For ϕ≠S⊆M,Γ¯S,C and Γ¯S,C of S with respect to aftersets are basically the two soft sets over M, defined as follows: (5)Γ¯Su=k∈M|kΓu⊆SandΓ¯Su=k∈M|kΓu∩S≠∅Further,Γ¯S,C and Γ¯S,C of S with respect to foreset are basically the two soft sets over M, defined as follows: Γ¯Su=k∈M|Γuk⊆S and Γ¯Su=k∈M|Γuk∩S≠∅.For allu∈C, kΓu=r∈M:k,r∈Γu is called afterset of k and Γuk=r∈M:r,k∈Γu is called foreset of k. Moreover, LAP and UAP are defined as Γ¯S:C⟶PMandΓ¯S:C⟶PM for aftersets. Γ¯S:C⟶PMandΓ¯S:C⟶PM for foresets for each S∈PM. Generally, kΓu≠Γuk, Γ¯Su≠Γ¯Su, and Γ¯Su≠Γ¯Su. However, they are equal if Γu is a symmetric relation. This is justified in the next example.


## 3.2. Example

LetKd=⊥,r,s,T and C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,T,T,⊥,r and Γu2=⊥,⊥,r,r,s,s,T,T,r,TThus, the aftersets of elementsM are as follows.⊥Γu1=⊥,r, rΓu1=r, sΓu1=s, TΓu1=T and ⊥Γu2=⊥, rΓu2=r,T, sΓu2=s, and TΓu2=T, and the foresets of elements of M are as follows: Γu1⊥=⊥, Γu1r=⊥,r, Γu1s=s, Γu1T=T and Γu2⊥=⊥, Γu2r=r, Γu2s=s, Γu2T=r,T. Let S=⊥,T. Then, LAP and UAP of S⊆M with respect to aftersets and foresets are as follows: Γ¯Su1=T, Γ¯Su2=⊥,T and Γ¯Su1=⊥,T, Γ¯Su2=⊥,r,T, Γ¯Su1=⊥,T, Γ¯Su2=⊥ and Γ¯Su1=⊥,r,T, Γ¯Su2=⊥,T. This shows that Γ¯S,C≠Γ¯S,C and Γ¯S,C≠Γ¯S,C.


## 3.3. Definition

A SBRΓ on a quantale module M is called soft compatible relation (SCRE), if it satisfies the following conditions: ∀i∈I if riΓsi⇒∨i∈IriΓ∨i∈Isi and rΓs⇒k⊕rΓk⊕s for any r,s∈M,ri,si⊆M,i∈I and k∈Kd.


## 3.4. Example

LetKd=⊥,r,s,T be a complete lattice as shown in Figure 3, and the operation ⊛ on Kd is a⊛b=⊥. Then, it is easy to verify that Kd,⊛ is a quantale. Let ⊕:Kd×M⟶M be the left action of Kd on M as shown in Table 3. In this case, M=Kd. Then, it is easy to check that Kd is a Kd-module over Kd and represented by M. Let C=u1,u2. Define Γ:C⟶PM×M by(6)Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,Γu2=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ.Figure 3
Illustration of M.Table 3
Binary operation subject to⊕.
⊕⊥rsT⊥⊥rsTr⊥rsTs⊥rsTT⊥rsTThen,Γ,C is a soft compatible relation (SCRE) and soft reflexive relation (SRRE) on M.


## 3.5. Remark

LetΓ,C be a SCRE on a Kd-module M. Then, it is easily verified that k⊕sΓu⊆k⊕sΓu and rΓu∨sΓu⊆r∨sΓu for all r,s∈M and k∈Kd.


## 3.6. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,Τ,s,r,Τ,Τ,r and Γu2=⊥,⊥,r,r,s,s,Τ,Τ. Then, Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows:(7)⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,Τ.ΤΓu1=r,s,Τ. Thus, we haverΓu1∨sΓu1=r,Τ∨s,Τ=Τ⊆r,s,Τ=r∨sΓu1. Hence, rΓu1∨sΓu1⊆r∨sΓu1 Also, similarly we can check that k⊕xΓu⊆k⊕xΓu∀k∈Kd,x∈M∀u∈C.


## 3.7. Remark

IfΓ,C is a SCRE on a Kd-module M, then k⊕Γus⊆Γuk⊕s and Γur∨Γus⊆Γur∨s with respect to foresets.


## 3.8. Definition

A SCREΓ,C on a Kd-module M is called soft join-complete with respect to to aftersets if pΓu∨qΓu=p∨qΓu and is called soft complete with respect to ⊕ if k⊕pΓu=k⊕pΓu∀k∈Kdp,q∈M∀u∈C.A SCREΓ,C which is both join-complete and ⊕-complete with respect to aftersets is called soft complete relation (SCTR) with respect to aftersets.


## 3.9. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by(8)Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,r,ΤandΓu2=⊥,⊥,r,r,s,s,Τ,Τ.Then,Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,Τ,ΤΓu1=Τ. It is easily checked that pΓu1∨qΓu1=p∨qΓu1​∀p,q∈M That is, rΓu1∨sΓu1=r,Τ∨s,Τ=Τ=r∨sΓu1 etc. Further, we can check that k⊕pΓu1=k⊕pΓu1∀k∈Kd,p∈M∀u∈C.So,Γ,C is a SCTR with respect to aftersets.


## 3.10. Definition

A SCREΓ,C on a Kd-module M is called soft join-complete with respect to foresets if Γup∨Γuq=Γup∨q and is called soft complete with respect to ⊕ if k⊕Γup=Γuk⊕p∀k∈Kdp,q∈M∀u∈C.A SCREΓ,C 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

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,s,Τ,r,Τ and Γu2=⊥,⊥,r,r,s,s,Τ,Τ. Then, Γ,C is a SCRE and SRRE on M. The aftersets and foresets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,Τ,ΤΓu1=Τ, Γu1⊥=⊥,Γu1r=r,Γu1s=s,Γu1Τ=r,s,Τ. It is observed that rΓu1∨sΓu1=r,Τ∨s,Τ=Τ=r∨sΓu1. Likewise, we can check that k⊕pΓu1=k⊕pΓu1∀k∈Kd,p∈M,∀u∈C So, Γ,C is a SCTR with respect to aftersets. But, Γu1r∨Γu1s=r∨s=Τ≠r,s,Τ=Γu1r∨s So, Γ,C 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∅≠S,∅≠R be the subsets of Kd-module M and Γ,C and γ,C be SRRE on M. Then, the following hold for all u∈C:(1)
Γ¯Su⊆S⊆Γ¯Su(2)
S⊆R⇒Γ¯Su⊆Γ¯Ru(3)
S⊆R⇒Γ¯Su⊆Γ¯Ru(4)
Γ¯S,C∩Γ¯R,C=Γ¯S∩R,C(5)
Γ¯S,C∩Γ¯R,C⊇Γ¯S∩R,C(6)
Γ¯S,C∪Γ¯R,C⊆Γ¯S∪R,C(7)
Γ¯S,C∪Γ¯R,C=Γ¯S∪R,C(8)
Γ,C⊆γ,C⇒Γ¯S,C⊇γ¯S,C(9)
Γ,C⊆γ,C⇒Γ¯S,C⊆γ¯S,C


## 3.14. Theorem (see [31])

Let∅≠S,∅≠R be the subset of Kd-module M and Γ,C and γ,C be SRRE on M. Then, the following hold for all u∈C:(10)
Γ¯Su⊆S⊆Γ¯Su(11)
S⊆R⇒Γ¯Su⊆Γ¯Ru(12)
S⊆R⇒Γ¯Su⊆Γ¯Ru(13)
Γ¯S,C∩Γ¯R,C=Γ¯S∩R,C(14)
Γ¯S,C∩Γ¯R,C⊇Γ¯S∩R,C(15)
Γ¯S,C∪Γ¯R,C⊆Γ¯S∩R,C(16)
Γ¯S,C∪Γ¯R,C=Γ¯S∪R,C(17)
Γ,C⊆γ,C⇒Γ¯S,C⊇γ¯S,C(18)
Γ,C⊆γ,C⇒Γ¯S,C⊆γ¯S,C


