Generalized roughness of fuzzy substructures in quantales with respect to soft relations

Abstract

In this work, we have proposed a new relationship among rough set, soft set and quantales with the help of soft compatible relation. This typical relationship is used to approximate the fuzzy substructures in quantales in association with soft compatible relations by using aftersets and foresets. This type of approximation is extended notation of rough quantales, rough fuzzy subquantales and soft subquantales. We have corroborated this work by considering some test examples containing soft compatible relations over quantales. Moreover, by using soft compatible relations, we will describe the relationship between upper (lower) generalized rough fuzzy soft substructures of quantale and the upper (lower) approximations of their homomorphic images with the help of weak quantale homomorphism. The comparison of this type approximations and their results affirms the superiority of our new approximation method over current methods on the topic.

Keywords

Quantale ideals soft relations aftersets forsets roughness of fuzzy substructures in Quantales

1 Introduction

The rough approximations theory presented by Pawlak [22], finds its application in various problems such as image processing, data mining, medical informatics, pattern recognition, extracting interesting patterns in databases, in conjunction with other uncertain reasoning methods such as fuzzy logic, neural network and genetic algorithm. A rough set, is a formal approximation of a crisp set in terms of a pair of sets which give the lower and the upper approximation of the original set and these lower and upper approximation sets are crisp sets itself. In a rough set theory an equivalence relation palys a vital role while approximating a undefinable subset of universe. A rough fuzzy set is a pair of fuzzy sets resulting from the approximation of a fuzzy set in a crisp approximation space. In rough fuzzy set presented by Dubois and Prade [7], universal set and equivalence relation on it, is necessary. Generalized rough set [38] contains the concept of set valued mapping between two different sets instead of equivalence relation. That is T : U → P (V). For X ⊆ V, $\bar{T} (X)$ and $\underline{T} (X)$ are called the upper and lower generalized approximations of X and the pair $(\underline{T} (X), \bar{T} (X))$ is referred to as a generalized rough set. This generalized rough set is generalization of Pawlak rough sets. In this work, we have applied a way of carrying out approximation in a different way, which is again a generalization of Pawlak rough set and rough fuzzy set.

Thus, in this work by keeping above methods, we have taken Ω : H → P (K₁ × K₂), where H is a subset of E (parametersset). Then (Ω, H) is called a soft binary relation from a quantale K₁ to K₂. For a fuzzy subset ψ of K₂, the lower approximation $({\underline{Ω}}^{ψ}, H)$ and the upper approximation $({\bar{Ω}}^{ψ}, H)$ of ψ with respect to the aftersets are the two fuzzy soft sets over K₁, defined as follows: $\begin{matrix} {\underline{Ω}}^{ψ} (e) (k_{1}) & = & {\begin{matrix} \land_{a \in k_{1} Ω (u)} ψ (a) if k_{1} Ω (u) \neq ϕ 0 if k_{1} Ω (u) = ϕ \end{matrix} and \\ {\bar{Ω}}^{ψ} (e) (k_{1}) & = & {\begin{matrix} \lor_{a \in k_{1} Ω (u)} ψ (a) if k_{1} Ω (u) \neq ϕ 0 if k_{1} Ω (u) = ϕ \end{matrix} \end{matrix}$

Thus, rough fuzzy soft sets are obtained in association with soft compatible relations powered by aftersets. Further, these concepts are applied to quantales K₁ and K₂.

Quantales introduced by Mulvey [21], are a generalisation of locales. The aim behind this is to provide a constructive formulation of the foundations of quantum mechanics. There is a relation between quantale theory and linear logic introduced by Yetter [40] showed a complete class of models for linear intuitionistic logic. There are many research topics where quantales are employed like functional analysis, rough set theory [23 , 39], theoretical computer science [28], algebraic theory [16], linear logic [8] and topological theory [9].

Molodtsov’s soft set theory [19] was proposed as a general scientific tool to manage vulnerability. Several algebraic operations in soft set theory, [18] were examined by Maji et al. and to consolidate the algebraic aspects of soft sets, some new operations were discussed by Ali et al., [5]. Soft set theory has been applied to many algebraic structures including soft modules [35], soft groups [2] and some contributions to soft groups [3], soft rings [1], soft ordered semigroups [4] etc.

