Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations

List of Acronyms*

1.   Introduction

Molodtsov ^[6] proposed the soft set (𝑆-set) theory which has many applications to find solutions of problems in economics, medicine, engineering and social sciences. A 𝑆-set over a set $U$ under consideration is a pair $(F, A) = \{F\left(\alpha \right)\subseteq U:\forall \alpha \in A\}$ where $F$ is a function from $A$ (set of parameter) to $P\left(U\right)$. The 𝑆-sets are generalization of conventional sets. Molodstov also discussed how this approach could be used to elaborate a variety of problems. The specification of a parameter is not required in 𝑆-set theory. This makes 𝑆-set theory a natural mathematical framework for approximate logic. Numerous theories were proposed to deal with uncertainty and imprecision following the development of 𝑆-set theory. Some of those are extensions of 𝑆-sets, while others strive to deal with uncertainty in another suitable way. To get better and precise result, rough set ($R$-set) presented by Pawlak ^[27] is combined with 𝑆-set named as rough soft set. In $R$-set theory an important part is played by an equivalence relation. There is always a question left whether an equivalence relation is simple to obtain. Thus, the combined $R$-set with 𝑆-set is termed as rough soft set defined by Feng et al., ^[7] is as follows: Let $G = (f, A)$ be a soft set over $U$. Then the pair $P = (U, G)$ is called soft approximation space. Based on $P$, following are defined as ${\underline {apr}}_{P}\left(X\right) = \{u\in U:\exists a\in A[u\in f\left(a\right)\subseteq X\left]\right\}$ and ${\overline{apr}}_{P}\left(X\right) = \{u\in U:\exists a\in A[u\in f\left(a\right)\cap X\ne \phi \left]\right\}$ where $X\subseteq U$, ${\underline {apr}}_{P}\left(X\right)$ and ${\overline{apr}}_{P}\left(X\right)$ are called lower and upper soft rough sets. In this way, further approximation of 𝑆-sets in a different way was proposed by Shabir et al., ^[14] to give a proper illustration of the information and allow a greater degree of freedom and flexibility in representing uncertainty which is as follows: Let $(F, V)$ be a soft binary relation from ${K}_{1}$ to ${K}_{2}$ (where ${K}_{1}$ to ${K}_{2}$ are universal sets under consideration). Thus, $({\underline {{F^M}} }, V)$ and $({\overline{F}}^{M}, V)$ are the lower and upper approximation of 𝑆-set $(M, V)$ over ${K}_{2}$ with respect to aftersets, are essentially two soft sets over ${K}_{1}$ defined as ${\underline {{F^M}} }\left(v\right) = \{k\in {K}_{1}:\phi \ne kF(v)\subseteq M(v\left)\right\}$ and $\left({\overline{F}}^{M}\right(v) = \{k\in {K}_{1}:kF\left(v\right)\cap M\left(v\right)\ne \phi \}$. From above discussion it is clear to understand that approximation of soft sets with respect to either aftersets or foresets by soft relations are simple and more suitable to handle different situations in different field of sciences.

1.1.   Background and importance of quantale module

The quantale module, introduced by Abramsky and Vickers ^[20], engaged a large number of researchers. By replacing rings by quantale and abelian group with complete lattices in the module over ring, the concept of the quantale module was developed. Abramsky and Vickers used the concept of a quantale module for the unified treatment of process semantics. The concept of modules over a commutative unital quantaleas by Rosenthal ^[11] provided a family of models of full linear logic. Russo ^[4] introduced an approach to data compression algorithms using quantale module homomorphism as an application. A quantale-theoretic approach to propositional deductive ^[5] systems has been developed in recent years, based on the notion that any propositional deductive system may be represented as a quantale module. However, despite of their multiple applications, the first systematic studies on the categories of quantale modules are rather recent ^[2,3,4,22]. On the other hand, the results presented in ^[15] and ^[21] clearly suggest that the algebraic categories of quantales, unital quantales, and quantale modules are worth to be further investigated.

1.2.   Literature review

The categories of modules over unital quantales were introduced by Russo. The main categorical properties were established and a special class of operators, called Q-module transforms, was defined ^[2]. Some applications of quantale modules with applications to logic and image processing were introduced Russo ^[4] including Free modules, hom-sets, products and coproducts, Q-module transforms, projective and injective Q-modules. Further decomposition and projectivity of quantale modules were discussed by Slesinger ^[18] and showed that every quantale module join-generated by its sub-set of join-irreducible elements can be uniquely decomposed into a collection of further indecomposable submodules. Further, he characterized regular projective essential modules that admitted this product decomposition as products of such cyclic quantale modules. The concept of Q-P quantale modules was defined by Liang and discussed categorical properties of Q-P quantale modules ^[23]. Algebraic properties of the category of Q-P quantale modules were defined by ^[23] and the structure of the free Q-P quantale modules generated by a set were obtained. Modules on involutive quantales were defined by Heymans and Stubbe ^[9]. They defined Canonical Hilbert structure, an application of sheaf theory. In 2018, rough set was applied to substructures of quantale modules and defined rough Q-submodule. Further the concepts of set-valued homomorphism and strong set-valued homomorphism of Q-modules were introduced, and related properties are investigated ^[25]. With the help of soft relations, substructures of quantale modules were approximated by Qurashi et al., ^[24]. An application of rough set theory based on Multi source information fusion was presented by Zhang et al., ^[16]. Zhang et al., introduces further heterogeneous feature selection dependent on neighborhood combination entropy ^[17].

1.3.   Research gap in the current literature and motivation of the study

The literature overview above highlights some developments in both classical and R-set theory. Furthermore, even though several results about rough submodule and rough submodule ideals of quantale module and generalized lower and upper approximations operators based on set-valued homomorphism of quantale modules have been demonstrated but there are still some open questions remain to be answered.

(1) In classical quantale module theory, there is a lot of contributions but there is no attention on its generalization, for example soft quantale module, different characterization of fuzzy substructures in quantale modules like $(ϵ, ϵ\bigvee q)$-fuzzy substructures, $(ϵ, ϵ\bigvee {q}_{k})$-fuzzy substructures of quantale modules and $({ϵ}_{\gamma }, {ϵ}_{\gamma }\bigvee {q}_{\delta })$-fuzzy substructures of Quantale modules. Rough neutrosophic soft substructures in quantale modules, fuzzy bipolar soft substructures in quantale modules.

(2) Study of different substructures of quantale modules by soft relations is present in existing literature. Further, approximation of soft ideals by soft relations in semigroups was proposed by Kanwal and Shabir ^[19]. Since soft substructures in quantale modules are generalization of its substructures so therefore, it is important to understand the characterization of soft substructures in quantale modules dependent on soft relations.

(3) Roughness of substructures with the help of congruence relations and set-valued homomorphism is in the lecturer ^[25]. A natural question comes into mind, what will be the behavior of roughness of soft substructures by soft relations is a logical question to ask.

(4) Some fundamental and important theorems of quantale module homomorphism are discussed in ^[24]. Therefore, it is necessary to discuss these remarkable theorems in the context of soft quantale module homomorphism.

(5) Numerous algebraic aspects of substructures of quantale module with and without by soft relations have been studied in the literature. In the context of soft substructures of quantale modules, these studies have yet to be examined from a broader perspective.

The ultimate goal of this research is to address the aforementioned open problems and fulfil the knowledge gap in the existing literature.

1.4.   Comparative study and limitations of the current research

The results proved in this paper are valid for substructures in quantale modules. Moreover, every fuzzy set is an IFS, so the present study can also be applied to fuzzy substructures and intuitionistic fuzzy substructures in quantale modules by soft relations. Further, approximations of Pythagorean fuzzy sets by soft binary relations were presented by Bilal and Shabir ^[13]. So, we can define approximation of Pythagorean fuzzy substructures in quantale modules by soft relations. However, we cannot apply these results directly to q-rung orthopair fuzzy ideals, picture fuzzy ideals and fuzzy soft hyper substructures in quantale modules. Therefore, separate studies are recommended for these generalized structures. This is the main limitation of our research.

The detail of paper is as follows. In Section 2, some necessary definitions related to substructures of quantale module are presented. Further, rough set, soft sets, soft substructures in quantale module and soft binary relations are discussed. In Section 3, some characterization of subsets of quantale modules are described. Moreover, different rough soft substructures with respect to aftersets and foresets will be expressed in Section 4. In the last section, soft quantale module homomorphism with its relation to upper (lower) approximations and homomorphic images are described.

2.   Preliminaries

In this section, we define soft substructures of quantale modules and present some basic notions of soft sets, rough sets and substructures of quantale modules, which are the main tools in our study.

Definition 2.1. ^[1] Let $Ǫ$ be a ${C}_{ltc}.$ Define an associative binary operation $\otimes \;\mathrm{o}\mathrm{n}\;Ǫ$ satisfying,

1) $\mathfrak{r}\otimes \left({\bigvee }_{l\in L}{\mathcal{z}}_{l}\right) = {\bigvee }_{l\in L}\left(\mathfrak{r}\otimes {\mathcal{z}}_{l}\right);$

2) $\left({\bigvee }_{l\in L}{\mathfrak{r}}_{l}\right)\otimes \mathcal{z} = {\bigvee }_{l\in L}\left({\mathfrak{r}}_{l}\otimes \mathcal{z}\right)$.

$\forall \mathfrak{r}, \mathcal{z}\in \mathcal{Ǫ}$ and $\left\{{\mathcal{z}}_{l}\right\}, \left\{{\mathfrak{r}}_{l}\right\}\subseteq Ǫ\left(l\in L\right).\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}\left(Ǫ, \otimes \right)\;\mathrm{i}\mathrm{s}\;\mathrm{a}\;\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{e}.$

Let ${\mathcal{R}}_{l}, {\mathcal{R}}_{1}, {\mathcal{R}}_{2}\subseteq Ǫ.$ Then the following are defined

Definition 2.2. ^[21] Let $Ǫ$ be a quantale and $\mathcal{Q}$ be a sup-lattice equipped with a left action $\star :\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{Ǫ}\times$ $\mathcal{Q} \to \mathcal{Q}.$ Then $\mathcal{Q}$ is called left $\mathcal{Ǫ}$-module over the quantale $\mathcal{Ǫ}$, if it satisfies the following criteria,

1) ${(\vee }_{l\in L}{\mathcal{a}}_{l}) \star \mathcal{x} = {\vee }_{l\in L}\left({\mathcal{a}}_{l}\star \mathcal{x}\right);$

2) $\mathcal{a}\star \left({\vee }_{i\in I}{\mathcal{x}}_{i}\right) = {\vee }_{i\in I}(\mathcal{a}\star {\mathcal{x}}_{i})$;

3) $\left(\mathcal{a}\otimes \mathcal{b}\right)\star \mathcal{x} = \mathcal{a}\star \left(\mathcal{b}\star \mathcal{x}\right).$

for any $\mathcal{a}, \mathcal{b}\in \mathcal{Ǫ}$, $\left\{{\mathcal{a}}_{l}\right\}\subseteq Ǫ\left(l\in L\right), \mathcal{x}\in \mathcal{Q}$, and $\left\{{\mathcal{x}}_{i}\right\}\subseteq \mathcal{Q}(i\in I)$

way. We write $\mathcal{Q}$ for left $\mathcal{Ǫ}$-module over the quantale throughout in this thesis. For a $\mathcal{Ǫ}$-module $\mathcal{Q}$, $A\subseteq Ǫ \;and \; \mathcal{m}\in \mathcal{Q}$ we have,

where $B\subseteq \mathcal{Q}$. $For \;A, B, {A}_{l}\subseteq \mathcal{Q}(l\in L$), We write

Example 2.3. Let $Ǫ = \left\{\mathcal{L}, \mathcal{y}, z, \mathcal{Ŧ}\right\}$ be the ${C}_{ltc}$ shown in Figure 1 and operation $\otimes$ on $Ǫ$ is shown in Table 1. Then $\left(Ǫ, \otimes \right)$ is a quantale. Let $\mathcal{Q} = \left\{\mathcal{L}, \mathcal{x}, \mathcal{Ŧ}\right\}$ be a sup lattice. The order relation of $\mathcal{Q}$ is given in Figure 2. And the left action on $\mathcal{Q}$ i.e., $\star :\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{Ǫ}\mathcal{ }\times Q\to \mathcal{Q}$ is shown in Table 2. Then it is easy to verify that $\mathcal{Q}$ is $\mathcal{Ǫ}$-module.

Example 2.4. Every quantale $\mathcal{Ǫ}$ is a certainly a $\mathcal{Ǫ}$-module over $\mathcal{Ǫ}$.

Definition 2.5. ^[20] Let $\mathcal{Q}$ be a $\mathcal{Ǫ}$-module. If a subset ${\mathcal{Q}}_{1}\subseteq \mathcal{Q}$ satisfies the following axioms for any $\mathcal{m}\in {\mathcal{Q}}_{1}, \left\{{\mathcal{m}}_{i}\right\}\subseteq {\mathcal{Q}}_{1}\;and\;\gamma \in Ǫ$, we have

1) ${\vee }_{i\in I}{\mathcal{m}}_{i}\in {\mathcal{Q}}_{1}\forall {\mathcal{m}}_{i}\in {\mathcal{Q}}_{1}$;

2) $\gamma \star \mathcal{m}\in {\mathcal{Q}}_{1}\forall \mathcal{m}\in {\mathcal{Q}}_{1}, \gamma \in Ǫ.$

Then ${\mathcal{Q}}_{1}$ is called $Ǫ$-submodule (${Ǫ}_{S})$ of $\mathcal{Q}$.

Definition 2.6. ^[20] Let $\mathcal{I}\ne \mathcal{\varnothing }$ be a subset of $\mathcal{Ǫ}$-module $\mathcal{Q}$. Then $\mathcal{I}$ is called $\mathcal{Ǫ}$-sub module ideal (${\mathcal{Ǫ}}_{\mathrm{I}})$ of $\mathcal{Q}$ if following holds;

1) $A\subseteq \mathcal{I}$ implies $\vee A\subseteq \mathcal{I};$

2) $\mathcal{x}\in \mathcal{I}$ and $\mathcal{b}\le \mathcal{x}$ implies $\mathcal{b}\in \mathcal{I}$;

3) $\mathcal{x}\in \mathcal{I}$ implies $\gamma \star \mathcal{x}\in \mathcal{I}, \mathcal{ }\forall \gamma \in Ǫ.$

Example 2.7. Consider the quantale module given in Example 2.3. Then $\left\{\mathcal{L}\right\}, \left\{\mathcal{L}, \mathcal{x}\right\}, \left\{\mathcal{L}, \mathcal{x}, \mathcal{Ŧ}\right\}$ are ${\mathcal{Ǫ}}_{I}$ of $\mathcal{Q}.$

Definition 2.8. ^[5] If $Л$ is a mapping given by $Л :V\to P\left(\mathcal{Q}\right)$ where $V$ $\subseteq$ E (S.P), then the pair $(Л, V)$ is called a soft set over $\mathcal{Q}$.

Definition 2.9. ^[8] Assume ($\mathcal{M}, {V}_{1})$ and ($\mathcal{N}, {V}_{2})$ be two soft sets over $\mathcal{Q}$. Then we called ($\mathcal{M}, {V}_{1})$ soft subset ($\mathcal{N}, {V}_{2})$ if the conditions listed below are satisfied,

● ${V}_{1}\subseteq {V}_{2}$;

● $\mathcal{M}\left(v\right)\subseteq \mathcal{N}\left(v\right)\forall v\in {V}_{1}.$

We will represent soft subsets defined in above manner by ($\mathcal{M}, {V}_{1})\subseteq \left(\mathcal{N}, {V}_{2}\right).$

Definition 2.10. ^[8] Let $\left(Л, V\right)$ be a soft set over $\mathcal{Q}\times \mathcal{Q}$, i.e, $\mathcal{Л}\mathcal{ }:V\to P(\mathcal{Q}\times \mathcal{Q})$. Then $\left(\mathcal{Л}, V\right)$ is called a soft binary relation (SBIR) over $\mathcal{Q}$. A SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ is a soft set $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. That is $Л :V\to P\left({\mathcal{Q}}_{1}\times {\mathcal{Q}}_{2}\right).$

Definition 2.11. Let $\left(Л, V\right)$ be a soft set over quantale module $\mathcal{Q}$. Then the soft substructures of quantale modules are defined as,

1) $\left(\mathcal{Л}, V\right)$ is called soft quantale submodule $\left({Ǫ}_{S}\right)$ over $\mathcal{Q}$ iff $\mathcal{Л}\left(v\right)$ is a ${Ǫ}_{S}$ of $\mathcal{Q}$, $\forall v\in V$.