## 3.15. Theorem (see [31])

Let∅≠S,∅≠R be the subsets of Kd-module M and Γ,C and γ,C be SRRE on M. Then, the following hold for all u∈C:(1)
Γ∩γ¯S,C⊆Γ¯S,C∩γ¯S,C(2)
Γ∩γ¯S,C⊇Γ¯S,C∪γ¯S,C(3)
Γ∩γ¯S,C⊆Γ¯S,C∩γ¯S,C(4)
Γ∩γ¯S,C⊇Γ¯S,C∪γ¯S,C


## 3.16. Theorem

LetΓ,C be SRRE and SCRE with respect to the aftersets on a Kd-module M. Then, for non-empty subset S and R of M, we have k⊕Γ¯Su⊆Γ¯k⊕Su and Γ¯Su∨Γ¯Ru⊆Γ¯S∨Ru∀​u∈C.Proof.
Letx∈k⊕Γ¯Su Then, x=k⊕p where p∈Γ¯Su such that pΓu∩S≠∅. Thus, there exist a∈pΓu∩S such that a∈S. That is, p,a∈Γu. Since Γ,C is SCRE, we have k⊕p,k⊕a∈Γu and k⊕a∈k⊕S, ⇒k⊕a∈k⊕pΓu∩k⊕S, ⇒∅≠k⊕pΓu∩k⊕S⇒k⊕p∈Γ¯k⊕Su. So, x∈Γ¯k⊕Su. Thus, k⊕Γ¯Su⊆Γ¯k⊕Su
Lety∈Γ¯Su∨Γ¯Ru. Then, y=p∨q where p∈Γ¯Su and q∈Γ¯Ru such that pΓu∩S≠∅ and qΓu∩R≠∅. That is, a∈pΓu∩S and b∈qΓu∩R. So, a∈pΓu, a∈S and b∈qΓu, b∈R, ⇒p,a∈Γu,a∈S and q,b∈Γu,b∈R. Since Γ,C is SCRE, we have p∨q,a∨b∈Γuand a∨b∈S∨R⇒a∨b∈p∨qΓu∩S∨R, ⇒∅≠p∨qΓu∩S∨R⇒p∨q∈Γ¯S∨Ru So, y∈Γ¯S∨Ru. Thus, Γ¯Su∨Γ¯Ru⊆Γ¯S∨Ru.


## 3.17. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ and Γu2=⊥,⊥,r,r,s,s,Τ,Τ. Then, Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,ΤΓu1=Τ, ⊥Γu2=⊥,rΓu2=r,sΓu2=s,ΤΓu2=Τ. Let R=s and S=T. Then, Γ¯Ru1=s,Γ¯Su1=r,T, and Γ¯Ru1∨Γ¯Su1=T. Now, S∨R=Tand Γ¯S∨Ru1=r,T. So, Γ¯Su1∨Γ¯Ru1⊆Γ¯S∨Ru1. Similarly, we can prove that k⊕Γ¯Su⊆Γ¯k⊕Su.


## 3.18. Theorem

LetΓ,C be SRRE and SCRE with respect to the foresets on a Kd-module M. Then, for non-empty subset S and R of M, we have k⊕Γ¯Su⊆Γ¯k⊕Su and Γ¯Su∨Γ¯Ru⊆Γ¯S∨Ru∀u∈C.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∅≠S⊆M and Γ,C be a SBR on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to aftersets if Γ¯S,C is a soft sub-Kd-module of M.


### 4.2. Theorem

LetΓ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to aftersets if S is a soft sub-Kd-module of M.Proof.
(1)
Suppose thatri∈Γ¯Su for u∈C and for some i∈I. Then, riΓu∩S≠∅. There are xi∈riΓu∩S such that xi∈riΓu and xi∈S. That is, ri,xi∈Γu and xi∈S, ∀i∈I. Since S is a sub-Kd-module of M and Γu is a SCRE∀u∈C. Thus, we have ∨i∈Iri,∨i∈Ixi∈Γu and ∨i∈Ixi∈S. That is, ∨i∈Ixi∈∨i∈IriΓu and ∨i∈Ixi∈S. So, ∨i∈Ixi∈∨i∈IriΓu∩S. Hence, ∨i∈IriΓu∩S≠∅. This shows that ∨i∈Iri∈Γ¯Su.(2)
Letr∈M such that r∈Γ¯Su. This shows rΓu∩S≠∅. Then, there is y∈rΓu∩S such that y∈rΓu and y∈S. That is, r,y∈Γu and y∈S. Since S is a sub-Kd-module of M and Γu is a SRRE and SCRE. Thus, we have k⊕y∈S and k⊕r,k⊕y∈Γu, ⇒k⊕y∈k⊕rΓu and k⊕y∈S, ⇒k⊕y∈k⊕rΓu∩S. Hence, k⊕rΓu∩S≠∅. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a sub Kd-module of M. That is, S is a generalized upper soft sub-Kd-module of M with respect to aftersets.
It is mentioned in the next example that the converse is not true.


### 4.3. Example

SupposeKd=⊥,r,s,t,T be a complete lattice as shown in Figure 4 and the operation ⊛ on Kd is that a⊛b=⊥. Then, it is easily checked that Kd,⊛ is a quantale. Suppose ⊛:Kd×M⟶M be the left action of Kd on M as shown in Table 4. In this case, M=Kd; then, it is easy to check that Kd is a quantale module M. Let C=u1,u2. Then, Γ:C⟶PM×M defined by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T,⊥,T,t,T,r,t,r,⊥,t,s,t,t and Γu2=⊥,⊥,r,r,s,s,t,t,Τ,Τ,T,t,t,r. Then, Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows:(9)⊥Γu1=⊥,T,rΓu1=⊥,r,t,Τ,sΓu1=s,T,tΓu1=s,t,T,TΓu1=T,⊥Γu2=⊥,rΓu2=r,sΓu2=s,tΓu2=r,t,TΓu2=t,T.Figure 4
Illustration of M.Table 4
Binary operation subject to⊕.
⊕⊥rstT⊥⊥rstTr⊥rstTs⊥rstTt⊥rstTT⊥rstTLetS=T,r,s. Then, S is not a sub-Kd-module of M. But, Γ¯Su1=⊥,r,s,t,T and Γ¯Su2=r,s,t,T are sub-Kd-module of M. Hence, S is a generalized upper soft sub-Kd-module of M with respect to aftersets.


### 4.4. Definition

Let∅≠S⊆M and Γ,C be a SBR on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to foresets if Γ¯S,C is a soft sub-Kd-module of M.


### 4.5. Theorem

LetΓ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to foresets if S is a soft sub-Kd-module of M.Proof.
The proof is similar to the proof of Theorem 4.2.


### 4.6. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized lower soft sub-Kd-module of M with respect to aftersets if Γ¯S,C is a soft sub-Kd-module of M.


### 4.7. Theorem

LetΓ,C be a SRRE and SCTE on a Kd-module M. Then, S is said to be generalized lower soft sub Kd-module of M with respect to aftersets if S is a soft sub-Kd-module of M.Proof.
(1)
Suppose thatri∈Γ¯Su for u∈C and for i∈I. This shows that riΓu⊆S. Hence, S is a sub-Kd-module of M and Γu is SCTR ∀u∈C. Thus, ∨i∈IriΓu=∨i∈IriΓu and ∨i∈IriΓu=∨i∈IriΓu⊆S. Hence, ∨i∈IriΓu⊆S. This shows that ∨i∈Iri∈Γ¯Su.(2)
Letr∈M be such that r∈Γ¯Su. This shows rΓu⊆S for u∈C. Hence, S is a sub-Kd-module of M and Γu is a SRRE and SCTE. Thus, we have k⊕rΓu=k⊕rΓu. So, we can write k⊕rΓu=k⊕rΓu⊆S. Hence, k⊕rΓu⊆S. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a sub Kd-module of M. That is, S is a generalized lower soft sub-Kd-module of M with respect to aftersets.
Observe in the next example that the converse is not true.