Approximations by soft binary relations with the help of aftersets and foresets are extensively and considerably utilized these days. Kanwal and Shabir [11], at first, introduced approximation of ideals in semigroups by soft relations. Rough approximation of fuzzy substructures based on soft relations was introduced by Kanwal and Shabir [12]. Further, generalized approximation of substructures of quantales by soft relation, [13] was introduced Kanwal et al. Soft substructures of semigroups were approximated by soft binary relation by Kanawal and Shabir [14]. Roughness applied to a set by (α, β)-indiscernibility of Bipolar fuzzy relation were presented by Gul and Shabir [29]. Modified rough bipolar soft sets was introduced by Shabir and Gul [32].

The fuzzy set, introduced by Zadeh [41], had been applied to generalize the fundamental ideas of algebra. Fuzzy substructures in quantales were first investigated in [17]. They defined rough fuzzy substructures of quantale. Generalized rough fuzzy substructures in quantales were introduced by Qurashi and Shabir [23]. Generalization of approximation of fuzzy substructures in quantales in the form of (∈ , ∈ ∨ q) and (∈ _γ, ∈ _γ ∨ q_δ) were studied by Qurashi and Shabir [25, 26]. Some results related to fuzzy hyperideals of hyperquantales [20], were introduced by Farooq et al. Several authors related fuzzy set theory to different algebraic structures like groups, rings, modules, semirings, semigroups and ordered semigroups, etc. Rough Pythagorean fuzzy ideals in semigroups, [10] were discussed by Hussain et al. Fuzzy idempotent near-ring was studied by Hussain and Shabir [33]. Some studies about regular and intra-regular semirings in terms of bipolar fuzzy ideals [34], was investigated by Shabir et al. Rough fuzzy ternary subsemigroups based on fuzzy ideals with three-dimensional congruence relation was studied by Bashir et al., [6].

In this paper, we have explored the approximation of fuzzy substructures of quantales with respect to soft binary relations powered by foresets and aftersets. In recent years, several methods have been proposed to delve into approximations of algebraic and fuzzy algebraic structures but as far as we know, this technique is novel and has not been employed for quantale.

The whole paper is organized as follows. After introduction, some related definitions and results are presented in Section 2 while Section 3 presents approximations by soft compatible relations with respect to a fuzzy subset with the help of aftersets and foresets and in this way fuzzy soft sets are obtained. These concepts are applied to fuzzy substructures of quantales in Section 4 and generalized rough fuzzy soft substructures in quantales are obtained. Moreover, under quantale homomorphism, relation between lower and upper approximations of fuzzy soft substructures of quantales and the lower (upper) approximations of their images under quantale homomorphism are discussed in Section 5.

2 Preliminaries

Some definitions which are basic in nature and are related to next sections, are given in the following.

Definition 1. [30] A complete lattice K having associative binary operation ⊗ is called a quantale if for all w, z ∈ K and {z_i} , {w_i} ⊆ K (i ∈ I). The following hold; $\begin{matrix} w \otimes (\lor_{i \in I} z_{i}) & = & \lor_{i \in I} (w \otimes z_{i}) and (\lor_{i \in I} w_{i}) \\ \otimes z & = & \lor_{i \in I} (w_{i} \otimes z) . \end{matrix}$ Let X_i, X₁, X₂ ⊆ K. Then the following are defined; $\begin{matrix} X_{1} \otimes X_{2} & = & {a_{1} \otimes a_{2} : a_{1} \in X_{1}, a_{2} \in X_{2}}; \\ X_{1} \lor X_{2} & = & {a_{1} \lor a_{2} : a_{1} \in X_{1}, a_{2} \in X_{2}}; \\ \lor_{i \in I} X_{i} & = & {\lor_{i \in I} a_{i} : a_{i} \in X_{i}} . \end{matrix}$ Let K be a quantale and ∅ ≠ K₁ ⊆ K. Then K₁ is a subquantale of K if it is closed under arbitrary sup and ⊗.