2) $\left(Л, V\right)$ is called soft quantale submodule ideal $\left({Ǫ}_{I}\right)$ over $\mathcal{Q}$ iff $\mathcal{Л}\left(v\right)$ is a ${Ǫ}_{I}$ of $\mathcal{Q}$, $\forall v\in V$.

Definition 2.12. ^[27] Let $\mathrm{Ҩ}$ be an equivalence relation on a non-empty finite set $\mathcal{Q}$. Then $(\mathcal{Q}, \mathcal{Ҩ})$ is called an approximation space. Let $\mathcal{C}$ be a subset of $\mathcal{Q}$. Then $\mathcal{C}$ may or may not be written as union of the equivalence classes of $\mathcal{Q}$. We say that $\mathcal{C}$ is definable, if $\mathcal{C}$ can be written as union of some equivalence classes of $\mathcal{Q}$. Otherwise, it is called not definable. In case, if $\mathcal{C}$ is not definable, then $\mathcal{C}$ can be approximated by two definable subsets called the lower and upper approximations of $\mathcal{C}$. These approximations are defined as follows,

3.   Approximation of soft subsets of quantale module by soft relation

In this section, we approximate the subsets of quantale module by using soft relations.

Definition 3.1. ^[14] Assume $V$ is the subset of E (S.P) and $(Л, V)$ be a SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ i.e., $Л:V\to P$(${\mathcal{Q}}_{1}\times {\mathcal{Q}}_{2}$). Thus, the ${LO}_{ap}({\underline {Л}}^{\mathcal{M}}, V)$ and ${UP}_{ap}({\overline{Л}}^{\mathcal{M}}, V)$ w.r.t the afterset of soft set $(\mathcal{M}, V)$ over ${\mathcal{Q}}_{2}$ are essentially two soft sets over ${\mathcal{Q}}_{1}$ defined as

and

The ${LO}_{ap}\left({}_{}{}^{\mathcal{N}}{\underline {{\mathcal{Л}}}}, V\right)$ and ${UP}_{ap}$(${}_{}{}^{\mathcal{N}}\overline{\mathcal{Л}}, V$) of w.r.t the foreset of a soft set $(\mathcal{N}, V)$ over ${\mathcal{Q}}_{1}$ are two soft sets over ${\mathcal{Q}}_{2}$ defined as,

and

where ${\gamma }_{1}Л\left(v\right) = \{{\gamma }_{2}\in {\mathcal{Q}}_{2}:({\gamma }_{1}, {\gamma }_{2})\in Л(v\left)\right\}$ is called the afterset of ${\gamma }_{1}$ and $Л\left(v\right){\gamma }_{2} = \{{\gamma }_{1}\in {\mathcal{Q}}_{1}: ({\gamma }_{1}, {\gamma }_{2})\in Л(v\left)\right\}$ is called the foreset of $\gamma$₂.

Remark 3.2. (1) For each soft set $(\mathcal{M}, V)$ over ${\mathcal{Q}}_{2}$, ${\underline {Л}}^{\mathcal{M}}:V\to P$(${\mathcal{Q}}_{1}$) and ${\overline{Л}}^{\mathcal{M}}:V\to P$(${\mathcal{Q}}_{1}$).

(2) For each soft set $(\mathcal{N}, V)$ over ${\mathcal{Q}}_{1}$, ${}_{}{}^{\mathcal{N}}{\underline {{\mathcal{Л}}}}:V\to P$(${\mathcal{Q}}_{2}$) and ${}_{}{}^{\mathcal{N}}\overline{\mathcal{Л}}:V\to P$(${\mathcal{Q}}_{2}$).

Theorem 3.3. ^[14] Let $(Л, V)$ and $(Ҩ, V$) be two SBIR from a non-empty set ${\mathcal{Q}}_{1}$ to a non-empty set ${\mathcal{Q}}_{2}$ and consider $({\mathcal{M}}_{1}, V)$ and $({\mathcal{M}}_{2}, V)$ be two soft set over ${\mathcal{Q}}_{2}$. Then

(1) $({\mathcal{M}}_{1}, V) \subseteq ({\mathcal{M}}_{2}, V) \Rightarrow ({\underline {Л}}^{{\mathcal{M}}_{1}}, V) \subseteq ({\underline {Л}}^{{\mathcal{M}}_{2}}, V),$

(2) $\left({\mathcal{M}}_{1}, V\right)\subseteq \left({\mathcal{M}}_{2}, V\right)\Rightarrow \left({\overline{Л}}^{{\mathcal{M}}_{1}}, V\right)\subseteq \left({\overline{Л}}^{{\mathcal{M}}_{2}}, V\right),$

(3) $\left({\underline {Л}}^{{\mathcal{M}}_{1}}, V\right)\cap \left({\underline {Л}}^{{\mathcal{M}}_{2}}, V\right) = \left({\underline {Л}}^{{\mathcal{M}}_{1}\cap {\mathcal{M}}_{2}}, V\right),$

(4) $\left({\overline{Л}}^{{\mathcal{M}}_{1}}, V\right)\cap \left({\overline{Л}}^{{\mathcal{M}}_{2}}, V\right)\supseteq \left({\overline{Л}}^{{\mathcal{M}}_{1}\cap {\mathcal{M}}_{2}}, V\right),$

(5) $\left({\underline {Л}}^{{\mathcal{M}}_{1}}, V\right)\cup \left({\underline {Л}}^{{\mathcal{M}}_{2}}, V\right)\subseteq \left({\underline {Л}}^{{\mathcal{M}}_{1}\cup {\mathcal{M}}_{2}}, V\right),$

(6) $\left({\overline{Л}}^{{\mathcal{M}}_{1}}, V\right)\cup \left({\overline{Л}}^{{\mathcal{M}}_{2}}, V\right) = \left({\overline{Л}}^{{\mathcal{M}}_{1}\cup {\mathcal{M}}_{2}}, V\right),$

(7) $\left(Л, V\right)\subseteq \left(Ҩ, V\right)\Rightarrow \left({\underline {Ҩ}}^{{\mathcal{M}}_{1}}, V\right)\subseteq \left({\underline {Л}}^{{\mathcal{M}}_{1}}, V\right),$

(8) $\left(Л, V\right)\subseteq \left(Ҩ, V\right)\Rightarrow \left({\overline{Ҩ}}^{{\mathcal{M}}_{1}}, V\right)\supseteq \left({\overline{Л}}^{{\mathcal{M}}_{1}}, V\right).$

Theorem 3.4. Let $(Л, V)$ and $(Ҩ, V)$ are two SBIR from ${\mathcal{Q}}_{1}$ $\ne \varnothing$ to ${\mathcal{Q}}_{2}$ $\ne \varnothing$. Then for any soft set $(\mathcal{M}, V)$ over ${\mathcal{Q}}_{2},$ we have

(1) $\left({\overline{Л\cap Ҩ}}^{\mathcal{M}}, V\right)\subseteq \left({\overline{Л}}^{\mathcal{M}}, V\right)\cap \left({\overline{Ҩ}}^{\mathcal{M}}, V\right).$

(2) $\left({\underline {{Л\cap Ҩ}}}^{\mathcal{M}}, V\right)\supseteq \left({\underline {Л}}^{\mathcal{M}}, V\right)\cup \left({\underline {Ҩ}}^{\mathcal{M}}, V\right).$

Proof. The proof is obvious and will be immediately concluded from part (7) and (8) of Theorem 3.3. In general converse of above Theorem is not true we will present an example to justify this as follows.

Example 3.5. Assume $Ǫ_1=\{L,r,Ŧ\}$ and $Ǫ_2=\{L^{'},w,x,y,z,Ŧ^{'}\}$ be two C_ltc as shown in Figures 3 and 4 respectively. The associative binary operation $\otimes _1$ and $\otimes _2$ on $Ǫ_1$ and $Ǫ_2$ is defined as,

(1) $a{\otimes }_{1}b = a˄b$

(2) $a{\otimes }_{2}b = \mathcal{ }\mathcal{L}\mathcal{՛}$

Then ${\mathcal{Ǫ}}_{1}$ and ${Ǫ}_{2}$ are quantales by (1) and (2), and ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ are quantale modules by Tables 3 and 4.

Consider $V = \left\{{v}_{1}, {v}_{2}\right\}$ and define $Л:V \to P$(${\mathcal{Q}}_{1}\times$ ${\mathcal{Q}}_{2}$) and $Ҩ:V \to P$(${\mathcal{Q}}_{1}\times$ ${\mathcal{Q}}_{2}$) by,

and

and

Following are the aftersets corresponding to $\mathcal{Л}\left({v}_{1}\right)$ and $Ҩ\left({v}_{1}\right)$,

$\mathcal{L}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{ }\mathcal{Ŧ}\mathcal{՛}\right\}, \mathfrak{r}\mathfrak{Л}\left({v}_{1}\right) = \left\{\mathcal{x}, \mathcal{w}, \mathcal{y}\right\}$ and $\mathcal{Ŧ}\mathcal{ }\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{x}, \mathcal{z}, \mathcal{Ŧ}\mathcal{՛}\right\},$

$\mathcal{L}\mathcal{Ҩ}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{ }\mathcal{Ŧ}\mathcal{՛}\right\}, \mathfrak{r}\mathfrak{Ҩ}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}\right\}$ and $\mathcal{Ŧ}\mathcal{ }\mathcal{Ҩ}\left({v}_{1}\right) = \left\{\mathcal{x}, \mathcal{z}\right\}.$

Also, $\mathcal{L}\left(\mathcal{Л}\mathcal{ }\cap \mathcal{ }\mathcal{Ҩ}\right)\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{Ŧ}\mathcal{՛}\right\}, \mathfrak{r}\left(\mathfrak{Л}\mathfrak{ }\cap \mathfrak{ }\mathfrak{Ҩ}\right)\left({v}_{1}\right) = \varnothing$ and $Ŧ (Л \cap Ҩ)\left({v}_{1}\right) = \left\{\mathcal{x}, \mathcal{z}\right\}.$ Now, we define soft set $({\mathcal{M}}_{1}, V)$ over ${\mathcal{Q}}_{2}$ by, ${\mathcal{M}}_{1}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{x}\right\}$ and ${\mathcal{M}}_{1}\left({v}_{2}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{x}, \mathcal{z}, \mathcal{Ŧ}\mathcal{՛}\right\}$. Thus, ${\overline{\mathcal{Л}}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right)\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ } = \mathcal{ }\{\mathcal{L}\mathcal{ }, \mathfrak{r}, \mathfrak{ }\mathfrak{Ŧ}\}$, ${\overline{\mathfrak{Ҩ}}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathfrak{r}, \mathfrak{ }\mathfrak{Ŧ}\right\}$ and ${\overline{\left(\mathfrak{Л}\cap \mathfrak{ }\mathfrak{Ҩ}\right)}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{ }\mathcal{Ŧ}\right\}$ $\Rightarrow {\overline{\mathcal{Л}}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right)\cap {\overline{Ҩ}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathfrak{r}, \mathfrak{ }\mathfrak{Ŧ}\right\}.$ This shows that ${\overline{\mathfrak{Л}}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right)\cap {\overline{Ҩ}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right)⊈{\overline{\left(Л\cap \mathrm{ }\mathrm{Ҩ}\right)}}^{{\mathcal{M}}_{1}}\left({v}_{1}\right).$

Now, consider $V = ({v}_{1}, {v}_{2})$ and define $Л :V \to P$(${\mathcal{Q}}_{1}$ $\times$ ${\mathcal{Q}}_{2}$) and $Ҩ:V \to P$(${\mathcal{Q}}_{1}$ $\times$ ${\mathcal{Q}}_{2}$) by,

and

So, $\left(\mathcal{Л}\mathcal{ }\cap \mathcal{ }\mathcal{Ҩ}\right)\left({v}_{1}\right) = \left\{\left(\mathcal{L}, \mathcal{L}\mathcal{՛}\right), \left(\mathcal{Ŧ}, \mathcal{L}\mathcal{՛}\right), \left(\mathfrak{r}, \mathfrak{Ŧ}\mathfrak{՛}\right)\right\}$ and $\left(\mathfrak{Л}\mathfrak{ }\cap \mathfrak{ }\mathfrak{Ҩ}\right)\left({v}_{2}\right) = \left\{\left(\mathcal{L}, \mathcal{L}\mathcal{՛}\right), \left(\mathcal{L}, \mathcal{w}\right), \left(\mathcal{Ŧ}, \mathcal{Ŧ}\mathcal{՛}\right)\right\}.$

Following are the aftersets corresponding to $\mathcal{Л}\left({v}_{1}\right)$ and $Ҩ\left({v}_{1}\right)$,

Also, $\mathcal{L}\left(\mathcal{Л}\cap \mathcal{ }\mathcal{Ҩ}\right)\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}\right\}, \mathfrak{r}\left(\mathfrak{Л}\mathfrak{ }\cap \mathfrak{ }\mathfrak{Ҩ}\right)\left({v}_{1}\right) = \left\{Ŧ՛\right\}$ and $Ŧ(Л \cap Ҩ)\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}\right\}.$

Now, we define soft set $({\mathcal{M}}_{2}, V)$ over ${\mathcal{Q}}_{2}$ by, ${\mathcal{M}}_{2}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{Ŧ}\mathcal{՛}\right\}$ and ${\mathcal{M}}_{2}\left({v}_{2}\right) = \mathcal{ }\left\{\mathcal{L}\mathcal{՛}\right\}$. Thus, ${{\underline {{\mathcal{Л}}}}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \varnothing, {\underline {Ҩ}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{Ŧ\right\}$ and $({\underline {{Л\cap \mathrm{ }\mathrm{Ҩ}}}})^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathfrak{r}, \mathfrak{ }\mathfrak{Ŧ}\right\}.$ Thereby, ${{\underline {{\mathfrak{Л}}}}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right)\cup {\underline {Ҩ}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{Ŧ\right\}.$ This shows that, ${\underline {Л}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right)\cup {\underline {Ҩ}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right)⊉{\underline{\left(Л\cap \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{Ҩ}\right)}}^{{\mathcal{M}}_{2}}\left({v}_{1}\right).$

Definition 3.6. For SBIR $(Л, V)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ i.e., $Л :V \to P$(${\mathcal{Q}}_{1}$ $\times$ ${\mathcal{Q}}_{2}$) the soft compatible relation is defined as, for all $\in$ ${\mathcal{Q}}_{1}$, $w\in$ ${\mathcal{Q}}_{2}$, ${f}_{l}\subseteq {\mathcal{Q}}_{1}$ and ${g}_{l}\subseteq$ ${\mathcal{Q}}_{2}$ for $(l\in L)$ we have

(1) $\left({f}_{l}, {g}_{l}\right)\in Л\left(v\right)\Rightarrow \left({\vee }_{l\in L}{f}_{l}, {\vee }_{l\in L}{g}_{l}\right)\in Л\left(v\right),$

(2) $\left(u, w\right)\in Л\left(v\right)\Rightarrow \left({\gamma }_{1}{\star }_{1}u, {\gamma }_{2}{\star }_{2}w\right)\in Л\left(v\right)\forall {\gamma }_{1}\in {Ǫ}_{1}, {\gamma }_{2}\in {Ǫ}_{2}.$

Example 3.7. Let ${{Ǫ}}_{1} = \left\{\mathcal{L}, \mathcal{j}, \mathcal{k}, \mathcal{ }\mathcal{Ŧ}\right\}$ and ${\mathcal{Ǫ}}_{2}$ $= \left\{\mathcal{L}\mathcal{՛}, x՛, y՛, z՛, {Ŧ}\mathit{՛}\right\}$ be two ${C_{lts}}$ described in Figures 5 and 6 respectively. The associative binary operation ${\otimes }_{1}$ and ${\otimes }_{2}$ on ${{Ǫ}}_{1}$ and ${{Ǫ}}_{2}$ is defined as,

(1) $a{\otimes }_{1}b = a˄b$

(2) $a{\otimes }_{2}b = \mathcal{ }\mathcal{L}\mathcal{՛}$

We define ${\star }_{1}$ and ${\star }_{2}$ the left action on ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$, respectively as shown in Tables 5 and 6. Then, ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ are quantale modules. Let $V = \left\{{v}_{1}, {v}_{2}\right\}$ and the SBIR $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ be defined by,

Then $\left(Л, V\right)$ is an SCMR.