### 4.8. Example

LetKd,⊕ be a Kd-module M as given in example 4.3 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T and Γu2=⊥,⊥,r,r,s,s,t,t,Τ,Τ,T,t,r,T,s,⊥. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,T,ΤΓu1=Τ, ⊥Γu2=⊥,rΓu2=r,T,sΓu2=⊥,s,ΤΓu2=r,Τ. Then, Γ,C is a SCTR and SRRE with respect to aftersets. Let S=⊥,r,s. Then, S is not a sub-Kd-module of M. However, Γ¯Su1=⊥,Γ¯Su2=⊥ are sub-Kd-module of M. Hence, S is generalized lower soft sub-Kd-module of M.


### 4.9. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized lower soft sub-Kd-module of M with respect to foresets if Γ¯S,C is a soft sub-Kd-module of M.


### 4.10. Theorem

LetΓ,C be a SCTR and SRRE on a Kd-module M. Then, S is said to be generalized lower soft sub Kd-module of M with respect to foresets if S is a soft sub-Kd-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∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized upper soft Kd-module ideal of M with respect to aftersets.


### 4.13. Theorem

Let∅≠S⊆M and Γ,C be a SRRE, SCRE and soft join-complete relation on a Kd-module M. Then, S is said to be generalized upper soft Kd-module ideal of M with respect to aftersets if S is an Kd-module ideal of M.Proof.
(1)
Suppose thatr,s∈M such that r,s∈Γ¯Su for u∈C. Therefore, rΓu∩S≠∅ and sΓu∩S≠∅. Then ,there are x∈rΓu∩S and y∈sΓu∩S such that x∈rΓu, y∈sΓu, and x∈S,y∈S. This implies r,x∈Γu, s,y∈Γu, and x,y∈S. S is a Kd-module ideal of M and Γu is a SCRE ∀u∈C. Hence, we have r∨s,x∨y∈Γu and x∨y∈S. That is, x∨y∈r∨sΓu and x∨y∈S. So, x∨y∈r∨sΓu∩S. Hence, r∨sΓu∩S≠∅. This shows that r∨s∈Γ¯Su.(2)
Lets∈Γ¯Su and r≤s. Therefore, sΓu∩S≠∅. Then, there is y∈sΓu∩S such that y∈sΓu and y∈S. Since Γu is a SCRE and soft join-complete relation, we have sΓu=r∨sΓu=rΓu∨sΓu. Then, there is u∈rΓu and v∈sΓu such that y=u∨v.S is a Kd-module ideal of M and u≤y∈S. We have u∈S so r∈Γ¯Su(3)
Letk∈M be such that r∈Γ¯Su. This shows rΓu∩S≠∅. Then, there are y∈rΓu∩S such that y∈rΓu and y∈S. That is, r,y∈Γu and y∈S. S is a Kd module ideal of M and Γu is a SRRE and SCRE. We have k⊕y∈S and k⊕r,k⊕y∈Γu⇒k⊕y∈k⊕rΓu and k⊕y∈S, ⇒k⊕y∈k⊕rΓu∩S. Hence, k⊕rΓu∩S≠∅. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a Kd-module ideal of M. That is, S is a generalized upper soft Kd-module ideal of M with respect to aftersets.
It is observed in the next example that the converse is not true.


### 4.14. Example

LetM be the quantale module as given in example 4.3 and C=u1,u2. Then, Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T,⊥,T,t,T,r,t,r,⊥,t,s,t,t and Γu2=⊥,⊥,r,r,s,s,t,t,Τ,Τ,T,t,t,r.The aftersets calculated by the elements ofM are as follows:(10)⊥Γu1=⊥,T,rΓu1=⊥,r,t,Τ,sΓu1=s,T,tΓu1=s,t,T,ΤΓu1=Τ⊥Γu2=⊥,rΓu2=r,T,sΓu2=s,ΤΓu2=Τ,tΓu2=t,then Γ,C is a SCRE and SRRE and soft join-complete relation with respect to aftersets on M. Let S=T,t,s,⊥. Then, S is not a Kd-module ideal of M. However, Γ¯Su1=⊥,r,s,t,T and Γ¯Su1=⊥,r,s,t,T are Kd-module ideal of M. Hence, S is a generalized upper soft Kd-module ideal of M with respect to aftersets.


### 4.15. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized upper soft Kd-module of M with respect to foresets if S is a Kd-module ideal of M.


### 4.16. Theorem

Let∅≠S⊆M and Γ,C be a SRRE, SCRE, and soft join-complete relation on a Kd-module M. Then, S is said to be generalized upper soft Kd-module ideal of M with respect to foresets if S is an Kd-module ideal of M.Proof.
The proof is obvious.


### 4.17. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized lower soft Kd-module of M with respect to aftersets.


### 4.18. Theorem

Let∅≠S⊆M and Γ,C be a SRRE and SCTR on a Kd-module M. Then, S is said to be generalized lower soft Kd-module ideal of M with respect to aftersets if S is an Kd-module ideal of M.Proof.
(1)
Suppose thatr,s∈M such that r,s∈Γ¯Su for u∈C. This shows rΓu⊆S and sΓu⊆S. Hence, S is a Kd-module ideal of M and Γu is SCTR ∀u∈C. So, we have r∨sΓu=rΓu∨sΓu and rΓu∨sΓu=r∨sΓu⊆S. Hence, r∨sΓu⊆S. This shows that r∨s∈Γ¯Su.(2)
Lets∈Γ¯Su and r≤s. Therefore, sΓu⊆S. For w∈rΓu, we have r,w∈Γu. Since r,w∈Γu is a SRRE and SCRE, we have r∨s,w∨s∈Γu. That is, s,w∨s∈Γu. Thus, w∨s∈S. Since S is a Kd-module ideal of M, we have w∈S. Thus, rΓu⊆S and r∈Γ¯Su.(3)
Letk∈M be such that r∈Γ¯Su. This shows rΓu⊆S for u∈C. Hence, S is a Kd-module ideal of M. Then, for all k∈M, we have k⊕rΓu⊆k⊕S⊆S. Also given that Γu is a SRRE and SCTE. Thus, we can write k⊕rΓu=k⊕rΓu⊆S. Hence, k⊕rΓu=k⊕rΓu⊆S. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a Kd-module ideal of M. That is, S is a generalized lower soft Kd-module ideal of M with respect to aftersets.
Observe in the next example that the converse is not true.


### 4.19. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T and Γu2=⊥,⊥,r,r,s,s,Τ,Τ,T,r,r,T,s,⊥. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,T,ΤΓu1=Τ, ⊥Γu2=⊥,rΓu2=r,T,sΓu2=⊥,s,ΤΓu2=r,Τ. Then, Γ,C is a SCTR and SRRE with respect to aftersets. Let S=⊥,r,s. Then, S is not an ideal of Kd-module of M. However, Γ¯Su1=⊥,Γ¯Su2=⊥ are Kd-module ideal of M. Hence, S is generalized lower soft Kd-module ideal of M with respect to aftersets.


### 4.20. Definition

Let∅≠S⊆M and Γ,C be a SBR on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized lower soft Kd-module of M with respect to foresets.


### 4.21. Theorem

Let∅≠S⊆M and Γ,C be a SRRE and SCTR on a Kd-module M. Then, S is said to be generalized lower soft Kd-module ideal of M with respect to foresets if S is an Kd-module ideal of M.Proof.
The proof is simple.


## 4.1. Definition

Let∅≠S⊆M and Γ,C be a SBR on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to aftersets if Γ¯S,C is a soft sub-Kd-module of M.