Throughout the paper, the symbol K₁ and K₂ will be utilized for quantales, the symbol ⊤ will denote the top element and ⊥ will stand for the bottom one for quantales, unless stated otherwise.

Now, some definitions about ideals, fuzzy ideals of quantale and some examples of them are given in the following.

Definition 2. [36] Let K be a quantale and ∅ ≠ I ⊆ K is called an ideal of K if the conditions below are satisfied:

(i) If z,w ∈ I, then z∨ w ∈ I ;

(ii) for all z, w ∈ K and w ∈ I such that z ⩽ w implies z∈ I ;

(iii) for all z ∈ K and w ∈ I implies z ⊗ w ∈ I and w ⊗ z ∈ I.

Let I be an ideal of a quantale K. Then, I is said to be a prime ideal if z ⊗ w ∈ I implies z ∈ I or w ∈ I for all z, w ∈ K. An ideal I is said to be a semi prime ideal if z ⊗ z ∈ I implies z ∈ I for each z ∈ K.

Definition 3. [41] A function ψ : K ⟶ [0, 1] is known as fuzzy subset of K. For α ∈ [0, 1], the sets ψ_α ={ k ∈ K ∣ ψ (k) ≥ α } ; ψ_{α
⁺} ={ k ∈ K ∣ ψ (k) > α } are called, α-cut and strong α-cut of the fuzzy subset ψ, respectively.

Definition 4. [7] Let (X, Ω) be an approximation space where X is a non-empty set and Ω is an equivalence relation on X. A fuzzy subset ψ is a mapping from X to [0, 1] then for x ∈ X, we define $\underline{Ω} (ψ) (z) = \underset{p \in {[z]}_{Ω}}{⋀} ψ (p) and \bar{Ω} (ψ) (z) = \underset{p \in {[z]}_{Ω}}{⋁} ψ (p)$

They are called, the lower and upper approximations of ψ, respectively. If $\underline{Ω} (ψ) \neq \bar{Ω} (ψ)$ , then $Ω (ψ) = (\underline{Ω} (ψ), \bar{Ω} (ψ))$ is called a rough fuzzy set with respect to Ω.

If ψ₁ and ψ₂ are the fuzzy subsets of K. Then, ψ₁⊆ ψ₂ if and only if ψ₁ (z) ≤ψ₂ (z) for all z ∈ K. Also (ψ₁ ∩ ψ₂) (z) = Min {ψ₁ (z) , ψ₂ (z)} and (ψ₁ ∪ ψ₂) (z) = Max {ψ₁ (z) , ψ₂ (z)} for all z ∈ K. Throughout this paper, Min for infimum and Max for maximum in [0, 1] will be employed, while the symbols ∧ and ∨ will be utilized for infimum and supremum for the elements of K. Moreover, we will express fuzzy subset, fuzzy subquantales, fuzzy ideals, fuzzy prime and fuzzy semi-prime ideals by fsst, fsq, fid, fpid and fspid, respectively unless stated otherwise.

Definition 5. [25] Let ψ be a fsst of a quantale K. Then ψ is a fsq of K if

(i) ψ (∨ _i∈Iz_i) ≥ $\underset{i \in I}{Min}$ ψ (z_i) ;

(ii) ψ (y ⊗ z) ≥ Min {ψ (y) , ψ (z)} for all {z_i} ⊆ K (i ∈ I) and for all z, y ∈ K.

Definition 6. [17] A non-empty fsst ψ of K is called a fid of K, if the following conditions are satisfied;

(i) z≤ w ⇒ ψ (w) ≤ ψ (z) ;

(ii) Min {ψ (z) , ψ (w)} ≤ψ (z∨ w) ;

(iii) Max {ψ (z) , ψ (w)} ≤ψ (z ⊗ w) for all z, w ∈ K.

From (i) and (ii) in Definition 6, it is observed that ψ (z ∨ w) = Min {ψ (z) , ψ (w)} for all z, w ∈ K. Thus, a fsst ψ of K is a fid of K if and only if ψ (z ∨ w) = Min {ψ (z) , ψ (w)} and ψ (z ⊗ w) ≥ Max {ψ (z) , ψ (w)} for all z, w ∈ K.