Throughout in this paper, we consider that $\left(Л, V\right)$ be a SBIR from ${Ǫ}_{1}\;to\;{Ǫ}_{2}$.

Lemma 3.8. If $\left(Л, V\right)$ is a SCMR form quantale module ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$, then for ${\gamma }_{1}\in {Ǫ}_{1}$ and $g, \mathcal{h}\in {\mathcal{Q}}_{1}$ we have

(1) ${\gamma }_{1}Л\left(v\right){\star }_{2}\mathcal{g}\mathcal{Л}\left(v\right)\subseteq \left({\gamma }_{1}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right).$

(2) $\mathcal{g}\mathcal{Л}\left(v\right)\vee \mathcal{h}\mathcal{Л}\left(v\right)\subseteq \left(\mathcal{g}\vee \mathcal{h}\right)\mathcal{Л}\left(v\right)$.

Proof. (1) Let $f\in {\gamma }_{1}Л\left(v\right){\star }_{2}\mathcal{g}\mathcal{Л}\left(v\right).$ Then for some $\mathcal{m}\in {\gamma }_{1}Л\left(v\right)$ and $\mathcal{n}\in \mathcal{g}\mathcal{Л}\left(v\right),$ we have $f = m{\star }_{2}n$. Thereby, $\left({\gamma }_{1}, \mathcal{m}\right)\in \mathcal{Л}\left(v\right)\mathrm{a}\mathrm{n}\mathrm{d}\left(\mathcal{g}, \mathcal{n}\right)\in \mathcal{Л}\left(v\right).$ By SCMR , we have $\left({\gamma }_{1}{\star }_{1}\mathcal{g}, \mathcal{m}{\star }_{2}\mathcal{n}\right)\in$ $\mathcal{Л}\left(v\right).\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}, \left(\mathcal{m}{\star }_{2}\mathcal{n}\right)\in \left({\gamma }_{1}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right).$ Consequently, ${\gamma }_{1}Л\left(v\right){\star }_{2}\mathcal{g}\mathcal{Л}\left(v\right)\subseteq \left({\gamma }_{1}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right).$

(2) Let $f\in \mathcal{g}\mathcal{Л}\left(v\right)\vee \mathcal{h}\mathcal{Л}\left(v\right)$. Then for some $\mathcal{m}\in \mathcal{g}\mathcal{Л}\left(v\right)$ and $\mathcal{n}\in \mathcal{h}\mathcal{Л}\left(v\right),$ we have $f = \mathcal{m}\vee \mathcal{n}.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{b}\mathrm{y}, \left(\mathcal{g}, \mathcal{m}\right)\in \mathcal{Л}\left(v\right)\mathrm{a}\mathrm{n}\mathrm{d}\left(\mathcal{h}, \mathcal{n}\right)\in \mathcal{Л}\left(v\right).$ By SCMR, we have $\left(\mathcal{g}\vee \mathcal{h}, \mathcal{m}\vee \mathcal{n}\right)\in \mathcal{Л}\left(v\right).\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}, \left(\mathcal{m}\vee \mathcal{n}\right)\in \left(\mathcal{g}\vee \mathcal{h}\right)\mathcal{Л}\left(v\right).$ Consequently, $\mathcal{g}\mathcal{Л}\left(v\right)\vee \mathcal{h}\mathcal{Л}\left(v\right)\subseteq \left(\mathcal{g}\vee \mathcal{h}\right)\mathcal{Л}\left(v\right).$

Lemma 3.9. If $\left(Л, V\right)$ is a SCMR from a quantale module ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$, then for ${\gamma }_{2}\in {Ǫ}_{2}$ and $\mathcal{x}, \mathcal{y}\in {\mathcal{Q}}_{2}$ we have

(1) $Л\left(v\right){\gamma }_{2}{\star }_{1}Л\left(v\right)\mathcal{y}\subseteq \mathcal{ }\mathcal{Л}\left(v\right)\left({\gamma }_{2}{\star }_{2}\mathcal{y}\right).$

(2) $\mathcal{Л}\left(v\right)\mathcal{x}\vee \mathcal{ }\mathcal{Л}\left(v\right)\mathcal{y}\subseteq \mathcal{ }\mathcal{Л}\left(v\right)\left(\mathcal{x}\vee \mathcal{y}\right)$.

Proof. (1) Assume $p\in Л\left(v\right){\gamma }_{2}{\star }_{1}Л\left(v\right)\mathcal{y}.$ Then for some $\mathcal{u}\in \mathcal{Л}\left(v\right){\gamma }_{2}$ and $\mathcal{w}\in \mathcal{Л}\left(v\right)\mathcal{y}$, we have $p = \mathcal{u}{\star }_{1}\mathcal{w}.$ Thereby, $\left(\mathcal{u}, {\gamma }_{2}\right)\in Л\left(v\right)\mathrm{a}\mathrm{n}\mathrm{d}\left(\mathcal{w}, \mathcal{y}\right)\in \mathcal{Л}\left(v\right).$ By SCMR, we have $\left(\mathcal{u}{\star }_{1}\mathcal{w}, {\gamma }_{2}{\star }_{2}\mathcal{y}\right)\in \mathcal{Л}\left(v\right).\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}, \left(\mathcal{u}{\star }_{1}\mathcal{w}\right)\in \mathcal{Л}\left(v\right)\left({\gamma }_{2}{\star }_{2}\mathcal{y}\right).$

$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}, Л\left(v\right){\gamma }_{2}{\star }_{1}Л\left(v\right)\mathcal{y}\subseteq \mathcal{Л}\left(v\right)\left({\gamma }_{2}{\star }_{2}\mathcal{y}\right)$

(2) Assume $p\in Л\left(v\right)\mathcal{x}\vee \mathcal{Л}\left(v\right)\mathcal{y}.$ Then for some $\mathcal{u}\in \mathcal{Л}\left(v\right)\mathcal{x}$ and $\mathcal{w}\in \mathcal{Л}\left(v\right)\mathcal{y}$, we have $p = \mathcal{u}\vee \mathcal{w}.$ Thereby, $\left(\mathcal{u}, \mathcal{x}\right)\in \mathcal{Л}\left(v\right)$ and $\left(\mathcal{w}, \mathcal{y}\right)\in \mathcal{Л}\left(v\right).\mathrm{B}\mathrm{y}\;\mathrm{ }\mathrm{S}\mathrm{C}\mathrm{M}\mathrm{R}, \mathrm{ }\mathrm{w}\mathrm{e}\;\mathrm{ }\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}\;\left(\mathcal{u}\vee \mathcal{w}, \mathcal{x}\vee \mathcal{y}\right)\in \mathcal{Л}\left(v\right).\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}, \left(\mathcal{u}\vee \mathcal{w}\right)\in \mathcal{Л}\left(v\right)\left(\mathcal{x}\vee \mathcal{y}\right).$

$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}, Л\left(v\right)\mathcal{x}\vee \mathcal{Л}\left(v\right)\mathcal{y}\subseteq \mathcal{ }\mathcal{Л}\left(v\right)\left(\mathcal{x}\vee \mathcal{y}\right).$

Definition 3.10. A SCMR $\left(\mathcal{Л}, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ w.r.t aftersets is called soft complete relation (SCMPR) if $\forall$ $\mathcal{u}, \mathcal{w}\in$ ${\mathcal{Q}}_{1}{, \gamma }_{1}\in {Ǫ}_{1}$ we have

(1) $\mathcal{u}\mathcal{Л}\left(v\right)\vee \mathcal{w}\mathcal{Л}\left(v\right) = \left(\mathcal{u}\vee \mathcal{w}\right)\mathcal{Л}\left(v\right).$

(2) ${\gamma }_{1}Л\left(v\right){\star }_{2}\mathcal{u}\mathcal{Л}\left(v\right) = \left({\gamma }_{1}{\star }_{1}\mathcal{u}\right)\mathcal{Л}\left(v\right)\forall$ $v\in V$.

If a SCMR $\left(Л, V\right)$ w.r.t the aftersets satisfies condition (1) only, then it is called $\vee$-complete. If a SCMR $\left(Л, V\right)$ w.r.t the aftersets satisfies condition (2) only, then it is called $\star$-complete.

A SCMR $\left(Ҩ, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ w.r.t foresets is called soft complete relation (SCMPR) if $\forall$ $\mathcal{s}, \mathcal{t}\in$ ${\mathcal{Q}}_{2}$, ${\gamma }_{2}\in {Ǫ}_{2}$ we have,

(1) $Ҩ\left(v\right)\mathcal{s}\vee \mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{Ҩ}\left(v\right)\mathcal{t} = \mathcal{Ҩ}\left(v\right)\left(\mathcal{s}\vee \mathcal{t}\right).$

(2) $\mathcal{Л}\left(v\right){\gamma }_{2}{\star }_{1}Л\left(v\right)\mathcal{t} = \mathcal{Л}\left(v\right)\left({\gamma }_{2}{\star }_{2}\mathcal{t}\right)$ $\forall v\in V.$

If a SCMR $\left(Л, V\right)$ w.r.t the foresets satisfies condition (1) only, then it is called $\vee$-complete. If a SCMR $\left(Л, V\right)$ w.r.t the foresets satisfies condition (2) only, then it is called $\star$-complete.

Example 3.11. Consider ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ be quantale modules described in Example 3.5. Let $V = \left({v}_{1}, {v}_{2}\right)$ and the SBIR $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ be defined by,

Then $\left(\mathfrak{Л}, V\right)$ is SCMR. Following are the aftersets corresponding to $Л\left({v}_{1}\right)$ and $Л\left({v}_{2}\right).$

$\mathcal{L}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{x}, \mathcal{z}\right\}$, $\mathfrak{r}\mathfrak{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{x}, \mathcal{z}\right\}, \mathcal{ }\mathcal{Ŧ}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{x}, \mathcal{z}\right\}.$

$\mathcal{L}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{y}, \mathcal{w}, \mathcal{Ŧ}\mathcal{՛}\right\}\mathfrak{r}\mathfrak{Л}\left({v}_{2}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{y}, \mathcal{w}, \mathcal{Ŧ}\mathcal{՛}\right\}, \mathcal{Ŧ}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{y}, \mathcal{w}, \mathcal{Ŧ}\mathcal{՛}\right\}$.

Then it is very easy to verify that $\left(\mathcal{Л}, V\right)$ w.r.t the aftersets is SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$.

Now, consider $V = ({v}_{1}, {v}_{2})$ and the SBIR $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ is defined by,

Then $\left(\mathfrak{Л}, V\right)$ is SCMR. Following are foresets corresponding to $Л\left({v}_{1}\right)$ and $Л\left({v}_{2}\right),$

$Л\left({v}_{1}\right)\mathcal{L}\mathcal{՛} = \left\{\mathcal{L}, \mathfrak{r}\right\},$$\mathfrak{ }\mathfrak{Л}\left({v}_{1}\right)\mathcal{x} = \left\{\mathcal{L}, \mathfrak{r}\right\}, \mathfrak{ }\mathfrak{Л}\left({v}_{1}\right)\mathcal{y} = \left\{\mathcal{L}, \mathfrak{r}\right\},$$\mathfrak{ }\mathfrak{Л}\left({v}_{1}\right)\mathcal{z} = \left\{\mathcal{L}, \mathfrak{r}\right\}, \mathfrak{ }\mathfrak{Л}\left({v}_{1}\right)\mathcal{w}$$= \left\{\mathcal{L}, \mathfrak{r}\right\}, \mathfrak{ }\mathfrak{Л}\left({v}_{1}\right)Ŧ՛ = \left\{\mathcal{L}, \mathfrak{r}\right\}.$

$\mathfrak{Л}\left({v}_{2}\right)\mathcal{L}\mathcal{՛} = \left\{\mathfrak{r}, \mathfrak{Ŧ}\right\}$, $\mathfrak{ }\mathfrak{Л}\left({v}_{2}\right)\mathcal{x} = \left\{\mathfrak{r}, \mathfrak{Ŧ}\right\}, \mathfrak{ }\mathfrak{Л}\left({v}_{2}\right)\mathcal{y} = \left\{\mathfrak{r}, \mathfrak{Ŧ}\right\}$, $\mathfrak{ }\mathfrak{Л}\left({v}_{2}\right)\mathcal{z} = \left\{\mathfrak{r}, \mathfrak{Ŧ}\right\}, \mathfrak{ }\mathfrak{Л}\left({v}_{2}\right)\mathcal{w} = \left\{\mathfrak{r}, \mathfrak{Ŧ}\right\}$, $\mathfrak{ }\mathfrak{Л}\left({v}_{2}\right)Ŧ՛ = \left\{\mathfrak{r}, \mathfrak{Ŧ}\right\}.$

Then it is very easy to verify that $\left(\mathfrak{Л}, V\right)$ w.r.t the foresets is SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$.

Remark 3.12. Generally, neither SCMPR w.r.t aftersets implies SCMPR w.r.t foresets nor SCMPR w.r.t foresets implies SCMPR w.r.t aftersets.

Theorem 3.13. Let $\left(Л, V\right)$ be a SCMR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. Then for any soft set $\left({\mathcal{M}}_{2}, V\right)$ over ${\mathcal{Q}}_{2}$ and $\left({\mathbb{Q}}_{2}, V\right)$ over ${Ǫ}_{2}$, we have

(1) $\left({\overline{Л}}^{{\mathbb{Q}}_{2}}, V\right){\star }_{1}\left({\overline{Л}}^{{\mathcal{M}}_{2}}, V\right)\subseteq \left({\overline{Л}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}, V\right).$

(2) ${\bigvee }_{l\in L}\left({\overline{Л}}^{{\mathcal{M}}_{l}}, V\right)\subseteq \left({\overline{Л}}^{{\vee }_{l\in L}{\mathcal{M}}_{l}}, V\right).$

Proof. (1) For arbitrary $v\in V$, let $f\in {\overline{Л}}^{{\mathbb{Q}}_{2}}\left(v\right){\star }_{1}{\overline{Л}}^{{\mathcal{M}}_{2}}\left(v\right)$. Then, for some $\mathcal{e}\in {\overline{\mathcal{Л}}}^{{\mathbb{Q}}_{2}}\left(v\right)$ and $\mathcal{g}\in {\overline{\mathcal{Л}}}^{{\mathcal{M}}_{2}}\left(v\right)$ we have $f = \mathcal{e}{\star }_{1}\mathcal{g}$. Thereby, $\mathcal{e}\mathcal{Л}\left(v\right)\cap {\mathbb{Q}}_{2}\left(v\right)\ne \varnothing$ and $\mathcal{g}\mathcal{Л}\left(v\right)\cap {\mathcal{M}}_{2}\left(v\right)\ne \varnothing$. So, for $\mathcal{j}\in {\mathcal{Ǫ}}_{2}$ and $\mathcal{k}\in$ ${\mathcal{Q}}_{2}$, we have $\mathcal{j}\in \mathcal{e}\mathcal{Л}\left(v\right)\cap {\mathbb{Q}}_{2}\left(v\right)$ and $\mathcal{k}\in \mathcal{g}\mathcal{Л}\left(v\right)\cap {\mathcal{M}}_{2}\left(v\right)\Rightarrow \mathcal{j}\in \mathcal{e}\mathcal{Л}\left(v\right)$, $\mathcal{k}\in \mathcal{g}\mathcal{Л}\left(v\right)$, $\mathcal{j}\in {\mathbb{Q}}_{2}\left(v\right)$ and $\mathcal{k}\in {\mathcal{M}}_{2}\left(v\right)$. Thereby, $\left(\mathcal{e}, \mathcal{j}\right)\in \mathcal{Л}\left(v\right)$, $\left(\mathcal{g}, \mathcal{k}\right)\in \mathcal{Л}\left(v\right)$. Since $Л\left(v\right)$ is SCMR thus, $\left(\mathcal{e}{\star }_{1}\mathcal{g}, \mathcal{j}{\star }_{2}\mathcal{k}\right)\in \mathcal{Л}\left(v\right)$ i.e., $\left(\mathcal{j}{\star }_{2}\mathcal{k}\right)\in \left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right)$ and $\left(\mathcal{j}{\star }_{2}\mathcal{k}\right)\in {\mathbb{Q}}_{2}\left(v\right){\star }_{2}{\mathcal{M}}_{2}\left(v\right)$. Thus, $\left(\mathcal{j}{\star }_{2}\mathcal{k}\right)\in \left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right)\cap {\mathbb{Q}}_{2}\left(v\right){\star }_{2}{\mathcal{M}}_{2}\left(v\right)\Rightarrow \left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right)\cap {\mathbb{Q}}_{2}\left(v\right){\star }_{2}{\mathcal{M}}_{2}\left(v\right)\ne \varnothing$. Hence, $f = \left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\in {\overline{\mathcal{Л}}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}\left(v\right)$.