## 4.2. Theorem

LetΓ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to aftersets if S is a soft sub-Kd-module of M.Proof.
(1)
Suppose thatri∈Γ¯Su for u∈C and for some i∈I. Then, riΓu∩S≠∅. There are xi∈riΓu∩S such that xi∈riΓu and xi∈S. That is, ri,xi∈Γu and xi∈S, ∀i∈I. Since S is a sub-Kd-module of M and Γu is a SCRE∀u∈C. Thus, we have ∨i∈Iri,∨i∈Ixi∈Γu and ∨i∈Ixi∈S. That is, ∨i∈Ixi∈∨i∈IriΓu and ∨i∈Ixi∈S. So, ∨i∈Ixi∈∨i∈IriΓu∩S. Hence, ∨i∈IriΓu∩S≠∅. This shows that ∨i∈Iri∈Γ¯Su.(2)
Letr∈M such that r∈Γ¯Su. This shows rΓu∩S≠∅. Then, there is y∈rΓu∩S such that y∈rΓu and y∈S. That is, r,y∈Γu and y∈S. Since S is a sub-Kd-module of M and Γu is a SRRE and SCRE. Thus, we have k⊕y∈S and k⊕r,k⊕y∈Γu, ⇒k⊕y∈k⊕rΓu and k⊕y∈S, ⇒k⊕y∈k⊕rΓu∩S. Hence, k⊕rΓu∩S≠∅. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a sub Kd-module of M. That is, S is a generalized upper soft sub-Kd-module of M with respect to aftersets.
It is mentioned in the next example that the converse is not true.


## 4.3. Example

SupposeKd=⊥,r,s,t,T be a complete lattice as shown in Figure 4 and the operation ⊛ on Kd is that a⊛b=⊥. Then, it is easily checked that Kd,⊛ is a quantale. Suppose ⊛:Kd×M⟶M be the left action of Kd on M as shown in Table 4. In this case, M=Kd; then, it is easy to check that Kd is a quantale module M. Let C=u1,u2. Then, Γ:C⟶PM×M defined by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T,⊥,T,t,T,r,t,r,⊥,t,s,t,t and Γu2=⊥,⊥,r,r,s,s,t,t,Τ,Τ,T,t,t,r. Then, Γ,C is a SCRE and SRRE on M. The aftersets calculated by the elements of M are as follows:(9)⊥Γu1=⊥,T,rΓu1=⊥,r,t,Τ,sΓu1=s,T,tΓu1=s,t,T,TΓu1=T,⊥Γu2=⊥,rΓu2=r,sΓu2=s,tΓu2=r,t,TΓu2=t,T.Figure 4
Illustration of M.Table 4
Binary operation subject to⊕.
⊕⊥rstT⊥⊥rstTr⊥rstTs⊥rstTt⊥rstTT⊥rstTLetS=T,r,s. Then, S is not a sub-Kd-module of M. But, Γ¯Su1=⊥,r,s,t,T and Γ¯Su2=r,s,t,T are sub-Kd-module of M. Hence, S is a generalized upper soft sub-Kd-module of M with respect to aftersets.


## 4.4. Definition

Let∅≠S⊆M and Γ,C be a SBR on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to foresets if Γ¯S,C is a soft sub-Kd-module of M.


## 4.5. Theorem

LetΓ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized upper soft sub Kd-module of M with respect to foresets if S is a soft sub-Kd-module of M.Proof.
The proof is similar to the proof of Theorem 4.2.


## 4.6. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized lower soft sub-Kd-module of M with respect to aftersets if Γ¯S,C is a soft sub-Kd-module of M.


## 4.7. Theorem

LetΓ,C be a SRRE and SCTE on a Kd-module M. Then, S is said to be generalized lower soft sub Kd-module of M with respect to aftersets if S is a soft sub-Kd-module of M.Proof.
(1)
Suppose thatri∈Γ¯Su for u∈C and for i∈I. This shows that riΓu⊆S. Hence, S is a sub-Kd-module of M and Γu is SCTR ∀u∈C. Thus, ∨i∈IriΓu=∨i∈IriΓu and ∨i∈IriΓu=∨i∈IriΓu⊆S. Hence, ∨i∈IriΓu⊆S. This shows that ∨i∈Iri∈Γ¯Su.(2)
Letr∈M be such that r∈Γ¯Su. This shows rΓu⊆S for u∈C. Hence, S is a sub-Kd-module of M and Γu is a SRRE and SCTE. Thus, we have k⊕rΓu=k⊕rΓu. So, we can write k⊕rΓu=k⊕rΓu⊆S. Hence, k⊕rΓu⊆S. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a sub Kd-module of M. That is, S is a generalized lower soft sub-Kd-module of M with respect to aftersets.
Observe in the next example that the converse is not true.


## 4.8. Example

LetKd,⊕ be a Kd-module M as given in example 4.3 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T and Γu2=⊥,⊥,r,r,s,s,t,t,Τ,Τ,T,t,r,T,s,⊥. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,T,ΤΓu1=Τ, ⊥Γu2=⊥,rΓu2=r,T,sΓu2=⊥,s,ΤΓu2=r,Τ. Then, Γ,C is a SCTR and SRRE with respect to aftersets. Let S=⊥,r,s. Then, S is not a sub-Kd-module of M. However, Γ¯Su1=⊥,Γ¯Su2=⊥ are sub-Kd-module of M. Hence, S is generalized lower soft sub-Kd-module of M.


## 4.9. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M. Then, S is said to be generalized lower soft sub-Kd-module of M with respect to foresets if Γ¯S,C is a soft sub-Kd-module of M.


## 4.10. Theorem

LetΓ,C be a SCTR and SRRE on a Kd-module M. Then, S is said to be generalized lower soft sub Kd-module of M with respect to foresets if S is a soft sub-Kd-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∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized upper soft Kd-module ideal of M with respect to aftersets.


## 4.13. Theorem

Let∅≠S⊆M and Γ,C be a SRRE, SCRE and soft join-complete relation on a Kd-module M. Then, S is said to be generalized upper soft Kd-module ideal of M with respect to aftersets if S is an Kd-module ideal of M.Proof.
(1)
Suppose thatr,s∈M such that r,s∈Γ¯Su for u∈C. Therefore, rΓu∩S≠∅ and sΓu∩S≠∅. Then ,there are x∈rΓu∩S and y∈sΓu∩S such that x∈rΓu, y∈sΓu, and x∈S,y∈S. This implies r,x∈Γu, s,y∈Γu, and x,y∈S. S is a Kd-module ideal of M and Γu is a SCRE ∀u∈C. Hence, we have r∨s,x∨y∈Γu and x∨y∈S. That is, x∨y∈r∨sΓu and x∨y∈S. So, x∨y∈r∨sΓu∩S. Hence, r∨sΓu∩S≠∅. This shows that r∨s∈Γ¯Su.(2)
Lets∈Γ¯Su and r≤s. Therefore, sΓu∩S≠∅. Then, there is y∈sΓu∩S such that y∈sΓu and y∈S. Since Γu is a SCRE and soft join-complete relation, we have sΓu=r∨sΓu=rΓu∨sΓu. Then, there is u∈rΓu and v∈sΓu such that y=u∨v.S is a Kd-module ideal of M and u≤y∈S. We have u∈S so r∈Γ¯Su(3)
Letk∈M be such that r∈Γ¯Su. This shows rΓu∩S≠∅. Then, there are y∈rΓu∩S such that y∈rΓu and y∈S. That is, r,y∈Γu and y∈S. S is a Kd module ideal of M and Γu is a SRRE and SCRE. We have k⊕y∈S and k⊕r,k⊕y∈Γu⇒k⊕y∈k⊕rΓu and k⊕y∈S, ⇒k⊕y∈k⊕rΓu∩S. Hence, k⊕rΓu∩S≠∅. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a Kd-module ideal of M. That is, S is a generalized upper soft Kd-module ideal of M with respect to aftersets.
It is observed in the next example that the converse is not true.