(2) Now for arbitrary $v\in V$, assume $f\in {\vee }_{l\in L}{\overline{Л}}^{{\mathcal{M}}_{l}}\left(v\right)$. Then $f = {\vee }_{l\in L}{\mathcal{g}}_{l}$ for some ${\mathcal{g}}_{l}\in {\overline{Л}}^{{\mathcal{M}}_{l}}\left(v\right)$. Thereby, ${\mathcal{g}}_{l}Л\left(v\right)\cap {\mathcal{M}}_{l}\left(v\right)\ne \varnothing$. So there exists ${\mathcal{e}}_{l}\in {\mathcal{g}}_{l}Л\left(v\right)\mathrm{i}.\mathrm{e}., \left({\mathcal{g}}_{l}, {\mathcal{e}}_{l}\right)\in Л\left(v\right)\;and\;{\mathcal{e}}_{l}\in {\mathcal{M}}_{l}\left(v\right).$ Since $Л\left(v\right)$ is SCMR thus, $({{\vee }_{l\in L}\mathcal{g}}_{l}, {\vee }_{l\in L}{\mathcal{e}}_{l})\in Л\left(v\right)\Rightarrow {\vee }_{l\in L}{\mathcal{e}}_{l}\in {{\vee }_{l\in L}\mathcal{g}}_{l}Л\left(v\right)$ and ${{\vee }_{l\in L}\mathcal{e}}_{l}\in {\vee }_{l\in L}{\mathcal{M}}_{l}\left(v\right).$ Thus, ${\vee }_{l\in L}{\mathcal{e}}_{l}\in {{\vee }_{l\in L}\mathcal{g}}_{l}Л\left(v\right)\cap {\vee }_{l\in L}{\mathcal{M}}_{l}\left(v\right)$. So, ${{\vee }_{l\in L}\mathcal{g}}_{l}Л\left(v\right)\cap {\vee }_{l\in L}{\mathcal{M}}_{l}\left(v\right)\ne \varnothing$. Hence, $f = {\vee }_{l\in L}{\mathcal{g}}_{l}\in {\overline{Л}}^{{\vee }_{l\in L}{\mathcal{M}}_{l}}\left(v\right)$.

Example 3.14. Consider ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ be two quantale modules described in Example 3.7. Let $V = \left\{{v}_{1}, {v}_{2}\right\}$ and the SBIR $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ be defined by,

Then $\left(Л, V\right)$ is SCMR. Following are the aftersets corresponding to $Л\left({v}_{1}\right)$ and $Л\left({v}_{2}\right).$

$\mathcal{L}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, y՛, z՛, Ŧ՛\right\}, \mathcal{j}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, y՛, z՛\right\}, \mathcal{k}\mathcal{Л}\left({v}_{1}\right) = \left\{y՛, Ŧ՛\right\}, ŦЛ\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{Ŧ}\right\}.$

$\mathcal{L}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{L}\mathcal{՛}, x՛, y՛, z՛\right\}, \mathcal{j}\mathcal{Л}\left({v}_{2}\right) = \left\{x՛, z՛\right\}, \mathcal{k}\mathcal{Л}\left({v}_{2}\right) = \left\{y՛, z՛\right\}$, $\mathrm{Ŧ}Л\left({v}_{2}\right) = \left\{z՛\right\}$.

Now let $V = \left\{{v}_{1}, {v}_{2}\right\}$ and define soft set $\left({\mathbb{Q}}_{2}, V\right)$ over ${Ǫ}_{2}$ and $\left({\mathcal{M}}_{2}, V\right)$ over ${\mathcal{Q}}_{2}$.

${\mathbb{Q}}_{2}\left({v}_{1}\right) = \left\{z՛\right\}, {\mathbb{Q}}_{2}\left({v}_{2}\right) = \left\{x՛\right\}$ and ${\mathcal{M}}_{2}\left({v}_{1}\right) = \left\{x՛, y՛\right\}$, ${\mathcal{M}}_{2}\left({v}_{2}\right) = \left\{y՛\right\}.$ Then ${\overline{Л}}_{}^{{\mathbb{Q}}_{2}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{j}\right\}$ and ${\overline{\mathcal{Л}}}_{}^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{j}, \mathcal{k}\right\}.$ So, ${\overline{\mathcal{Л}}}_{}^{{\mathbb{Q}}_{2}}\left({v}_{1}\right){\star }_{1}{\overline{Л}}_{}^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{j}\right\}{\star }_{1}\left\{\mathcal{L}, \mathcal{j}, \mathcal{k}\right\} = \left\{\mathcal{L}\right\}$ and ${\mathbb{Q}}_{2}\left({v}_{1}\right){\star }_{2}{\mathcal{M}}_{2}\left({v}_{1}\right) = \left\{z՛\right\}{\star }_{2}\left\{x՛, y՛\right\} = \left\{x՛, y՛\right\}$. UP_ap of ${\mathbb{Q}}_{2}\left({v}_{1}\right){\star }_{2}{\mathcal{M}}_{2}\left({v}_{1}\right)$ is ${\overline{Л}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{j}, \mathcal{k}\right\}.$ Therefore ${\overline{\mathcal{Л}}}_{}^{{\mathbb{Q}}_{2}}\left({v}_{1}\right){\star }_{1}{\overline{Л}}_{}^{{\mathcal{M}}_{2}}\left({v}_{1}\right) = \left\{\mathcal{L}, j\right\}{\star }_{1}\left\{\mathcal{L}, \mathcal{j}, \mathcal{k}\right\} = \left\{\mathcal{L}\right\}\subseteq \left\{\mathcal{L}, \mathcal{j}, \mathcal{k}\right\} = {\overline{\mathcal{Л}}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}\left({v}_{1}\right).$

Also ${\overline{Л}}_{}^{{\mathbb{Q}}_{2}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{j}\right\}$ and ${\overline{\mathcal{Л}}}_{}^{{\mathcal{M}}_{2}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{k}\right\}.$ Thus ${\overline{\mathcal{Л}}}_{}^{{\mathbb{Q}}_{2}}\left({v}_{2}\right){\star }_{1}{\overline{Л}}_{}^{{\mathcal{M}}_{2}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{j}\right\}{\star }_{1}\left\{\mathcal{L}, \mathcal{k}\right\} = \left\{\mathcal{L}\right\}$ and ${\mathbb{Q}}_{2}\left({v}_{2}\right){\star }_{2}{\mathcal{M}}_{2}\left({v}_{2}\right) = \left\{\mathcal{x}\mathcal{՛}\right\}{\star }_{2}\left\{y՛\right\} = \left\{y՛\right\}.$ UP_ap of ${\mathbb{Q}}_{2}\left({v}_{2}\right){\star }_{2}{\mathcal{M}}_{2}\left({v}_{2}\right)$ is ${\overline{Л}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{k}\right\}.$ Therefore ${\overline{\mathcal{Л}}}_{}^{{\mathbb{Q}}_{2}}\left({v}_{2}\right){\star }_{1}{\overline{Л}}_{}^{{\mathcal{M}}_{2}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{j}\right\}{\star }_{1}\left\{\mathcal{L}, \mathcal{k}\right\} = \left\{\mathcal{L}\right\}\subset \left\{\mathcal{L}, \mathcal{k}\right\} = {\overline{\mathcal{Л}}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}\left({v}_{2}\right).$ Hence, $\left({\overline{Л}}^{{\mathbb{Q}}_{2}}, V\right){\star }_{1}\left({\overline{Л}}^{{\mathcal{M}}_{2}}, V\right)\subseteq \left({\overline{Л}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}, V\right).$ Similarly, we can find an example that supports the argument for Theorem 3.13 (2).

Theorem 3.15. Let $\left(Л, V\right)$ be a SCMR w.r.t foresets from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. Then for any soft set $\left({\mathcal{N}}_{1}, V\right)$ over ${\mathcal{Q}}_{1}$ and $\left({\mathbb{Q}}_{1}, V\right)$ over ${Ǫ}_{1}$, we have

(1) $\left({}_{}{}^{{\mathbb{Q}}_{1}}\overline{Л}, V\right){\star }_{2}\left({}_{}{}^{{\mathcal{N}}_{1}}\overline{Л}, V\right)\subseteq \left({}_{}{}^{{\mathbb{Q}}_{1}{\star }_{1}{\mathcal{N}}_{1}}\overline{Л}, V\right).$

(2) ${\bigvee }_{l\in L}\left({}_{}{}^{{\mathcal{N}}_{l}}\overline{Л}, V\right)\subseteq \left({}_{}{}^{{\vee }_{l\in L}{\mathcal{N}}_{l}}\overline{Л}, V\right).$

Proof. Similar to proof of Theorem 3.13.

Theorem 3.16. Let $\left(Л, V\right)$ be a SCMPR w.r.t afterset from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. Then for any soft set $\left({\mathcal{M}}_{2}, V\right)$ over ${\mathcal{Q}}_{2}$ and $\left({\mathbb{Q}}_{2}, V\right)$ over ${Ǫ}_{2}$, we have

(1) $\left({\underline {Л}}^{{\mathbb{Q}}_{2}}, V\right){\star }_{1}\left({\underline {Л}}^{{\mathcal{M}}_{2}}, V\right)\subseteq \left({\underline {Л}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}, V\right).$

(2) ${\bigvee }_{l\in L}\left({\underline {Л}}^{{\mathcal{M}}_{l}}, V\right)\subseteq \left({\underline {Л}}^{{\vee }_{l\in L}{\mathcal{M}}_{l}}, V\right).$

Proof. (1) For arbitrary $v\in V$, let $f\in {\underline {Л}}^{{\mathbb{Q}}_{2}}\left(v\right){\star }_{1}{\underline {Л}}^{{\mathcal{M}}_{2}}\left(v\right)$. Then for some $\mathcal{e}\in {{\underline {{\mathcal{Л}}}}}^{{\mathbb{Q}}_{2}}\left(v\right)$ and $\mathcal{g}\in {{\underline {{\mathcal{Л}}}}}^{{\mathcal{M}}_{2}}(v$), we have $f = \mathcal{e}{\star }_{1}\mathcal{g}$. Thereby, $\mathcal{\varnothing }\ne \mathcal{e}\mathcal{Л}\left(v\right)\subseteq {\mathbb{Q}}_{2}\left(v\right)$ and $\mathcal{\varnothing }\ne \mathcal{g}\mathcal{Л}\left(v\right)\subseteq {\mathcal{M}}_{2}\left(v\right)$. Then by SCMPR, $\left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right) = \mathcal{e}\mathcal{Л}\left(v\right){\star }_{2}\mathcal{g}\mathcal{Л}\left(v\right)\subseteq {\mathbb{Q}}_{2}\left(v\right){\star }_{2}{\mathcal{M}}_{2}\left(v\right)\Rightarrow \left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\mathcal{Л}\left(v\right)\subseteq {\mathbb{Q}}_{2}\left(v\right){\star }_{2}{\mathcal{M}}_{2}\left(v\right)$. Thus, $f = \left(\mathcal{e}{\star }_{1}\mathcal{g}\right)\in {{\underline {{\mathcal{Л}}}}}^{{\mathbb{Q}}_{2}{\star }_{2}{\mathcal{M}}_{2}}\left(v\right)$.

(2) Now for arbitrary $v\in V$, assume $f\in {\vee }_{l\in L}{\underline {Л}}^{{\mathcal{M}}_{l}}\left(v\right)$. Then, $f = {\vee }_{l\in L}{\mathcal{g}}_{l}$ for some ${\mathcal{g}}_{l}\in {\underline {Л}}^{{\mathcal{M}}_{l}}\left(v\right)$. Thereby, $\varnothing \ne {\mathcal{g}}_{l}Л\left(v\right)\subseteq {\mathcal{M}}_{l}\left(v\right)$. By SCMPR, we have ${{\vee }_{l\in L}(\mathcal{g}}_{l}Л\left(v\right)) = {(\vee }_{l\in L}{\mathcal{g}}_{l})Л\left(v\right)\subseteq {{\vee }_{l\in L}\mathcal{M}}_{l}\left(v\right)$. So, ${{(\vee }_{l\in L}\mathcal{g}}_{l})Л\left(v\right)\subseteq {\vee }_{l\in L}{\mathcal{M}}_{l}(v)$. Hence, $f = {\vee }_{l\in L}{\mathcal{g}}_{l}\in {\underline {Л}}^{{\vee }_{l\in L}{\mathcal{M}}_{l}}\left(v\right)$.

Theorem 3.17. Let $\left(Л, V\right)$ be a SCMPR w.r.t foresets from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. Then for any soft set $\left({\mathcal{N}}_{1}, V\right)$ over ${\mathcal{Q}}_{1}$ and $\left({\mathbb{Q}}_{1}, V\right)$ over ${Ǫ}_{1}$, we have

(1) $\left({}_{}{}^{{\mathbb{Q}}_{1}}{\underline {Л}}, V\right){\star }_{2}\left({}_{}{}^{{\mathcal{N}}_{1}}{\underline {Л}}, V\right)\subseteq \left({}_{}{}^{{\mathbb{Q}}_{1}{\star }_{1}{\mathcal{N}}_{1}}{\underline {Л}}, V\right).$

(2) ${\bigvee }_{l\in L}\left({}_{}{}^{{\mathcal{N}}_{l}}{\underline {Л}}, V\right)\subseteq \left({}_{}{}^{{\vee }_{l\in L}{\mathcal{N}}_{l}}{\underline {Л}}, V\right).$

Proof. Similar to proof of Theorem 3.16.

4.   Approximation of soft substructures in quantale modules

In this section, we use SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ to approximate different soft substructures of quantale modules ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$. We will show that by using SCMR, UP_ap of a soft substructure of quantale module is again a soft substructure of quantale module, and we will provide counter examples to show that the converse is not true. We'll also show that by using SCMPR, LO_ap of a soft substructure of quantale modules is again a soft substructure of quantale module and provide a counter-example to show that the converse is not true.

Throughout in this section, we consider $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ and $bЛ\left(v\right)\ne \varnothing$ for all $\in$ ${\mathcal{Q}}_{1}$, $v\in V$ and $Л\left(v\right)c\ne \varnothing \forall c\in$ ${\mathcal{Q}}_{2}$, $v\in V$ unless otherwise specified. Consider two soft sets $\left(\mathcal{N}, V\right)$ and $\left(\mathcal{M}, V\right)$ over ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ respectively, by the rule,

$\mathcal{N}\left(v\right) =$ ${\mathcal{Q}}_{1}$ and $\mathcal{M}\left(v\right) =$ ${\mathcal{Q}}_{2}$ $\forall$ $v\in V$. Then,

${}_{}{}^{\mathcal{N}}{\underline {{\mathcal{Л}}}}\left(v\right) =$ {$\gamma \in$ ${\mathcal{Q}}_{2}$ $: Л\left(v\right)\gamma \subseteq$ ${\mathcal{Q}}_{1}$} $\subseteq$ ${\mathcal{Q}}_{2}$,

${}_{}{}^{\mathcal{N}}\overline{\mathcal{Л}}\left(v\right) =$ {$\gamma \in$ ${\mathcal{Q}}_{2}$ $: Л\left(v\right)\gamma \cap$ ${\mathcal{Q}}_{1}$ $\ne \varnothing$} $\subseteq$ ${\mathcal{Q}}_{2}$,

${\underline {Л}}_{}^{\mathcal{M}}\left(v\right) =$ {$\rho \in$ ${\mathcal{Q}}_{1}: Л\left(v\right)\rho \subseteq$ ${\mathcal{Q}}_{2}$ } $\subseteq$ ${\mathcal{Q}}_{1}.$

${\overline{Л}}_{}^{\mathcal{M}}\left(v\right) =$ {$\rho \in$ ${\mathcal{Q}}_{1}: Л\left(v\right)\rho \cap$ ${\mathcal{Q}}_{2}$ $\ne \varnothing$} $\subseteq {\mathcal{Q}}_{1}$,

Definition 4.1. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{M}, V\right)$ over ${\mathcal{Q}}_{2}$ is called GU_rS (generalized upper soft) ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ w.r.t aftersets if ${UP}_{ap}$ $\left({\overline{Л}}_{}^{\mathcal{M}}, V\right)$ is soft ${Ǫ}_{s}$ of ${\mathcal{Q}}_{1}$.

Definition 4.2. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{M}, V\right)$ over ${\mathcal{Q}}_{2}$ is called GU_rS left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ w.r.t aftersets if ${UP}_{ap}$ $\left({\overline{Л}}_{}^{\mathcal{M}}, V\right)$ is soft left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$.

Definition 4.3. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{N}, V\right)$ over ${\mathcal{Q}}_{1}$ is called GU_rS quantale sub-module of ${\mathcal{Q}}_{2}$ w.r.t foresets if ${UP}_{ap}$ $\left({}_{}{}^{\mathcal{N}}\overline{\mathcal{Л}}, V\right)$ is soft ${Ǫ}_{s}$ of ${\mathcal{Q}}_{2}$.

Definition 4.4. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{N}, V\right)$ over ${\mathcal{Q}}_{1}$ is called GU_rS left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ w.r.t foresets if ${UP}_{ap}$ $\left({}_{}{}^{\mathcal{N}}\overline{\mathcal{Л}}, V\right)$ is soft left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$.

Theorem 4.5. Let $\left(Л, V\right)$ be a SCMR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. If $\left(\mathcal{M}, V\right)$ is a soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$, then $\left(\mathcal{M}, V\right)$ is a GU_rS ${Ǫ}_{s}$ of ${\mathcal{Q}}_{1}$ w.r.t the aftersets.

Proof. (1) Assume that $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$, then $\varnothing \ne {\overline{Л}}^{\mathcal{M}}\left(v\right)$ for any $v\in V.$ Let ${\mu }_{l}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)forl\in L$. Thereby, ${\mu }_{l}Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. So there exist, ${w}_{l}\in {\mu }_{l}Л\left(v\right)i.e., ({\mu }_{l}, {w}_{l})\in Л\left(v\right)\;and\;{w}_{l}\in \mathcal{M}\left(v\right)$. By SCMR, $\left({{\vee }_{l\in L}\mu }_{l}, {\vee }_{l\in L}{w}_{l}\right)\in Л\left(v\right)\Rightarrow {\vee }_{l\in L}{w}_{l}\in {{\vee }_{l\in L}\mu }_{l}Л\left(v\right).$ As $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ so, we have ${{\vee }_{l\in L}w}_{l}\in \mathcal{M}\left(v\right)$. Thus, ${\vee }_{l\in L}{w}_{l}\in {{\vee }_{l\in L}\mu }_{l}Л\left(v\right)\cap \mathcal{M}\left(v\right)$. So, ${{\vee }_{l\in L}\mu }_{l}Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. Hence, ${\vee }_{l\in L}{\mu }_{l}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)$.

(2) Let $\gamma \in {\overline{Л}}^{\mathbb{Q}}\left(v\right)\subseteq {Ǫ}_{1}$ and $\mu \in {\overline{Л}}^{\mathcal{M}}\left(v\right),$ where $\left(\mathbb{Q}, V\right)\;is\;a\;subset\;of\;\left({Ǫ}_{2}, V\right).$ Thereby, $\gamma Л\left(v\right)\cap \mathbb{Q}\left(v\right)\ne \varnothing$ and $\mu Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. So, for $\hslash \in {Ǫ}_{2}$ and $\mathcal{g}\in$ ${\mathcal{Q}}_{2}$, we have $\hslash \in \gamma Л\left(v\right)\cap \mathbb{Q}\left(v\right)$ and $\mathcal{g}\in \mu Л\left(v\right)\cap \mathcal{M}\left(v\right)\Rightarrow$ $\hslash \in \gamma Л\left(v\right)$, $g\in \mu Л\left(v\right)$, $\hslash \in \mathbb{Q}\left(v\right), \mathcal{ }\mathcal{g}\in \mathcal{M}\left(v\right)$. Thereby, $\left(\gamma, \hslash \right)\in Л\left(v\right)$, $\left(\mu, \mathcal{ }\mathcal{g}\right)\in \mathcal{Л}\left(v\right)$. Since $Л\left(v\right)$ is SCMR thus, $\left(\gamma {\star }_{1}\mu, \hslash {\star }_{2}\mathcal{g}\right)\in \mathcal{Л}\left(v\right)$, i.e., $\left(\hslash {\star }_{2}\mathcal{g}\right)\in \left(\gamma {\star }_{1}\mu \right)Л\left(v\right)$ and $\left(\hslash {\star }_{2}\mathcal{g}\right)\in \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right)$. As $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ so, $\left(\hslash {\star }_{2}\mathcal{g}\right)\in \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Thus, $\left(\hslash {\star }_{2}\mathcal{g}\right)\in \left(\gamma {\star }_{1}\mu \right)Л\left(v\right)\cap \mathcal{M}\left(v\right)\Rightarrow \left(\gamma {\star }_{1}\mu \right)Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. Hence, $\left(\gamma {\star }_{1}\mu \right)\in {\overline{Л}}^{\mathcal{M}}\left(v\right)$.

Theorem 4.6. Let $\left(Л, V\right)$ be a SCMR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. If $\left(\mathcal{N}, V\right)$ is a soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$, then $\left(\mathcal{N}, V\right)$ is a GU_rS ${Ǫ}_{s}$ of ${\mathcal{Q}}_{2}$ w.r.t the foresets.

Proof. Similar to Proof of Theorem 4.5.

Theorem 4.7. Let $\left(Л, V\right)$ be a soft $\vee -$ complete relation from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ w.r.t aftersets. If $\left(\mathcal{M}, V\right)$ is a soft left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$, then $\left(\mathcal{M}, V\right)$ is a GU_rS left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ w.r.t the aftersets.

Proof. (1) Assume that $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$, then $\varnothing \ne {\overline{Л}}^{\mathcal{M}}\left(v\right)$ for any $v\in V.$ Let ${\rho }_{1}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)\;and\;{\rho }_{2}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)$. Thereby, ${\rho }_{1}Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing \;and\;{\rho }_{2}Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. So there exist ${w}_{1}\in {\rho }_{1}Л\left(v\right)i.e., \left({\rho }_{1}, {w}_{1}\right)\in Л\left(v\right)\;and\;{w}_{1}\in \mathcal{M}\left(v\right)$ and ${w}_{2}\in {\rho }_{2}Л\left(v\right)i.e., ({\rho }_{2}, {w}_{2})\in Л\left(v\right)\;and\;{w}_{2}\in \mathcal{M}\left(v\right)$. By SCMR, $({\rho }_{1}\vee {\rho }_{2}, {w}_{1}\vee {w}_{2})\in Л\left(v\right)\Rightarrow {w}_{1}\vee {w}_{2}\in {(\rho }_{1}\vee {\rho }_{2})Л\left(v\right).$ As $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$, so we have ${w}_{1}\vee {w}_{2}\in \mathcal{M}\left(v\right)$. Thus, ${w}_{1}\vee {w}_{2}\in {(\rho }_{1}\vee {\rho }_{2})Л\left(v\right)\cap \mathcal{M}(v)$. So, ${(\rho }_{1}\vee {\rho }_{2})Л\left(v\right)\cap \mathcal{M}(v)\ne \varnothing$. Hence, ${(\rho }_{1}\vee {\rho }_{2})\in {\overline{Л}}^{\mathcal{M}}(v)$.

(2) Let ${\rho }_{1}, {\rho }_{2}\in$ ${\mathcal{Q}}_{1}$, ${\rho }_{1}\le {\rho }_{2}\;\mathrm{a}\mathrm{n}\mathrm{d}\;{\rho }_{2}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)$. So, ${\rho }_{1}\vee {\rho }_{2} = {\rho }_{2}\in {\overline{Л}}^{\mathcal{M}}\left(v\right).\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}\;{\rho }_{2}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)\Rightarrow {\rho }_{2}Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. So there exist ${w}_{2}\in {\rho }_{2}Л\left(v\right)i.e., \left({\rho }_{2}, {w}_{2}\right)\in Л\left(v\right)\;and\;{w}_{2}\in \mathcal{M}\left(v\right)$. As $\left(Л, V\right)$ is a soft $\vee -$ complete relation, thereby, ${w}_{2}\in {\rho }_{2}Л\left(v\right) = {(\rho }_{1}\vee {\rho }_{2}\left)Л\right(v) = {\rho }_{1}Л(v)\vee {\rho }_{2}Л(v)$ $\Rightarrow {w}_{2} = \mathcal{u}\vee \mathcal{v},$ for some $\mathcal{u}\in {\rho }_{1}Л\left(v\right)\;and\;\mathcal{v}\in {\rho }_{2}Л\left(v\right).$ Thus, $\mathcal{u}\le {w}_{2}\;and\;{w}_{2}\in \mathcal{M}\left(v\right)$. As $M\left(v\right)$ is ${Ǫ}_{I}$ so, $\mathcal{u}\in \mathcal{M}\left(v\right)$. Thus, $\mathcal{u}\in {\rho }_{1}Л\left(v\right)\cap \mathcal{M}\left(v\right)$. Consequently, ${\rho }_{1}\in {\overline{Л}}^{\mathcal{M}}\left(v\right).$

(3) Let $\beta \in {\overline{Л}}^{\mathbb{Q}}\left(v\right)\subseteq {Ǫ}_{1}$ and $\rho \in {\overline{Л}}^{\mathcal{M}}\left(v\right)$, where $\left(\mathbb{Q}, V\right)\;is\;a\;subset\;of({Ǫ}_{2}, V)$. Thereby, $\beta Л\left(v\right)\cap \mathbb{Q}\left(v\right)\ne \varnothing$ and $\rho Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. So, for $\mathcal{u}\in {\mathcal{Ǫ}}_{2}$ and $\mathcal{v}\in$ ${\mathcal{Q}}_{2}$, we have $\mathcal{u}\in \beta Л\left(v\right)\cap \mathbb{Q}\left(v\right)$ and $\mathcal{v}\in \rho Л\left(v\right)\cap \mathcal{M}\left(v\right)$. $\Rightarrow \mathcal{u}\in \beta Л\left(v\right)$, $\mathcal{v}\in \rho Л\left(v\right)$, $\mathcal{u}\in \mathbb{Q}\left(v\right)$ and $\mathcal{v}\in \mathcal{M}\left(v\right)$. Thereby, $\left(\beta, \mathcal{u}\right)\in \mathcal{Л}\left(v\right)$, $\left(\rho, \mathcal{v}\right)\in \mathcal{Л}\left(v\right)$. Since $(Л, V)$ is SCMR thus, $\left(\beta {\star }_{1}\rho, \mathcal{u}{\star }_{2}\mathcal{v}\right)\in \mathcal{Л}\left(v\right)$, i.e., $\left(\mathcal{u}{\star }_{2}\mathcal{v}\right)\in \left(\beta {\star }_{1}\rho \right)Л\left(v\right)$ and $\left(\mathcal{u}{\star }_{2}\mathcal{v}\right)\in \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right)$. As $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ so, $\left(\mathcal{u}{\star }_{2}\mathcal{v}\right)\in \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Thus $\left(\mathcal{u}{\star }_{2}\mathcal{v}\right)\in \left(\beta {\star }_{1}\rho \right)Л\left(v\right)\cap \mathcal{M}\left(v\right)\Rightarrow \left(\beta {\star }_{1}\rho \right)Л\left(v\right)\cap \mathcal{M}\left(v\right)\ne \varnothing$. Hence, $\left(\beta {\star }_{1}\rho \right)\in {\overline{Л}}^{\mathcal{M}}\left(v\right)$.

Theorem 4.8. Let $\left(Л, V\right)$ be a soft $\vee -$ complete relation from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ w.r.t foresets. If $\left(\mathcal{N}, V\right)$ is a soft left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$, then $\left(\mathcal{N}, V\right)$ is a GU_rS left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ w.r.t the foresets.

Proof. Similar to proof of Theorem 4.7.

Definition 4.9. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{M}, V\right)$ over ${\mathcal{Q}}_{2}$ is called GL_rS (generalized lower soft) ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ w.r.t aftersets, if ${LO}_{ap}$ $\left({\underline {Л}}^{\mathcal{M}}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$.

Definition 4.10. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{M}, V\right)$ over ${\mathcal{Q}}_{2}$ is called GL_rS left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ w.r.t aftersets, if ${LO}_{ap}$ $\left({\underline {Л}}^{\mathcal{M}}, V\right)$ is soft left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$.

Definition 4.11. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{N}, V\right)$ over ${\mathcal{Q}}_{1}$ is called GL_rS quantale sub-module of ${\mathcal{Q}}_{2}$ w.r.t foresets, if ${LO}_{ap}$ $\left({}_{}{}^{\mathcal{N}}{\underline {{\mathcal{Л}}}}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}.$

Definition 4.12. Let $\left(Л, V\right)$ be the SBIR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. A soft set $\left(\mathcal{N}, V\right)$ over ${\mathcal{Q}}_{1}$ is called GL_rS left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ w.r.t foresets, if ${LO}_{ap}$ $\left({}_{}{}^{\mathcal{N}}{\underline {{\mathcal{Л}}}}, V\right)$ is soft left (right) ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$.

Theorem 4.13. Let $\left(Л, V\right)$ be a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. If $\left(\mathcal{M}, V\right)$ is a soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$, then $\left(\mathcal{M}, V\right)$ is a GL_rS ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ w.r.t the aftersets.

Proof. (1) Assume that $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$, then $\varnothing \ne {\underline {Л}}^{\mathcal{M}}\left(v\right)$ for any $v\in V.$ Let ${\mathcal{x}}_{l}\in {\underline {Л}}^{\mathcal{M}}\left(v\right)for\left(l\in L\right).$ Thereby, ${\mathcal{x}}_{l}Л\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Since $\left(Л, V\right)$ is a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ therefore, ${\vee }_{l\in L}\left({\mathcal{x}}_{l}Л\left(v\right)\right) = {(\vee }_{l\in L}{x}_{l})Л\left(v\right)\subseteq {\vee }_{l\in L}\mathcal{M}(v)$. As $\left(\mathcal{M}, V\right)$ is a soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ So, ${\vee }_{l\in L}\mathcal{M}\left(v\right)\mathcal{ }\subseteq \mathcal{M}\left(v\right)$. Thereby, ${(\vee }_{l\in L}{x}_{l})Л\left(v\right)\subseteq \mathcal{M}\left(v\right).$ Consequently, ${\vee }_{l\in L}{\mathcal{x}}_{l}\in {\underline {Л}}^{\mathcal{M}}\left(v\right)$.

(2) Let $\gamma \in {\underline {Л}}^{\mathbb{Q}}\left(v\right)\subseteq {Ǫ}_{1}$ and $\mathcal{x}\in {{\underline {{\mathcal{Л}}}}}^{\mathcal{M}}\left(v\right),$ where $\left(\mathbb{Q}, V\right)\;is\;a\;subset\;of\;({Ǫ}_{2}, V)$. Thereby, $\gamma Л\left(v\right)\subseteq \mathbb{Q}\left(v\right)$ and $\mathcal{x}\mathcal{Л}\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Since $\left(Л, V\right)$ is a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. Therefore, $\left(\gamma {\star }_{1}\mathcal{x}\right)\mathcal{Л}\left(v\right) =$ $\gamma Л\left(v\right){\star }_{2}\mathcal{x}\mathcal{Л}\left(v\right)\subseteq \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right).$ As $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ so, $\left(\gamma {\star }_{1}\mathcal{x}\right)\mathcal{Л}\left(v\right)\subseteq \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Thus, $\left(\gamma {\star }_{1}x\right)Л\left(v\right)\subseteq \mathcal{M}\left(v\right).$ Hence $\left(\gamma {\star }_{1}\mathcal{x}\right)\in {{\underline {{\mathcal{Л}}}}}^{\mathcal{M}}\left(v\right)$.