## 4.14. Example

LetM be the quantale module as given in example 4.3 and C=u1,u2. Then, Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T,⊥,T,t,T,r,t,r,⊥,t,s,t,t and Γu2=⊥,⊥,r,r,s,s,t,t,Τ,Τ,T,t,t,r.The aftersets calculated by the elements ofM are as follows:(10)⊥Γu1=⊥,T,rΓu1=⊥,r,t,Τ,sΓu1=s,T,tΓu1=s,t,T,ΤΓu1=Τ⊥Γu2=⊥,rΓu2=r,T,sΓu2=s,ΤΓu2=Τ,tΓu2=t,then Γ,C is a SCRE and SRRE and soft join-complete relation with respect to aftersets on M. Let S=T,t,s,⊥. Then, S is not a Kd-module ideal of M. However, Γ¯Su1=⊥,r,s,t,T and Γ¯Su1=⊥,r,s,t,T are Kd-module ideal of M. Hence, S is a generalized upper soft Kd-module ideal of M with respect to aftersets.


## 4.15. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized upper soft Kd-module of M with respect to foresets if S is a Kd-module ideal of M.


## 4.16. Theorem

Let∅≠S⊆M and Γ,C be a SRRE, SCRE, and soft join-complete relation on a Kd-module M. Then, S is said to be generalized upper soft Kd-module ideal of M with respect to foresets if S is an Kd-module ideal of M.Proof.
The proof is obvious.


## 4.17. Definition

Let∅≠S⊆M and Γ,C be a SRRE and SCRE on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized lower soft Kd-module of M with respect to aftersets.


## 4.18. Theorem

Let∅≠S⊆M and Γ,C be a SRRE and SCTR on a Kd-module M. Then, S is said to be generalized lower soft Kd-module ideal of M with respect to aftersets if S is an Kd-module ideal of M.Proof.
(1)
Suppose thatr,s∈M such that r,s∈Γ¯Su for u∈C. This shows rΓu⊆S and sΓu⊆S. Hence, S is a Kd-module ideal of M and Γu is SCTR ∀u∈C. So, we have r∨sΓu=rΓu∨sΓu and rΓu∨sΓu=r∨sΓu⊆S. Hence, r∨sΓu⊆S. This shows that r∨s∈Γ¯Su.(2)
Lets∈Γ¯Su and r≤s. Therefore, sΓu⊆S. For w∈rΓu, we have r,w∈Γu. Since r,w∈Γu is a SRRE and SCRE, we have r∨s,w∨s∈Γu. That is, s,w∨s∈Γu. Thus, w∨s∈S. Since S is a Kd-module ideal of M, we have w∈S. Thus, rΓu⊆S and r∈Γ¯Su.(3)
Letk∈M be such that r∈Γ¯Su. This shows rΓu⊆S for u∈C. Hence, S is a Kd-module ideal of M. Then, for all k∈M, we have k⊕rΓu⊆k⊕S⊆S. Also given that Γu is a SRRE and SCTE. Thus, we can write k⊕rΓu=k⊕rΓu⊆S. Hence, k⊕rΓu=k⊕rΓu⊆S. This shows that k⊕r∈Γ¯Su. Therefore, Γ¯Su is a Kd-module ideal of M. That is, S is a generalized lower soft Kd-module ideal of M with respect to aftersets.
Observe in the next example that the converse is not true.


## 4.19. Example

LetKd,⊕ be a Kd-module M as given in example 3.4 and let C=u1,u2. Define Γ:C⟶PM×M by Γu1=⊥,⊥,r,r,s,s,Τ,Τ,r,Τ,s,T and Γu2=⊥,⊥,r,r,s,s,Τ,Τ,T,r,r,T,s,⊥. The aftersets calculated by the elements of M are as follows: ⊥Γu1=⊥,rΓu1=r,Τ,sΓu1=s,T,ΤΓu1=Τ, ⊥Γu2=⊥,rΓu2=r,T,sΓu2=⊥,s,ΤΓu2=r,Τ. Then, Γ,C is a SCTR and SRRE with respect to aftersets. Let S=⊥,r,s. Then, S is not an ideal of Kd-module of M. However, Γ¯Su1=⊥,Γ¯Su2=⊥ are Kd-module ideal of M. Hence, S is generalized lower soft Kd-module ideal of M with respect to aftersets.


## 4.20. Definition

Let∅≠S⊆M and Γ,C be a SBR on a Kd-module M if Γ¯S,C is a soft Kd-module ideal M. Then, S is said to be generalized lower soft Kd-module of M with respect to foresets.


## 4.21. Theorem

Let∅≠S⊆M and Γ,C be a SRRE and SCTR on a Kd-module M. Then, S is said to be generalized lower soft Kd-module ideal of M with respect to foresets if S is an Kd-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 theKd- module, as well as the images of upper (lower) approximations under Kd-module homomorphism, is being discussed in this section. Further, we study some properties of these approximations.


### 5.1. Lemma

LetM1 and M2 be Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
β1,C is a SBR on M1(2)
β1,C is SRRE if β2,C is SRRE(3)
β1,C is SCRE if β2,C is SCREProof.
The proof is obvious.


### 5.2. Lemma

LetM1 and M2 be a Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C Then, β1,C is SCTR with respect to aftersets if β2,C is SCTR with respect to aftersets and Ω is one-one.Proof.
Clearlyrβ1u∨sβ1u⊆r∨sβ1u∀u∈C.
Conversely, suppose thatx∈r∨sβ1u. Then, by definition of aftersets r∨s,x∈β1u. Thus, Ωr∨s,Ωx∈β2u, that is, Ωr∨Ωs,Ωx∈β2u. This implies that Ωx∈Ωr∨Ωsβ2u. Since β2,C is SCTR with respect to aftersets, Ωx∈Ωrβ2u∨Ωsβ2u. Then, there is t1∈Ωrβ2u and t2∈Ωrβ2u such that Ωx=t1∨t2. As Ω is onto, we have s1,s2∈M1, Ωs1=t1 and Ωs2=t2. Thus, Ωx=Ωs1∨Ωs2. This implies, Ωx=Ωs1∨s2. As Ω is one-one, we have x=s1∨s2. Now, t2∈Ωrβ2u, ⇒, Ωs1∈Ωrβ2uww, ⇒Ωr,Ωs1∈β2u, ⇒r,s1∈β1u, ⇒s1∈rβ1u. Also now, t2∈Ωsβ2u, ⇒, Ωs2∈Ωsβ2u, ⇒Ωs,Ωs2∈β2u, ⇒s,s2∈β1u, ⇒s2∈sβ1u. Now, x=s1∨s2∈rβ1u∨sβ1u. That is, x∈rβ1u∨sβ1u. This implies that r∨sβ1u⊆rβ1u∨sβ1u. Hence, r∨sβ1u=rβ1u∨sβ1u∀u∈C. Similarly, we can prove that k⊕rβ1u=k⊕rβ1u∀u∈C. This show that β1,C is SCTR with respect to aftersets.


### 5.3. Lemma

LetM1 and M2 be a Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, β1,C is SCTR with respect to foresets if β2,C is SCTR with respect to foresets and Ω is one-one.Proof.
The proof is obvious.


### 5.4. Lemma

LetM1 and M2 be Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C Then, Ωβ¯1Su=β¯2ΩSu for S⊆M1 and ∀u∈C.Proof.
Lets∈β¯2ΩSu. Then, sβ2u∩ΩS≠∅. Then, there is r∈sβ2u∩ΩS such that r∈sβ2u and r∈ΩS. So, s,r∈β2u and r∈ΩS. Since Ω is onto, there exist x∈S and y∈M1 such that r=Ωx and s=Ωy. Thus, s,r=Ωy,Ωx∈β2u, that is, (Ωy,Ωx∈β2u. Hence, yx∈β1u, that is, x∈yβ1u and x∈S. This implies x∈yβ1u∩S. So, we have y∈β¯1Su, ⇒Ωy∈Ωβ¯1Su, ⇒s∈Ωβ¯1Su. This show that ⇒s∈Ωβ¯1Su.
Conversely, letz∈Ωβ¯1Su. Then, there is r∈β¯1Su such that rβ1u∩S≠∅ and z∈Ωr. Thus, x∈rβ1u∩S such that x∈rβ1u and x∈S. So, r,x∈β1u and x∈S, ⇒Ωx∈ΩS and (Ωr,Ωx∈β2u, ⇒Ωx∈Ωrβ2u and Ωx∈ΩS. Hence, Ωx∈Ωrβ2u∩ΩS, that is, ∅≠Ωrβ2u∩ΩS, ⇒Ωr∈β¯2ΩS, ⇒z∈β¯2ΩSu. This shows that Ωβ¯1Su⊆β¯2ΩSu. Hence, Ωβ¯1Su=β¯2ΩSu∀u∈C.