Theorem 4.14. Let $\left(Л, V\right)$ be a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. If $\left(\mathcal{N}, V\right)$ is a soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$, then $\left(\mathcal{N}, V\right)$ is a GL_rS ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ w.r.t the foresets.

Proof. Similar to proof of Theorem 4.13.

The converse of above Theorems is not true in general. To substantiate our claim, we will provide the following example:

Example 4.15. Let ${Ǫ}_{1} = \left\{\mathcal{L}, \mathcal{a}, \mathcal{b}, \mathcal{c}, \mathcal{d}, \mathcal{Ŧ}\right\}$ and ${\mathcal{Ǫ}}_{2}$ $= \left\{\mathcal{L}\mathcal{՛}, \mathcal{e}, \mathcal{f}, \mathcal{ }\mathcal{g}, \mathcal{h}, \mathcal{Ŧ}\mathcal{՛}\mathcal{՛}\right\}$ be two ${C_{lts}}$ described in Figures 7 and 8 respectively. The associative binary operation ⊗₁ and ⊗₂ on ${\mathcal{Ǫ}}_{1}$ and ${Ǫ}_{2}$ is defined as,

(1) $a{\otimes }_{1}b = \mathcal{ }\mathcal{L}$

(2) $a{\otimes }_{2}b = \mathcal{ }\mathcal{L}\mathcal{՛}$

We define ${\star }_{1}$ and ${\star }_{2}$ the left action on ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$, respectively as shown in Tables 7 and 8.

Then, ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ are quantale modules. Let $V = \left\{{v}_{1}, {v}_{2}\right\}$ and the SBIR $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ be defined by,

Then $\left(\mathcal{Л}, V\right)$ is SCMR. Following are the aftersets corresponding to $Л\left({v}_{1}\right)$ and $Л\left({v}_{2}\right),$

$\mathcal{L}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}\right\},$ $\mathcal{a}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}\right\}, \mathcal{b}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}\right\}$, $cЛ\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}\right\}$, $\mathrm{d}Л\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}\right\}, \mathcal{Ŧ}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}\right\},$

$\mathcal{L}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{f}, \mathcal{L}\mathcal{՛}\right\},$ $\mathcal{a}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{f}, \mathcal{L}\mathcal{՛}\right\}$, $\mathcal{b}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{f}, \mathcal{L}\mathcal{՛}\right\}$, $\mathcal{c}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{f}, \mathcal{L}\mathcal{՛}\right\}$, $dЛ\left({v}_{2}\right) = \left\{\mathcal{f}, \mathcal{L}\mathcal{՛}\right\}, \mathcal{Ŧ}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{f}, \mathcal{L}\mathcal{՛}\right\}.$

Then $\left(\mathcal{Л}, V\right)$ is SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ w.r.t aftersets. Define soft set $(\mathcal{M}, V)$ over ${\mathcal{Q}}_{2}$ by the rule, $\mathcal{M}\left({v}_{1}\right) = \left\{\mathcal{e}, \mathcal{h}, \mathcal{g}\right\}$ and $\mathcal{M}\left({v}_{2}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}, \mathcal{h}\right\}.$ Then $\left(\mathcal{M}, V\right)$ is not a soft ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$. But ${\underline {Л}}^{\mathcal{M}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{a}, \mathcal{b}, \mathcal{c}, \mathcal{d}, \mathcal{Ŧ}\right\}$ and ${{\underline {{\mathcal{Л}}}}}^{\mathcal{M}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{a}, \mathcal{b}, \mathcal{c}, \mathcal{d}, \mathcal{Ŧ}\right\}$ are ${\mathcal{Ǫ}}_{S}$ of ${\mathcal{Q}}_{1}$. So, $\left(\mathcal{M}, V\right)$ is a GL_rS ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ w.r.t aftersets.

Theorem 4.16. Let $\left(Л, V\right)$ be a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. If $\left(\mathcal{M}, V\right)$ is a soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$, then $\left(\mathcal{M}, V\right)$ is a GL_rS ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ w.r.t the aftersets.

Proof. (1) Assume that $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$, then $\varnothing \ne {\underline {Л}}^{\mathcal{M}}\left(v\right)$ for any $v\in V.$ Let ${\mathcal{x}}_{1}, {\mathcal{x}}_{2}\in {\underline {Л}}^{\mathcal{M}}\left(v\right)$. Thereby, ${\mathcal{x}}_{1}Л\left(v\right)\subseteq \mathcal{M}\left(v\right)$ and ${\mathcal{x}}_{2}Л\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Since $\left(Л, V\right)$ is a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ therefore ${\mathcal{x}}_{1}Л\left(v\right)\vee {\mathcal{x}}_{2}Л\left(v\right) = {(\mathcal{x}}_{1}\vee {\mathcal{x}}_{2})Л\left(v\right)\subseteq \mathcal{M}(v)\vee \mathcal{M}(v)$. As $\left(\mathcal{M}, V\right)$ is a soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$. So, $\mathcal{M}\left(v\right)\vee \mathcal{M}\left(v\right)\mathcal{ }\subseteq \mathcal{M}\left(v\right)$. Thereby, ${(\mathcal{x}}_{1}\vee {\mathcal{x}}_{2})Л\left(v\right)\subseteq \mathcal{M}(v).$ Consequently, ${\mathcal{x}}_{1}\vee {\mathcal{x}}_{2}\in {\underline {Л}}^{\mathcal{M}}\left(v\right)$. (2) Let ${\mathcal{x}}_{1}, {\mathcal{x}}_{2}\in$ ${\mathcal{Q}}_{1}$, ${\mathcal{x}}_{1}\le {\mathcal{x}}_{2}\;\mathrm{a}\mathrm{n}\mathrm{d}\;{\mathcal{x}}_{2}\in {\underline {Л}}^{\mathcal{M}}\left(v\right)$. So, ${\mathcal{x}}_{1}\vee {\mathcal{x}}_{2} = {\mathcal{x}}_{2}\in {\underline {Л}}^{\mathcal{M}}\left(v\right).$ Since ${\mathcal{x}}_{2}\in {\underline {Л}}^{\mathcal{M}}\left(v\right)\Rightarrow {\mathcal{x}}_{2}Л\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Suppose ${\mathcal{y}}_{1}\in {\mathcal{x}}_{1}Л\left(v\right)$ and ${\mathcal{y}}_{2}\in {\mathcal{x}}_{2}Л\left(v\right).$ As $\left(Л, V\right)$ is a SCMPR thereby, ${\mathcal{y}}_{1}\vee {\mathcal{y}}_{2}\in {\mathcal{x}}_{1}Л\left(v\right)\vee {\mathcal{x}}_{2}Л\left(v\right) = {(\mathcal{x}}_{1}\vee {\mathcal{x}}_{2}\left)Л\right(v).$ i.e., ${\mathcal{y}}_{1}\vee {\mathcal{y}}_{2}\in {\mathcal{x}}_{2}Л\left(v\right)\subseteq \mathcal{M}\left(v\right).$ As $M\left(v\right)is{Ǫ}_{I},$ so ${{\mathcal{y}}_{1}\le \mathcal{y}}_{1}\vee {\mathcal{y}}_{2}\in \mathcal{M}\left(v\right)$. Thus, ${\mathcal{y}}_{1}\in \mathcal{M}\left(v\right)$. Hence, ${\mathcal{x}}_{1}Л\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Consequently, ${\mathcal{x}}_{1}\in {\overline{Л}}^{\mathcal{M}}\left(v\right)$.

(3) Let $\beta \in {\underline {Л}}^{\mathbb{Q}}\left(v\right)\subseteq {Ǫ}_{1}$ and $\mathcal{x}\in {{\underline {{\mathcal{Л}}}}}^{\mathcal{M}}\left(v\right),$ where $\left(\mathbb{Q}, V\right)\;is\;a\;subset\;of\;\left({Ǫ}_{2}, V\right).$ Thereby, $\beta Л\left(v\right)\subseteq \mathbb{Q}\left(v\right)$ and $\mathcal{x}\mathcal{Л}\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Since $\left(Л, V\right)$ is a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$, therefore, $\left(\beta {\star }_{1}\mathcal{x}\right)\mathcal{Л}\left(v\right) =$ $\beta Л\left(v\right){\star }_{2}\mathcal{x}\mathcal{Л}\left(v\right)\subseteq \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right).$ As $\left(\mathcal{M}, V\right)$ is soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ so, $\left(\beta {\star }_{1}\mathcal{x}\right)\mathcal{Л}\left(v\right)\subseteq \mathbb{Q}\left(v\right){\star }_{2}\mathcal{M}\left(v\right)\subseteq \mathcal{M}\left(v\right)$. Thus, $\left(\beta {\star }_{1}\mathcal{x}\right)\mathcal{Л}\left(v\right)\subseteq \mathcal{M}\left(v\right).$ Hence, $\left(\beta {\star }_{1}\mathcal{x}\right)\in {{\underline {{\mathcal{Л}}}}}^{\mathcal{M}}\left(v\right)$.

Theorem 4.17. Let $\left(Л, V\right)$ be a SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$. If $\left(\mathcal{N}, V\right)$ is a soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$, then $\left(\mathcal{N}, V\right)$ is a GL_rS ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ w.r.t the foresets.

Proof. Similar to proof of Theorem 4.16.

The converse of above Theorem is not true in general. To substantiate our claim, we will provide the following example:

Example 4.18. Consider ${\mathcal{Q}}_{1}$ $= \left\{\mathcal{L}, \mathcal{a}, \mathcal{b}, \mathcal{c}, \mathcal{d}, \mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{Ŧ}\right\}$ and ${\mathcal{Q}}_{2} = \mathcal{ }\{\mathcal{L}\mathcal{՛}, \mathcal{ }\mathcal{e}, \mathcal{f}, \mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{g}, \mathcal{h}, \mathcal{ }\mathcal{Ŧ}\mathcal{՛}\}$ be two quantale modules described in Example 4.15. Let $V = \left\{{v}_{1}, {v}_{2}\right\}$ and the SBIR $\left(Л, V\right)$ from ${\mathcal{Q}}_{1}to$ ${\mathcal{Q}}_{2}$ be defined by,

Then $\left(\mathcal{Л}, V\right)$ is SCMR. Following are the aftersets corresponding to $Л\left({v}_{1}\right)$ and $Л\left({v}_{2}\right).$ $\mathcal{L}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}\right\}, \mathcal{a}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}\right\}, \mathcal{b}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}\right\},$ $\mathcal{c}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}\right\}, \mathcal{d}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}\right\}, \mathcal{Ŧ}\mathcal{Л}\left({v}_{1}\right) = \left\{\mathcal{L}\mathcal{՛}, \mathcal{f}, \mathcal{g}\right\},$

$\mathcal{L}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{g}, \mathcal{f}, \mathcal{h}\right\}, \mathcal{a}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{g}, \mathcal{f}, \mathcal{h}\right\},$ $\mathcal{b}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{g}, \mathcal{f}, \mathcal{h}\right\}$ $\mathcal{c}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{g}, \mathcal{f}, \mathcal{h}\right\}, \mathcal{d}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{g}, \mathcal{f}, \mathcal{h}\right\}, \mathcal{Ŧ}\mathcal{Л}\left({v}_{2}\right) = \left\{\mathcal{g}, \mathcal{f}, \mathcal{h}\right\}.$

Then $\left(\mathcal{Л}, V\right)$ is SCMPR from ${\mathcal{Q}}_{1}$ to ${\mathcal{Q}}_{2}$ w.r.t aftersets. A soft set $\left(\mathcal{M}, V\right)$ over ${\mathcal{Q}}_{2}$ is defined by, $\mathcal{M}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{f}, \mathcal{g}, \mathcal{h}\right\}$ and $\mathcal{M}\left({v}_{2}\right) = \left\{\mathcal{e}, \mathcal{f}, \mathcal{g}, \mathcal{h}\right\}.$ Then $\left(\mathcal{M}, V\right)$ is not a soft ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$. But ${\underline {Л}}^{\mathcal{M}}\left({v}_{1}\right) = \left\{\mathcal{L}, \mathcal{a}, \mathcal{b}, \mathcal{c}, \mathcal{d}, \mathcal{Ŧ}\right\}$ and ${{\underline {{\mathcal{Л}}}}}^{\mathcal{M}}\left({v}_{2}\right) = \left\{\mathcal{L}, \mathcal{a}, \mathcal{b}, \mathcal{c}, \mathcal{d}, \mathcal{Ŧ}\right\}$ are ${\mathcal{Ǫ}}_{I}$ of ${\mathcal{Q}}_{1}$. So, $\left(\mathcal{M}, V\right)$ is a GL_rS ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ w.r.t aftersets.

5.   Relationship between soft quantale module homomorphism and their approximation

We define soft weak quantale module homomorphism (SWMH) in this section and then by using SBIR, we established relationship between homomorphic images and their approximation.

Definition 5.1. ^[20] Consider two quantale modules (${\mathcal{Q}}_{1}$, ${\star }_{1}$) and (${\mathcal{Q}}_{2}$, ${\star }_{2}$) over $Ǫ$. A function $\mathrm{Ӻ}:{\mathcal{Q}}_{1}\to$ ${\mathcal{Q}}_{2}$ is called weak quantale module homomorphism (WMH) if

(1) $Ӻ\left(\mathcal{m}\vee \mathcal{n}\right) = \mathcal{Ӻ}\left(\mathcal{m}\right)\vee \mathcal{Ӻ}\left(\mathcal{n}\right);$

(2) $\mathcal{Ӻ}\left(\mathrm{\gamma }{\star }_{1}\mathcal{m}\right) = \mathrm{\gamma }{\star }_{2}\mathrm{Ӻ}\left(\mathcal{m}\right).$ For any $\mathrm{\gamma }\in \mathrm{Ǫ}$, $\mathcal{m}$, $\mathcal{n}\in$ ${\mathcal{Q}}_{1}$.

If $\mathrm{Ӻ}$ is one-one then $\mathrm{Ӻ}$ is monomorphism. If $\mathrm{Ӻ}$ is onto then $\mathrm{Ӻ}$ is called epimorphism and if $\mathrm{Ӻ}$ is bijective then $\mathrm{Ӻ}$ is called isomorphism between (${\mathcal{Q}}_{1}$, ${\star }_{1}$) and (${\mathcal{Q}}_{2}$, ${\star }_{2}$) over $Ǫ$.

Definition 5.2. Let $\left(\mathcal{M}, {V}_{1}\right)$ be a soft quantale module over ${\mathcal{Q}}_{1}$ and $\left(\mathcal{N}, {V}_{2}\right)$ be a soft quantale module over ${\mathcal{Q}}_{2}$. If there exist an ordered pair of functions $\left(Ӻ, \xi \right)$ satisfies the following,

(1) $\mathrm{Ӻ}:{\mathcal{Q}}_{1}\to$ ${\mathcal{Q}}_{2}$ is onto WMH.

(2) $\xi :{V}_{1}\to$ ${V}_{2}$ is surjective,

(3)$Ӻ\left(\mathcal{M}\left({v}_{1}\right)\right)$ = $\left(\mathcal{N}\left(\xi \right({v}_{1})\right)$ $\forall {v}_{1}\in {V}_{1}$.

Then $\left(\mathcal{M}, {V}_{1}\right)$ is said to be soft weak homomorphic to $\left(\mathcal{N}, {V}_{2}\right)$. The ordered pair $\left(Ӻ, \xi \right)$ of functions is SWMH. The pair $\left(Ӻ, \xi \right)$ is called soft week quantale module isomorphism (SWMI) and $\left(\mathcal{M}, {V}_{1}\right)$ is said to soft weak isomorphic to $\left(\mathcal{N}, {V}_{2}\right)$, if in ordered pair $\left(Ӻ, \xi \right)$ both $\mathrm{Ӻ}$ and $\xi$ are one-to-one functions.