### 5.5. Lemma

LetM1 and M2 be Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
Ωβ1¯Su⊆β2¯Ωsu∀u∈C; if Ω is one-one, then Ωβ1¯Su=β2¯Ωsu∀u∈C(2)
IfΩ:M1⟶M2 is one-one, then Ωx∈Ωβ¯1Su if and only if x∈β¯1Su∀u∈CProof.
(1)
Letr∈Ωβ1¯Su for u∈C. Then, there is s∈β1¯Su such that sβ1u⊆S and r=Ωx. Suppose that y∈rβ2u. Then, there is x∈M1 such that Ωx=y and Ωx∈Ωsβ2u, that is, (Ωs,Ωx)∈β2u. Hence, s,x∈β1u by definition of aftersets x∈sβ1u. Hence, sβ1u⊆S and x∈S. So, Ωx∈ΩS using above Ωx=y, ⇒y∈ΩS. Thus, rβ2u⊆ΩS. Hence, r∈β2¯Ωsuwhich shows that Ωβ1¯Su⊆β2¯Ωsu∀u∈C. Conversely, let s∈β2¯Ωsu∀u∈C. Then, sβ2u⊆ΩS. Suppose that Ω is one-one. Then, there is r∈M1 such that s=Ωr and Ωrβ2u⊆ΩS. Let x∈rβ1u by definition of aftersets, we have r,x∈β1u. Then, Ωr,Ωx∈β2u, that is, Ωx∈Ωrβ2u. Since Ωrβ2u⊆ΩS, Ωx∈ΩS, ⇒x∈S. Thus, rβ1u⊆S, that is, r∈β1¯Su, ⇒Ωr∈Ωβ1¯Su since s=Ωr. Hence, s∈Ωβ1¯Su. So,β2¯Ωsu⊆Ωβ1¯Su∀u∈C. Thus, Ωβ1¯Su=β2¯Ωsu∀u∈C.(2)
LetΩr∈Ωβ¯1Su. Then, there is s∈β¯1Su such that Ωr=Ωs. Since Ω is one-one, we get r=s. Thus, r∈β¯1Su. Conversely, let r∈β¯1Su∀u∈C. Then, Ωr∈Ωβ¯1Su∀u∈C.


### 5.6. Theorem

LetS⊆M1 and let Ω be a surjective Kd-module homomorphism on a Kd-modules M1,M2 and β2,C be a SBR and SCTR with respect to aftersets on M2. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
β¯1Su is a Kd-module ideal of M1 if and only if β¯2ΩSu is a Kd-module ideal of M2∀u∈C(2)
β¯1Su is a sub-Kd-module of M1 if and only if β¯2ΩSu is a sub-Kd-module of M2∀u∈CProof.
(1) Letβ¯1Su be a Kd-module ideal of M1. Then, we have to show that β¯2ΩSu is a Kd-module ideal of M2∀u∈C. By Lemma 5.4, we have Ωβ¯1Su=β¯2ΩSu for all S⊆M1 and ∀u∈C.
Letri∈Ωβ¯1Su∀u∈C. Then, there is ti∈β¯1Su such that Ωti=ri. Hence, Ω is a Kd-module homomorphism and β¯1Su is a Kd-module ideal of M1∀u∈C. So, we have ∨i∈Iti∈β¯1Su, ⇒Ω∨i∈Iti∈Ωβ¯1Su. Thus, ∨i∈IΩti∈Ωβ¯1Su. Hence, ∨i∈Iri∈Ωβ¯1Su.
Letr≤s and s∈Ωβ¯1Su. Then, there is t∈M1 and u∈β¯1Su such that Ωt=r and Ωu=s. Hence, r≤s, that is, Ωt≤Ωu and Ωt∨Ωu=Ωu. Further, Ω is a Kd-module homomorphism. Thus, we have Ωt∨u=Ωt∨Ωu, =Ωu∈Ωβ¯1Su, ⇒Ωt∨u∈Ωβ¯1Su. By Lemma 5.5(2), we have t∨u∈β¯1Su. Hence, β¯1Su is a lower set and t≤t∨u. Thus, t∈β¯1Su and Ωt∈Ωβ¯1Su so r∈Ωβ¯1Su. Hence, Ωβ¯1Su is a lower set.
Now, we show thatk⊕r∈Ωβ¯1Su for all r∈Ωβ¯1Su and ∀k∈Kd. Let k=Ωs∈Kd and r=Ωt∈Ωβ¯1Su. So, Ωt∈Ωβ¯1Su. Then, by Lemma 5.5(2), we have t∈β¯1Su. Hence, β¯1Su is an Kd-module ideal of M1. Thus, s⊕t∈β¯1Su. Thus, Ωs⊕t∈Ωβ¯1Su. Hence, Ω is an Kd-module homomorphism. Thus, Ωs⊕t=Ωs+Ωt, that is, Ωs⊕Ωt∈Ωβ¯1Su. Hence, k⊕r∈Ωβ¯1Su. Thus, Ωβ¯1Su=β¯2ΩSu is a Kd-module ideal of M2∀u∈C.
Conversely, supposeΩβ¯1Su=β¯2ΩSu is a Kd-module ideal of M2∀u∈C.
Supposesi∈β¯1Su∀u∈C for some i∈I. Then, Ωsi∈Ωβ¯1Su∀u∈C. Hence, Ωβ¯1Su is a Kd-module ideal and Ω is a Kd-module homomorphism. So, ∨i∈IΩsi=Ω∨i∈Isi∈Ωβ¯1Su, ∀u∈C. Then, by Lemma 5.5(2), we have ∨i∈Isi∈β¯1Su and β¯1Su is directed.
Letr≤sand s∈β¯1Su. Then, Ωr≤Ωs and Ωs∈Ωβ¯1Su. Since Ωβ¯1Su is a lower set, we have Ωr∈Ωβ¯1Su. Thus, by Lemma 5.5(2), we have r∈β¯1Su. So, β¯1Su is a lower set. Suppose k∈Kd and v∈β¯1Su Then, Ωk∈Kd and Ωv∈Ωβ¯1Su. Since Ωβ¯1Su is an Kd-module ideal of M2, we have Ωk⊕v=k⊕Ωv∈Ωβ¯1Su, ⇒Ωk⊕v∈Ωβ¯1Su∀u∈C and then by Lemma 5.5(2), we have k⊕v∈β¯1Su. By the above discussion, we have β¯1Su is Kd-module ideal of M1.(2)
Letβ¯1ΩSu is a sub-Kd-module of M1∀u∈C. Then, we have to show that β¯2ΩSu is sub-Kd-module of M2∀u∈C. By Lemma 5.4, we have Ωβ¯1Su=β¯2ΩSu for all S⊆M1 and ∀u∈C.
Letri∈Ωβ¯1Su∀u∈C Then, there is ti∈β¯1Su such that Ωti=ri. Hence, Ω is a Kd-module homomorphism and β¯1Su is a sub-Kd-module of M1, ∀u∈C. So, ∨i∈Iti∈β¯1Su, ⇒Ω∨i∈Iti∈Ωβ¯1Su. So, we have ∨i∈IΩti∈Ωβ¯1Su. Hence, ∨i∈Iri∈Ωβ¯1Su.
Now, we show thatk⊕r∈Ωβ¯1Su for all r∈Ωβ¯1Su and ∀k∈Kd. Letk=Ωs∈Kd and r=Ωt∈Ωβ¯1Su.So, Ωt∈Ωβ¯1Su. Then, by Lemma 5.5(2), we have t∈β¯1Su. Hence, β¯1Su is a sub-Kd-module of M1. This shows that k⊕t∈β¯1Su. Thus, Ωk⊕t∈Ωβ¯1Su. Hence, Ω is an Kd-module homomorphism. Thus, Ωk⊕t=Ωk+Ωt, that is, Ωk⊕Ωt∈Ωβ¯1Su. Hence, k⊕t∈Ωβ¯1Su. Thus, we have that Ωβ¯1Su=β¯2ΩSu is a sub-Kd-module of M2∀u∈C.
Conversely, supposeΩβ¯1Su=β¯2ΩSu is a sub-Kd-module of M2∀u∈C. Suppose, si∈β¯1Su∀u∈C for some i∈I.Then, Ωsi∈Ωβ¯1Su∀u∈C. Hence, Ωβ¯1Su is a sub-Kd-module and Ω is a Kd-module homomorphism. So, ∨i∈IΩsi=Ω∨i∈Isi∈Ωβ¯1Su, ∀u∈C. Then, by Lemma 5.5(2), we have ∨i∈Isi∈β¯1Su.
Supposek∈Kdand v∈β¯1Su. Then, Ωk∈Kd and Ωv∈Ωβ¯1Su. Since Ωβ¯1Su is an sub-Kd-module ideal of M, we have Ωk⊕v=k⊕Ωv∈Ωβ¯1Su⇒Ωk⊕v∈Ωβ¯1Su∀u∈C and then by Lemma 5.5(2) we have k⊕v∈β¯1Su. From the above discussion, we get β¯1Su is sub-Kd-module of M1.