Lemma 5.3. Let $\left(\mathcal{M}, {V}_{1}\right)$ be soft weak homomorphic to $\left(\mathcal{N}, {V}_{2}\right)$ with SWMH $\left(Ӻ, \zeta \right)$. Let $\left({Л}_{2}, {V}_{3}\right)$ be a SBIR over ${\mathcal{Q}}_{2}$ and $\left({\mathcal{M}}_{1}, {V}_{3}\right)\subseteq \left(\mathcal{M}, {V}_{1}\right)$. Define ${Л}_{1}\left({v}_{3}\right) =$ {$\left(\mathcal{a}, \mathcal{b}\right)\in {\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}$ : ($Ӻ$ ($\mathcal{a}), \mathcal{ }\mathcal{ }\mathcal{ }\mathcal{ }\mathcal{Ӻ}(\mathcal{b}\left)\right)\in {\mathcal{Л}}_{2}\left({v}_{3}\right)\}$ be a SBIR over ${\mathcal{Q}}_{1}$. Then the following holds,

(1) $\left({Л}_{1}, {V}_{3}\right)$ is SCMR if $\left({Л}_{2}, {V}_{3}\right)$ is SCMR.

(2) If $\left(Ӻ, \xi \right)$ is SWMI and $\left({Л}_{2}, {V}_{3}\right)$ is SCMR w.r.t the aftersets (w.r.t the foresets), then $\left({Л}_{1}, {V}_{3}\right)$ is SCMR w.r.t the aftersets (w.r.t the foresets).

(3) $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right) = {\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$

(4) $Ӻ\left({\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\subseteq {\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ and if $\left(Ӻ, \xi \right)$ is SWMI, then $Ӻ\left({\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right) = {\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$

(5) Let $\left(Ӻ, \xi \right)$ be a SWMI. Then $Ӻ\left(\mathcal{a}\right)\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\iff \mathcal{a}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ and $Ӻ\left(\mathcal{a}\right)\in \mathcal{Ӻ}\left({{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\iff \mathcal{a}\in {{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$.

Proof. (1) and (2) are obvious.

(3) Suppose $\left({\mathcal{M}}_{1}, {V}_{3}\right)\subseteq \left(\mathcal{M}, {V}_{1}\right)$. For any ${v}_{3}\in {V}_{3}$, let $\mathcal{p}\in {\mathcal{Ӻ}(\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right))$ for some $\mathcal{p}\in$ ${\mathcal{Q}}_{2}.$ Then there exists $\mathcal{q}\in$ ${\mathcal{Q}}_{1}$ such that, ${\mathcal{q}\in \overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ and $Ӻ\left(\mathcal{q}\right) = \mathcal{p}$. Thereby, ${\mathcal{q}\mathcal{Л}}_{1}\left({v}_{3}\right)\cap {\mathcal{M}}_{1}\left({v}_{3}\right)\ne \varnothing$. So, we have $\mathcal{k}{\in \mathcal{q}\mathcal{Л}}_{1}\left({v}_{3}\right)\cap {\mathcal{M}}_{1}\left({v}_{3}\right)$ i.e., $\mathcal{k}\in$ ${\mathcal{q}\mathcal{Л}}_{1}\left({v}_{3}\right)\;and\;\mathcal{k}\in {\mathcal{M}}_{1}\left({v}_{3}\right)$ which means, $(\mathcal{q}, \mathcal{k})\in$ ${\mathcal{Л}}_{1}\left({v}_{3}\right)$ $\Rightarrow$ $\left(Ӻ\right(\mathcal{q}), \mathcal{Ӻ}(\mathcal{k}\left)\right)\in$ ${\mathcal{Л}}_{2}\left({v}_{3}\right)$.Thereby, $Ӻ\left(\mathcal{k}\right)\in$ $\mathcal{Ӻ}\left(\mathcal{q}\right){\mathcal{Л}}_{2}\left({v}_{3}\right)\;and\;Ӻ\left(\mathcal{k}\right)\in \mathcal{Ӻ}\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right).$ Thus, $Ӻ\left(\mathcal{k}\right)\in$ $\mathcal{Ӻ}\left(\mathcal{q}\right){\mathcal{Л}}_{2}\left({v}_{3}\right)\cap Ӻ\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right)\Rightarrow Ӻ\left(\mathcal{q}\right){\mathcal{Л}}_{2}\left({v}_{3}\right)\cap Ӻ\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right)\ne \varnothing$. Hence, $\mathcal{p} = \mathcal{Ӻ}\left(\mathcal{q}\right)\in {\overline{\mathcal{Л}}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$ Consequently, $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\subseteq {\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$ Conversely, let $\mathcal{r}\in {\overline{\mathcal{Л}}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)\subseteq$ ${\mathcal{Q}}_{2}\Rightarrow {\mathcal{r}\mathcal{Л}}_{2}\left({v}_{3}\right)\cap Ӻ\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right)\ne \varnothing.$ So, there exists $\mathcal{t}{\in \mathcal{r}\mathcal{Л}}_{2}\left({v}_{3}\right)\cap {Ӻ(\mathcal{M}}_{1}\left({v}_{3}\right))$ such that $\mathcal{t}\in {\mathcal{r}\mathcal{Л}}_{2}\left({v}_{3}\right)\;and\;\mathcal{t}\in {\mathcal{Ӻ}(\mathcal{M}}_{1}\left({v}_{3}\right))$ i.e $(\mathcal{r}, \mathcal{t})\in$ ${\mathcal{Л}}_{2}\left({v}_{3}\right).$ Let 𝓊 $\in {\mathcal{M}}_{1}\left({v}_{3}\right)\subseteq \mathcal{M}\left({v}_{1}\right)\subseteq$ ${\mathcal{Q}}_{1}.$. So, $\mathcal{u}\in$ ${\mathcal{Q}}_{1}$ and $u՛\in$ ${\mathcal{Q}}_{1}$ such that $Ӻ$($\mathcal{u}) = \mathcal{t}$ and $\mathcal{Ӻ}$($\mathcal{u}\mathcal{՛}) = \mathcal{r}$ we have $\left(\mathcal{r}, \mathcal{t}\right) = \left(\mathcal{Ӻ}\left(\mathcal{u}\right), \mathcal{Ӻ}\left(\mathcal{u}\mathcal{՛}\right)\right)\in$ ${\mathcal{Л}}_{2}\left({v}_{3}\right).$ By above definition, $\left(\mathcal{u}, \mathcal{u}\mathcal{՛}\right)\in$ ${\mathcal{Л}}_{1}\left({v}_{3}\right)\Rightarrow$ $u\in$ ${\mathcal{u}\mathcal{՛}\mathcal{Л}}_{1}\left({v}_{3}\right).$ Thus, $\mathcal{u}\in$ $u՛{Л}_{1}\left({v}_{3}\right)\cap {\mathcal{M}}_{1}\left({v}_{3}\right)\Rightarrow \mathcal{u}\mathcal{՛}{\mathcal{Л}}_{1}\left({v}_{3}\right)\cap {\mathcal{M}}_{1}\left({v}_{3}\right)\ne \varnothing.$ So, $\mathcal{u}\mathcal{՛}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$. Hence $\mathcal{r}$ $=$ $\mathcal{Ӻ}$($\mathcal{u}\mathcal{՛})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Consequently, ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)\subseteq Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$.

(4) Suppose $\left({\mathcal{M}}_{1}, {V}_{3}\right)\subseteq \left(\mathcal{M}, {V}_{1}\right)$. For any ${v}_{3}\in {V}_{3}$, let $\mathcal{p}\in \mathcal{Ӻ}\left({{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ for some $\mathcal{p}\in$ ${\mathcal{Q}}_{2}$ then there exists $\mathcal{q}\in {\mathcal{Q}}_{1},$ such that ${\mathcal{q}\in {\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ and $Ӻ\left(\mathcal{q}\right) = \mathcal{p}$. Thereby, ${\mathcal{q}\mathcal{Л}}_{1}\left({v}_{3}\right)\subseteq {\mathcal{M}}_{1}\left({v}_{3}\right)$. Assume, $\mathcal{k}{\in \mathcal{p}\mathcal{Л}}_{2}\left({v}_{3}\right)\subseteq {\mathcal{Q}}_{2}$ there exists, $\mathcal{l}\in {\mathcal{Q}}_{1}$ such that $Ӻ\left(\mathcal{l}\right) = \mathcal{k}$. Thereby, $\mathcal{Ӻ}\left(\mathcal{l}\right)\in$ $\mathcal{Ӻ}\left(\mathcal{q}\right){\mathcal{Л}}_{2}\left({v}_{3}\right)\Rightarrow (Ӻ\left(\mathcal{q}\right), \mathcal{Ӻ}\left(\mathcal{l}\right))\in$ ${\mathcal{Л}}_{2}\left({v}_{3}\right)$. By given definition $\left(\mathcal{q}, \mathcal{l}\right)\in$ ${\mathcal{Л}}_{1}\left({v}_{3}\right)\Rightarrow \mathcal{l}\in$ ${\mathcal{q}\mathcal{Л}}_{1}\left({v}_{3}\right)\subseteq {\mathcal{M}}_{1}\left({v}_{3}\right)\Rightarrow$ $Ӻ\left(\mathcal{l}\right)\in \mathcal{Ӻ}({\mathcal{M}}_{1}\left({v}_{3}\right)$. Thus, $Ӻ\left(\mathcal{q}\right){\mathcal{Л}}_{2}\left({v}_{3}\right)\subseteq Ӻ({\mathcal{M}}_{1}\left({v}_{3}\right).$ Hence, $\mathcal{p} = \mathcal{Ӻ}\left(\mathcal{q}\right)\in {{\underline {{\mathcal{Л}}}}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$ Consequently, $Ӻ\left({\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\subseteq {\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$ Conversely, let $\mathcal{r}\in {{\underline {{\mathcal{Л}}}}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)\subseteq$ ${\mathcal{Q}}_{2}$ then there exists $\mathcal{s}\in$ ${\mathcal{Q}}_{1}$ such that, $Ӻ$($\mathcal{s})$ = $r$ and ${\mathcal{r}\mathcal{Л}}_{2}\left({v}_{3}\right)\subseteq Ӻ\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right).$ So, $Ӻ{\left(\mathcal{s}\right)\mathcal{Л}}_{2}\left({v}_{3}\right)\subseteq {Ӻ(\mathcal{M}}_{1}\left({v}_{3}\right)).$ Let $\mathcal{t}\in {\mathcal{s}\mathcal{Л}}_{1}\left({v}_{3}\right)$ i.e., $(\mathcal{s}, \mathcal{t})\in$ ${\mathcal{Л}}_{1}\left({v}_{3}\right).$ Thereby, $\left(Ӻ\right(\mathcal{s}), \mathcal{Ӻ}(\mathcal{t}\left)\right)\in {\mathcal{Л}}_{2}\left({v}_{3}\right)$ $\Rightarrow Ӻ\left(\mathcal{t}\right)\in \mathcal{Ӻ}\left(\mathcal{s}\right){\mathcal{Л}}_{2}\left({v}_{3}\right)\subseteq Ӻ\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right).$ So, $Ӻ\left(\mathcal{t}\right)\in \mathcal{Ӻ}\left({\mathcal{M}}_{1}\left({v}_{3}\right)\right)\Rightarrow \mathcal{t}\in \left({\mathcal{M}}_{1}\left({v}_{3}\right)\right).$ Thus, $\mathcal{s}{\mathcal{Л}}_{1}\left({v}_{3}\right)\subseteq$ ${\mathcal{M}}_{1}\left({v}_{3}\right)\Rightarrow \mathcal{s}\in {{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\Rightarrow Ӻ\left(\mathcal{s}\right)\in \mathcal{Ӻ}\left({{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Thereby, $\mathcal{r}$ $=$ $\mathcal{Ӻ}$($\mathcal{s})\in \mathcal{Ӻ}\left({{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Consequently, ${\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)\subseteq Ӻ\left({\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Hence, ${\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right) = Ӻ\left({\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$

(5) Let $\mathcal{r}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ for any ${v}_{3}\in {V}_{3}.$ Then, $Ӻ$($\mathcal{r})\in \mathcal{Ӻ}({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right))$. Conversely, suppose that $Ӻ$($\mathcal{r})\in \mathcal{Ӻ}({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right))$. As $Ӻ$ is SWMI so, $\mathcal{r}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$. Similarly, we can show that $Ӻ\left(\mathcal{a}\right)\in \mathcal{Ӻ}\left({{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\iff \mathcal{a}\in {{\underline {{\mathcal{Л}}}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$.

Theorem 5.4. Let $\left(\mathcal{M}, {V}_{1}\right)$ be soft weak isomorphic to $\left(\mathcal{N}, {V}_{2}\right)$ with SWMI $\left(Ӻ, \xi \right)$. Let $\left({Л}_{2}, {V}_{3}\right)$ be a SCMR over ${\mathcal{Q}}_{2}$ and $\left({\mathcal{M}}_{1}, {V}_{3}\right)\subseteq \left(\mathcal{M}, {V}_{1}\right)$. Define ${Л}_{1}\left({v}_{3}\right) =$ {$\left(\mathcal{a}, \mathcal{b}\right)\in {\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}$: ($Ӻ$($\mathcal{a}), \mathcal{Ӻ}(\mathcal{b}\left)\right)\in {\mathcal{Л}}_{2}\left({v}_{3}\right)\}$ for any ${v}_{3}\in {V}_{3}$. Then the following holds,

(1) ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}\iff$ ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ is ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ $\forall$ ${v}_{3}\in {V}_{3}$.

(2) ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}\iff$ ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ $\forall {v}_{3}\in {V}_{3}.$

Proof. (1) Let ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ be ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ for any ${v}_{3}\in {V}_{3},$ we will show that ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ is an ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}.$ By Lemma 5.3 we have $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right) = {\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$

ⅰ. Let $\mathcal{p}, \mathcal{q}\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Then there exists $r, \mathcal{s}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ such that $Ӻ$($r) = \mathcal{p}$ and $\mathcal{Ӻ}$($\mathcal{s}) = \mathcal{q}$. Since ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is an ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ so, $r\vee \mathcal{s}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$. By Lemma 5.3 (5) we have $Ӻ(r\vee \mathcal{s})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. As $\left(Ӻ, \xi \right)$ is SWMI, thus $\mathcal{p}\vee \mathcal{q}$ $= \mathcal{Ӻ}\left(r\right)\vee \mathrm{Ӻ}\left(\mathcal{s}\right) = \mathcal{Ӻ}(r\vee \mathcal{s})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Thus, $\mathcal{p}\vee \mathcal{q}\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$.