### 5.7. Remark

With similar arguments, Theorem 5.6 can be similarly proved but for the foresets.


### 5.8. Theorem

LetS⊆M1 and let Ω be a surjective Kd-module homomorphism on a Kd-module M and β2,C be a SBR and SCTR and with respect to aftersets on M2. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
β¯1Su is a Kd-module ideal of M1 if and only if β¯2ΩSu is a Kd-module ideal of M2∀u∈C(2)
β¯1Su is a sub-Kd-module of M1 if and only if β¯2ΩSu is a sub-Kd-module of M2∀u∈CProof.
The proof is similar in view of Theorem 5.6.


## 5.1. Lemma

LetM1 and M2 be Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
β1,C is a SBR on M1(2)
β1,C is SRRE if β2,C is SRRE(3)
β1,C is SCRE if β2,C is SCREProof.
The proof is obvious.


## 5.2. Lemma

LetM1 and M2 be a Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C Then, β1,C is SCTR with respect to aftersets if β2,C is SCTR with respect to aftersets and Ω is one-one.Proof.
Clearlyrβ1u∨sβ1u⊆r∨sβ1u∀u∈C.
Conversely, suppose thatx∈r∨sβ1u. Then, by definition of aftersets r∨s,x∈β1u. Thus, Ωr∨s,Ωx∈β2u, that is, Ωr∨Ωs,Ωx∈β2u. This implies that Ωx∈Ωr∨Ωsβ2u. Since β2,C is SCTR with respect to aftersets, Ωx∈Ωrβ2u∨Ωsβ2u. Then, there is t1∈Ωrβ2u and t2∈Ωrβ2u such that Ωx=t1∨t2. As Ω is onto, we have s1,s2∈M1, Ωs1=t1 and Ωs2=t2. Thus, Ωx=Ωs1∨Ωs2. This implies, Ωx=Ωs1∨s2. As Ω is one-one, we have x=s1∨s2. Now, t2∈Ωrβ2u, ⇒, Ωs1∈Ωrβ2uww, ⇒Ωr,Ωs1∈β2u, ⇒r,s1∈β1u, ⇒s1∈rβ1u. Also now, t2∈Ωsβ2u, ⇒, Ωs2∈Ωsβ2u, ⇒Ωs,Ωs2∈β2u, ⇒s,s2∈β1u, ⇒s2∈sβ1u. Now, x=s1∨s2∈rβ1u∨sβ1u. That is, x∈rβ1u∨sβ1u. This implies that r∨sβ1u⊆rβ1u∨sβ1u. Hence, r∨sβ1u=rβ1u∨sβ1u∀u∈C. Similarly, we can prove that k⊕rβ1u=k⊕rβ1u∀u∈C. This show that β1,C is SCTR with respect to aftersets.


## 5.3. Lemma

LetM1 and M2 be a Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, β1,C is SCTR with respect to foresets if β2,C is SCTR with respect to foresets and Ω is one-one.Proof.
The proof is obvious.


## 5.4. Lemma

LetM1 and M2 be Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C Then, Ωβ¯1Su=β¯2ΩSu for S⊆M1 and ∀u∈C.Proof.
Lets∈β¯2ΩSu. Then, sβ2u∩ΩS≠∅. Then, there is r∈sβ2u∩ΩS such that r∈sβ2u and r∈ΩS. So, s,r∈β2u and r∈ΩS. Since Ω is onto, there exist x∈S and y∈M1 such that r=Ωx and s=Ωy. Thus, s,r=Ωy,Ωx∈β2u, that is, (Ωy,Ωx∈β2u. Hence, yx∈β1u, that is, x∈yβ1u and x∈S. This implies x∈yβ1u∩S. So, we have y∈β¯1Su, ⇒Ωy∈Ωβ¯1Su, ⇒s∈Ωβ¯1Su. This show that ⇒s∈Ωβ¯1Su.
Conversely, letz∈Ωβ¯1Su. Then, there is r∈β¯1Su such that rβ1u∩S≠∅ and z∈Ωr. Thus, x∈rβ1u∩S such that x∈rβ1u and x∈S. So, r,x∈β1u and x∈S, ⇒Ωx∈ΩS and (Ωr,Ωx∈β2u, ⇒Ωx∈Ωrβ2u and Ωx∈ΩS. Hence, Ωx∈Ωrβ2u∩ΩS, that is, ∅≠Ωrβ2u∩ΩS, ⇒Ωr∈β¯2ΩS, ⇒z∈β¯2ΩSu. This shows that Ωβ¯1Su⊆β¯2ΩSu. Hence, Ωβ¯1Su=β¯2ΩSu∀u∈C.


## 5.5. Lemma

LetM1 and M2 be Kd-modules and β2,C be a SBR on M2. Let Ω:M1⟶M2 be a surjective Kd-module homomorphism. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
Ωβ1¯Su⊆β2¯Ωsu∀u∈C; if Ω is one-one, then Ωβ1¯Su=β2¯Ωsu∀u∈C(2)
IfΩ:M1⟶M2 is one-one, then Ωx∈Ωβ¯1Su if and only if x∈β¯1Su∀u∈CProof.
(1)
Letr∈Ωβ1¯Su for u∈C. Then, there is s∈β1¯Su such that sβ1u⊆S and r=Ωx. Suppose that y∈rβ2u. Then, there is x∈M1 such that Ωx=y and Ωx∈Ωsβ2u, that is, (Ωs,Ωx)∈β2u. Hence, s,x∈β1u by definition of aftersets x∈sβ1u. Hence, sβ1u⊆S and x∈S. So, Ωx∈ΩS using above Ωx=y, ⇒y∈ΩS. Thus, rβ2u⊆ΩS. Hence, r∈β2¯Ωsuwhich shows that Ωβ1¯Su⊆β2¯Ωsu∀u∈C. Conversely, let s∈β2¯Ωsu∀u∈C. Then, sβ2u⊆ΩS. Suppose that Ω is one-one. Then, there is r∈M1 such that s=Ωr and Ωrβ2u⊆ΩS. Let x∈rβ1u by definition of aftersets, we have r,x∈β1u. Then, Ωr,Ωx∈β2u, that is, Ωx∈Ωrβ2u. Since Ωrβ2u⊆ΩS, Ωx∈ΩS, ⇒x∈S. Thus, rβ1u⊆S, that is, r∈β1¯Su, ⇒Ωr∈Ωβ1¯Su since s=Ωr. Hence, s∈Ωβ1¯Su. So,β2¯Ωsu⊆Ωβ1¯Su∀u∈C. Thus, Ωβ1¯Su=β2¯Ωsu∀u∈C.(2)
LetΩr∈Ωβ¯1Su. Then, there is s∈β¯1Su such that Ωr=Ωs. Since Ω is one-one, we get r=s. Thus, r∈β¯1Su. Conversely, let r∈β¯1Su∀u∈C. Then, Ωr∈Ωβ¯1Su∀u∈C.