ⅱ. Let $p, \mathcal{q}\in {\mathcal{Q}}_{2}$ Consider $\mathcal{p}\le \mathcal{q}$ and $\mathcal{q}{\in \mathcal{Ӻ}(\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right))$. Then there exists $\mathcal{s}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ and $r\in$ ${\mathcal{Q}}_{1}$ such that $Ӻ$($r) = \mathcal{p}$ and $\mathcal{Ӻ}$($\mathcal{s}) = \mathcal{q}.$ Thus, $\mathcal{Ӻ}$($r)\le \mathrm{Ӻ}(\mathcal{s}$). As $\left(\mathcal{Ӻ}, \xi \right)$ is SWMI so, $Ӻ\left(r\vee \mathcal{s}\right) = \mathcal{Ӻ}$($r)\vee \mathrm{Ӻ}\left(\mathcal{s}\right) = \mathcal{Ӻ}\left(\mathcal{s}\right)\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\Rightarrow Ӻ\left(r\vee \mathcal{s}\right)\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ By Lemma 5.3 (5), $\left(r\vee \mathcal{s}\right)\in \left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Thereby, $r\le r\vee \mathcal{s}$. As ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{I}$, so $r\in {\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$. Thus, $Ӻ$($r)\in Ӻ({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right))$. Consequently, $\mathcal{p}\in$ $\mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$

ⅲ. Let $\gamma \in Ǫ$ and $\mathcal{p}\in$ $\mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Then there exists $r\in {\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ such that $Ӻ$($r) = \mathcal{p}$. As ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}$ so, $\gamma \star r\in$ ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$ Then by Lemma 5.3 (5) we have $Ӻ\left(\gamma \star r\right)\in Ӻ$(${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Since $\left(Ӻ, \xi \right)$ is SWMI thus, $\gamma \star \mathrm{Ӻ}\left(r\right) = Ӻ\left(\gamma \star r\right)\in Ӻ$(${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Thereby, $\gamma \star \mathrm{Ӻ}\left(r\right)\in Ӻ$ (${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$ Hence, $\gamma \star \mathcal{p}\in \mathcal{Ӻ}$ (${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$

Conversely, let ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ be a ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ for any ${v}_{3}\in {V}_{3}$, we will show that ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is a ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}.$ By Lemma 5.3 we have $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right) = {\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$

ⅰ. Let $\mathcal{p}, \mathcal{q}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\Rightarrow$ $Ӻ$($\mathcal{p})$, $\mathcal{Ӻ}$($\mathcal{q})\in$ $\mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Since $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ is an ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ so, $\mathrm{Ӻ}$ ($\mathcal{p})$ $\vee$ $\mathcal{Ӻ}$ ($\mathcal{q})\in$ $\mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. As $\left(Ӻ, \xi \right)$ is SWMI, thus $\mathrm{Ӻ}\left(\mathcal{p}\right)\vee \mathcal{Ӻ}\left(\mathcal{q}\right) = \mathcal{Ӻ}(\mathcal{p}\vee \mathcal{q})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Thereby, $Ӻ(\mathcal{p}\vee \mathcal{q})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. By Lemma 5.3 (5), $\mathcal{p}\vee \mathcal{q}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$.

ⅱ. Let $p, \mathcal{q}\in {\mathcal{Q}}_{1}$. Consider, $\mathcal{p}\le \mathcal{q}$ and $\mathcal{q}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$. Thus, $Ӻ$($\mathcal{p})\le \mathcal{Ӻ}(\mathcal{q}$)$\in$ $\mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Since $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ is an ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ so, $Ӻ$($\mathcal{p})\in \mathcal{Ӻ}({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right))$. By Lemma 5.3 (5) $\mathcal{p}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$.

ⅲ. Let $\gamma \in Ǫ$ and $\mathcal{p}\in$ ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$ Thereby, $Ӻ$($\mathcal{p})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Since $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ is ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ so, $\gamma \star \mathrm{Ӻ}\left(\mathcal{p}\right)\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Since $\left(Ӻ, \xi \right)$ is SWMI, thus, $\gamma \star \mathrm{Ӻ}\left(\mathcal{p}\right) = \mathcal{Ӻ}\left(\gamma \star \mathcal{p}\right)\in \mathcal{Ӻ}$(${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Then by Lemma 5.3 (5) we have $\gamma \star \mathcal{p}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$

(2) Let ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ be a ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ for any ${v}_{3}\in {V}_{3}$ we will show that ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}.$ By Lemma 5.3 we have $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right) = {\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$

ⅰ. Let ${\mathcal{q}}_{l}\in Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ for $l\in L$. Then there exists ${\mathcal{s}}_{l}\in {\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ such that $Ӻ$(${\mathcal{s}}_{l}) = {\mathcal{q}}_{l}$. Since ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is an ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ so, ${\vee }_{l\in L}{\mathcal{s}}_{l}\in {\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$. As $\left(Ӻ, \xi \right)$ is SWMI, thus ${\vee }_{l\in L}{\mathcal{q}}_{l}$ $= {\vee }_{l\in L}Ӻ\left({\mathcal{s}}_{l}\right) = \mathrm{Ӻ}\left({\vee }_{l\in L}{\mathcal{s}}_{l}\right)\in Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)\Rightarrow {\vee }_{l\in L}{\mathcal{q}}_{l}\in Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$.

ⅱ. Let $\beta \in Ǫ$ and $\mathcal{q}\in$ $\mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Then there exists $\mathcal{s}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ such that $Ӻ$($\mathcal{s}) = \mathcal{q}$. As ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}$ so, $\beta \star \mathcal{s}\in$ ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$ Then by Lemma 5.3 (5) we have $Ӻ(\beta \star \mathcal{s})\in$ $\mathcal{Ӻ}$ (${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Since $\left(Ӻ, \xi \right)$ is SWMI thus, $\beta \star \mathrm{Ӻ}\left(\mathcal{s}\right) = \mathcal{Ӻ}\left(\beta \star \mathcal{s}\right)\in \mathcal{Ӻ}$(${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Thereby, $\beta \star \mathrm{Ӻ}\left(\mathcal{s}\right)\in \mathcal{Ӻ}$(${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Hence, $\beta \star \mathcal{q}\in \mathcal{Ӻ}$(${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$

Conversely, let ${\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ be a ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ for any ${v}_{3}\in {V}_{3},$ we will show that ${\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is a ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}.$ By Lemma 5.3 we have $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right) = {\overline{Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right).$

ⅰ. Let ${\mathcal{q}}_{l}\in {\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ for $l\in L\Rightarrow$ $Ӻ$(${\mathcal{q}}_{l})\in$ $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Since $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ is an ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ so, ${\vee }_{l\in L}Ӻ$(${\mathcal{q}}_{l})\in$ $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. As $\left(Ӻ, \xi \right)$ is SWMI, thus ${\vee }_{l\in L}\mathrm{Ӻ}\left({\mathcal{q}}_{l}\right) = Ӻ\left({\vee }_{l\in L}{\mathcal{q}}_{l}\right)\in Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Thereby, $Ӻ\left({\vee }_{l\in L}{\mathcal{q}}_{l}\right)\in Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. By Lemma 5.3 (5), ${\vee }_{l\in L}{\mathcal{q}}_{l}\in {\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$.

ⅱ. Let $\beta \in Ǫ$ and $\mathcal{q}\in$ ${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$ Thereby, $Ӻ$($\mathcal{q})\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$. Since $Ӻ\left({\overline{Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ so, $\beta \star \mathrm{Ӻ}\left(\mathcal{q}\right)\in \mathcal{Ӻ}\left({\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)\right).$ Since $\left(Ӻ, \xi \right)$ is SWMI thus, $\beta \star \mathrm{Ӻ}\left(\mathcal{q}\right) = \mathcal{Ӻ}\left(\beta \star \mathcal{q}\right)\in \mathcal{Ӻ}$(${\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)).$ Then by Lemma 5.3 (5), we have $\beta \star \mathcal{q}\in {\overline{\mathcal{Л}}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right).$

Theorem 5.5. Let $\left(\mathcal{M}, {V}_{1}\right)$ be soft weak isomorphic to $\left(\mathcal{N}, {V}_{2}\right)$ with SWMI $\left(Ӻ, \xi \right)$. Let $\left({Л}_{2}, {V}_{3}\right)$ be a SCMPR over ${\mathcal{Q}}_{2}$ and $\left({\mathcal{M}}_{1}, {V}_{3}\right)\subseteq \left(\mathcal{M}, {V}_{1}\right)$. Define ${Л}_{1}\left({v}_{3}\right) =$ {$\left(\mathcal{a}, \mathcal{b}\right)\in {\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}$: ($Ӻ$($\mathcal{a}), \mathcal{Ӻ}(\mathcal{b}\left)\right)\in {\mathcal{Л}}_{2}\left({v}_{3}\right)\}$ for any ${v}_{3}\in {V}_{3}$. Then the following holds,

(1) ${\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{I}$ of ${\mathcal{Q}}_{1}\iff$ ${\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ is ${Ǫ}_{I}$ of ${\mathcal{Q}}_{2}$ $\forall$ ${v}_{3}\in {V}_{3}$.

(2) ${\underline {Л}}_{1}^{{\mathcal{M}}_{1}}\left({v}_{3}\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{1}\iff$ ${\underline {Л}}_{2}^{Ӻ\left({\mathcal{M}}_{1}\right)}\left({v}_{3}\right)$ is ${Ǫ}_{S}$ of ${\mathcal{Q}}_{2}$ $\forall$ ${v}_{3}\in {V}_{3}$.

Proof. Similar to proof of Theorem 5.4.

6.   An application of the decision-making approach

This section proposes decision-making techniques based on soft rough set theory based on soft binary relations. This strategy makes it possible to utilize the data provided by decision-makers without the need for additional information.

We obtain two values ${\underline {Л}}^{\mathcal{M}}\left({v}_{i}\right)$ and ${\overline{Л}}^{\mathcal{M}}\left({v}_{i}\right)$ which are most closed with respect to the aftersets by the soft lower and upper approximations of the soft set $\mathcal{M}$. Therefore, the choice value ${\delta }_{i}$ is redefined with respect to the aftersets as follows:

In a decision making problem, the maximum choice value ${\delta }_{i}$ is the optimum decision for the object ${x}_{i}\in U$ and the minimum choice value ${\delta }_{i}$ is the worst decision for the object ${x}_{i}\in U.$ For the given decision making problem, if the same maximum choice value ${\delta }_{i}$ belongs to two or more objects ${x}_{i}\in U$, then take one of them as the optimum decision randomly.

Algorithm

An algorithm is designed to approach a decision-making problem with respect to the aftersets is provided below. The decision algorithm is as follows:

(1) Compute the lower soft set approximation ${\underline {Л}}^{\mathcal{M}}$ and upper soft set approximation ${\overline{\mathcal{Л}}}^{\mathcal{M}}$ of a soft set $\mathcal{M}$ with respect to the aftersets.

(2) Corresponding to each ${x}_{i}\in U$, we calculate ${\underline {d}}_{ij}$ which is 0 if ${x}_{i}\notin {\underline {Л}}^{\mathcal{M}}\left({v}_{j}\right)$ and is 1 if ${x}_{i}\in {\underline {Л}}^{\mathcal{M}}\left({v}_{j}\right)$. Similarly, we calculate ${\overline{d}}_{ij}$ which is 0 if ${x}_{i}\notin {\overline{Л}}^{\mathcal{M}}\left({v}_{j}\right)$ and is 1 if ${x}_{i}\in {\overline{Л}}^{\mathcal{M}}\left({v}_{j}\right).$

(3) Compute the choice value ${\delta }_{i} = \sum _{j = 1}^{n}{\underline {d}}_{ij}+\sum _{j = 1}^{n}{\overline{d}}_{ij}$ with respect to the aftersets.

(4) The best decision is ${x}_{k}\in U$ if ${\delta }_{k} = ma{x}_{i}$ ${\delta }_{i}$, $i\text{ = 1, 2, …, }\left|U\right|.$

(5) The worst decision is ${x}_{k}\in U$ if ${\delta }_{k} = mi{n}_{i}$ ${\delta }_{i}$, $i\text{ = 1, 2, …, }\left|U\right|.$

(6) If the value of $k$ is more than one, then we can choose any one of ${x}_{k}$. In a similar way, we can define an algorithm for foresets.

By an example in this subsection, an application of the decision-making approach is given.

Example 6.1. Suppose that Mr. X wants to buy a shirt for his own use. Let $U = \left\{the \; set\; of\; all\; shirts\; designs\right\} = \left\{{d}_{1}, {d}_{2}, {d}_{3}, {d}_{4}, {d}_{5}, {d}_{6}\right\}$ and $W = \left\{the\; colors\; of\; all\; designs\right\} = \left\{{c}_{1}, {c}_{2}, {c}_{3}, {c}_{4}\right\}$ and the set of attributes be $V = \left\{{v}_{1}, {v}_{2}, {v}_{3}\right\} = \left\{the\; set\; of\; stores\; near\; his\; house\right\}.$ Define $Л:V \to P\left(U\times \mathrm{W}\right)$ by

Represents the relation between designs and colors available on store ${v}_{i}$ for $1\le i\le 3.$

Then

${d}_{1}Л\left({v}_{1}\right) = \left\{{c}_{1}, {c}_{2}, {c}_{3}\right\}, {d}_{2}Л\left({v}_{1}\right) = \left\{{c}_{2}, {c}_{4}\right\}, {d}_{3}Л\left({v}_{1}\right) = \varnothing, {d}_{4}Л\left({v}_{1}\right) = \left\{{c}_{2}, {c}_{3}\right\},$

${d}_{5}Л\left({v}_{1}\right) = \left\{{c}_{4}, {c}_{3}\right\}, {d}_{6}Л\left({v}_{1}\right) = \left\{{c}_{1}\right\}, and{d}_{1}Л\left({v}_{2}\right) = \left\{{c}_{3}\right\}, {d}_{2}Л\left({v}_{2}\right) = \left\{{c}_{3}\right\},$

${d}_{3}Л\left({v}_{2}\right) = \left\{\varnothing \right\}, {d}_{4}Л\left({v}_{2}\right) = \left\{{c}_{1}\right\}, {d}_{5}Л\left({v}_{2}\right) = \left\{{c}_{1}\right\}, {d}_{6}Л\left({v}_{2}\right) = \left\{{c}_{2}, {c}_{3}\right\}and$

${d}_{1}Л\left({v}_{3}\right) = \left\{\varnothing \right\}, {d}_{2}Л\left({v}_{3}\right) = \left\{{c}_{4}\right\}, {d}_{3}Л\left({v}_{3}\right) = \left\{{c}_{1}, {c}_{3}\right\}, {d}_{4}Л\left({v}_{3}\right) = \varnothing,$

${d}_{5}Л\left({v}_{3}\right) = \left\{{c}_{4}, {c}_{3}\right\}, {d}_{6}Л\left({v}_{3}\right) = \left\{\varnothing \right\}.$

Where ${d}_{i}Л\left({v}_{j}\right)$ represents the color of the design ${d}_{i}$ available on store ${v}_{j}.$ Further

Where $Л\left({v}_{j}\right){c}_{i}$ represents the design of the color ${c}_{i}$ available on store ${v}_{j}.$

Define $\mathcal{M}:V\to P\left(W\right)$ which represents the preference of the color given by Mr. X such that

$\mathcal{M}\left({v}_{1}\right) = \left\{{c}_{1}, {c}_{4}\right\}, \mathcal{M}\left({v}_{2}\right) = \left\{{c}_{2}, {c}_{5}\right\}, \mathcal{M}\left({v}_{3}\right) = \left\{{c}_{2}, {c}_{3}, {c}_{4}\right\}and$ define $H:V\to P\left(U\right)$ which represents the preference of the design given by Mr. X such that

$H\left({v}_{1}\right) = \left\{{d}_{2}, {d}_{3}, {d}_{6}\right\}, H\left({v}_{2}\right) = \left\{{d}_{1}, {d}_{3}\right\}, H\left({v}_{3}\right) = \left\{{d}_{1}, {d}_{2}, {d}_{5}, {d}_{6}\right\}.$

Consider the following table after applying the above algorithm (see Table 9).

Here the choice value ${\delta }_{i} = \sum _{j = 1}^{3}{\underline {d}}_{ij}+\sum _{j = 1}^{3}{\overline{d}}_{ij}$ is calculated with respect to aftersets. The shirt of design ${d}_{6}$ scores the maximum choice value ${\delta }_{k} = 4 = {\delta }_{6}$, and the decision is in favor of the shirt of design ${d}_{6}$ for selection. Moreover, the shirts of designs ${d}_{4}$ are totally ignored. Hence, Mr. X will choose the shirt of design ${d}_{6}$ for his personal use and he will not select the shirt of design ${d}_{4}$ with respect to the aftersets.

7.   Conclusions

There are many applications of soft set and rough set theories. Their combination involved many researchers to develop many concepts in mathematics. In this paper, major role of rough soft sets with substructures of quantale module are discussed. Some characterizations of soft substructures of quantale modules are introduced. The detailed study of approximations of soft substructure in quantale module are presented. During this process, we have made further detailed discussion how soft substructures of one quantale module can be related to that of another quantale module under soft quantale homomorphism. Additionally, we describe the algebraic relationships between the upper (lower) approximations of soft substructures of quantale modules and the upper (lower) approximations of their homomorphic images using the concept of soft quantale module homomorphism. For further work, one can proceed this type of approximations to different algebraic structures especially to substructures of hyperquantales and fuzzy hypersubstructures of hyperquantales.

Acknowledgments

Researchers Supporting Project number (RSP2023R440), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors state that they are not involved in any conflict of interest.