## 5.6. Theorem

LetS⊆M1 and let Ω be a surjective Kd-module homomorphism on a Kd-modules M1,M2 and β2,C be a SBR and SCTR with respect to aftersets on M2. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
β¯1Su is a Kd-module ideal of M1 if and only if β¯2ΩSu is a Kd-module ideal of M2∀u∈C(2)
β¯1Su is a sub-Kd-module of M1 if and only if β¯2ΩSu is a sub-Kd-module of M2∀u∈CProof.
(1) Letβ¯1Su be a Kd-module ideal of M1. Then, we have to show that β¯2ΩSu is a Kd-module ideal of M2∀u∈C. By Lemma 5.4, we have Ωβ¯1Su=β¯2ΩSu for all S⊆M1 and ∀u∈C.
Letri∈Ωβ¯1Su∀u∈C. Then, there is ti∈β¯1Su such that Ωti=ri. Hence, Ω is a Kd-module homomorphism and β¯1Su is a Kd-module ideal of M1∀u∈C. So, we have ∨i∈Iti∈β¯1Su, ⇒Ω∨i∈Iti∈Ωβ¯1Su. Thus, ∨i∈IΩti∈Ωβ¯1Su. Hence, ∨i∈Iri∈Ωβ¯1Su.
Letr≤s and s∈Ωβ¯1Su. Then, there is t∈M1 and u∈β¯1Su such that Ωt=r and Ωu=s. Hence, r≤s, that is, Ωt≤Ωu and Ωt∨Ωu=Ωu. Further, Ω is a Kd-module homomorphism. Thus, we have Ωt∨u=Ωt∨Ωu, =Ωu∈Ωβ¯1Su, ⇒Ωt∨u∈Ωβ¯1Su. By Lemma 5.5(2), we have t∨u∈β¯1Su. Hence, β¯1Su is a lower set and t≤t∨u. Thus, t∈β¯1Su and Ωt∈Ωβ¯1Su so r∈Ωβ¯1Su. Hence, Ωβ¯1Su is a lower set.
Now, we show thatk⊕r∈Ωβ¯1Su for all r∈Ωβ¯1Su and ∀k∈Kd. Let k=Ωs∈Kd and r=Ωt∈Ωβ¯1Su. So, Ωt∈Ωβ¯1Su. Then, by Lemma 5.5(2), we have t∈β¯1Su. Hence, β¯1Su is an Kd-module ideal of M1. Thus, s⊕t∈β¯1Su. Thus, Ωs⊕t∈Ωβ¯1Su. Hence, Ω is an Kd-module homomorphism. Thus, Ωs⊕t=Ωs+Ωt, that is, Ωs⊕Ωt∈Ωβ¯1Su. Hence, k⊕r∈Ωβ¯1Su. Thus, Ωβ¯1Su=β¯2ΩSu is a Kd-module ideal of M2∀u∈C.
Conversely, supposeΩβ¯1Su=β¯2ΩSu is a Kd-module ideal of M2∀u∈C.
Supposesi∈β¯1Su∀u∈C for some i∈I. Then, Ωsi∈Ωβ¯1Su∀u∈C. Hence, Ωβ¯1Su is a Kd-module ideal and Ω is a Kd-module homomorphism. So, ∨i∈IΩsi=Ω∨i∈Isi∈Ωβ¯1Su, ∀u∈C. Then, by Lemma 5.5(2), we have ∨i∈Isi∈β¯1Su and β¯1Su is directed.
Letr≤sand s∈β¯1Su. Then, Ωr≤Ωs and Ωs∈Ωβ¯1Su. Since Ωβ¯1Su is a lower set, we have Ωr∈Ωβ¯1Su. Thus, by Lemma 5.5(2), we have r∈β¯1Su. So, β¯1Su is a lower set. Suppose k∈Kd and v∈β¯1Su Then, Ωk∈Kd and Ωv∈Ωβ¯1Su. Since Ωβ¯1Su is an Kd-module ideal of M2, we have Ωk⊕v=k⊕Ωv∈Ωβ¯1Su, ⇒Ωk⊕v∈Ωβ¯1Su∀u∈C and then by Lemma 5.5(2), we have k⊕v∈β¯1Su. By the above discussion, we have β¯1Su is Kd-module ideal of M1.(2)
Letβ¯1ΩSu is a sub-Kd-module of M1∀u∈C. Then, we have to show that β¯2ΩSu is sub-Kd-module of M2∀u∈C. By Lemma 5.4, we have Ωβ¯1Su=β¯2ΩSu for all S⊆M1 and ∀u∈C.
Letri∈Ωβ¯1Su∀u∈C Then, there is ti∈β¯1Su such that Ωti=ri. Hence, Ω is a Kd-module homomorphism and β¯1Su is a sub-Kd-module of M1, ∀u∈C. So, ∨i∈Iti∈β¯1Su, ⇒Ω∨i∈Iti∈Ωβ¯1Su. So, we have ∨i∈IΩti∈Ωβ¯1Su. Hence, ∨i∈Iri∈Ωβ¯1Su.
Now, we show thatk⊕r∈Ωβ¯1Su for all r∈Ωβ¯1Su and ∀k∈Kd. Letk=Ωs∈Kd and r=Ωt∈Ωβ¯1Su.So, Ωt∈Ωβ¯1Su. Then, by Lemma 5.5(2), we have t∈β¯1Su. Hence, β¯1Su is a sub-Kd-module of M1. This shows that k⊕t∈β¯1Su. Thus, Ωk⊕t∈Ωβ¯1Su. Hence, Ω is an Kd-module homomorphism. Thus, Ωk⊕t=Ωk+Ωt, that is, Ωk⊕Ωt∈Ωβ¯1Su. Hence, k⊕t∈Ωβ¯1Su. Thus, we have that Ωβ¯1Su=β¯2ΩSu is a sub-Kd-module of M2∀u∈C.
Conversely, supposeΩβ¯1Su=β¯2ΩSu is a sub-Kd-module of M2∀u∈C. Suppose, si∈β¯1Su∀u∈C for some i∈I.Then, Ωsi∈Ωβ¯1Su∀u∈C. Hence, Ωβ¯1Su is a sub-Kd-module and Ω is a Kd-module homomorphism. So, ∨i∈IΩsi=Ω∨i∈Isi∈Ωβ¯1Su, ∀u∈C. Then, by Lemma 5.5(2), we have ∨i∈Isi∈β¯1Su.
Supposek∈Kdand v∈β¯1Su. Then, Ωk∈Kd and Ωv∈Ωβ¯1Su. Since Ωβ¯1Su is an sub-Kd-module ideal of M, we have Ωk⊕v=k⊕Ωv∈Ωβ¯1Su⇒Ωk⊕v∈Ωβ¯1Su∀u∈C and then by Lemma 5.5(2) we have k⊕v∈β¯1Su. From the above discussion, we get β¯1Su is sub-Kd-module of M1.


## 5.7. Remark

With similar arguments, Theorem 5.6 can be similarly proved but for the foresets.


## 5.8. Theorem

LetS⊆M1 and let Ω be a surjective Kd-module homomorphism on a Kd-module M and β2,C be a SBR and SCTR and with respect to aftersets on M2. Set β1u=s,t∈M1×M1:Ωs,Ωt∈β2u∀u∈C. Then, the following holds:(1)
β¯1Su is a Kd-module ideal of M1 if and only if β¯2ΩSu is a Kd-module ideal of M2∀u∈C(2)
β¯1Su is a sub-Kd-module of M1 if and only if β¯2ΩSu is a sub-Kd-module of M2∀u∈CProof.
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



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