Definition 7. [17] A non-constant fid ψ of a quantale K is called a fpid of K if ψ (z ⊗ w) = ψ (z) or ψ (z ⊗ w) = ψ (w) for all z, w ∈ K.

Definition 8. [17] Let ψ be a fid of a quantale K. Then ψ is called a fspid of K if ψ (z ⊗ z) = ψ (z) for all z ∈ K.

Definition 9. [31] A pair (Ω, A) is called a fuzzy soft set over U if Ω is a mapping given by Ω : A → ₣(U) and A is a subset of E (the set of parameters) and ₣(U) is the set of all fuzzy subsets of U.

Definition 10. [31] Let (Ω, A) and (Γ, B) be two fuzzy soft sets over a common universe U. Then (Ω, A) is a fuzzy soft subset of (Γ, B) if

(1) A ⊆ B and

(2) Ω (e) is a fuzzy subset of Γ (e) for all e ∈ A.

(Ω, A) and (Γ, B) over a common universe U are said to be fuzzy soft equal if (Ω, A) is a fuzzy soft subset of (Γ, B) and (Γ, B) is a fuzzy soft subset of (Ω, A).

For convenience, we employ FSS, FSTQ, FSTI, FSTPI and FSTSPI for fuzzy soft set, fuzzy soft quantale, fuzzy soft ideal, fuzzy soft prime ideal and fuzzy soft semi-prime ideal.

Definition 11. Let K be a quantale and (Ω, A) be a FSS over K. Then

(1) (Ω, A) is called a FSTQ over K if Ω (e) is a fsq of K for all e ∈ A with Ω (e) ≠ ϕ.

(2) A FSS, (Ω, A) over a quantale K is called a FSTI over K if Ω (e) is a fid of K for all e ∈ A with Ω (e) ≠ ϕ.

(3) A FSS, (Ω, A) over a quantale K is called a FSTPI over K, if Ω (e) is a fpid of K for all e ∈ A with Ω (e) ≠ ϕ. A FSS, (Ω, A) over a quantale K is called a FSTSPI over K, if Ω (e) is a fspid of K for all e ∈ A with Ω (e) ≠ ϕ.

Definition 12. [13] A soft binary relation Ω from a quantale K₁ to K₂ is called soft compatible if (a, b) , (c, d) ∈ Ω (u) ⇒ (a ⊗ c, b ⊗ d) ∈ Ω (u) and also (a_i, b_i) ∈ Ω (u) ⇒ (∨ _i∈Ia_i, ∨ _i∈Ib_i) ∈ Ω (u) for all a, c, a_i ∈ K₁ and b, d, b_i ∈ K₂ (i ∈ I) and for all u ∈ A.

Definition 13. [13] Let (Ω, H) be a soft compatible relation from a quantale K₁ to K₂. Then (Ω, H) is called soft ∨-complete and soft ⊗-complete with respect to aftersets if

(1) aΩ (u) ∨ bΩ (u) = (a ∨ b) Ω (u) and

(2) aΩ (u) ⊗ bΩ (u) = (a ⊗ b) Ω (u) for all a, b ∈ K₁ and for all u ∈ H.

If it is both ∨-complete and ⊗-complete, then it is called a soft complete relation with respect to aftersets. Similarly, one can define Definition 13 but for the foresets.

3 Roughness of a fuzzy set by soft relations

In this section, while using two quantales, foresets and aftersets are applied to find out approximations of fssts in two ways with the help of soft binary relations from K₁ and K₂. That is by taking the fsst of a quantale K₂ and then evaluating their lower and upper approximations by aftersets, the resulting fsst of K₁ are obtained. Likewise, for fsst of K₁ and by taking their lower and upper approximations by foresets, the resulting fsst of K₂ are obtained.

Definition 14. [15] Let Ω : H → P (K₁ × K₂), where H is a subset of E (parametersset). Then (Ω, H) is called a soft binary relation from a quantale K₁ to K₂. For a fsst ψ of K₂, the lower approximation $({\underline{Ω}}^{ψ}, H)$ and the upper approximation $({\bar{Ω}}^{ψ}, H)$ of ψ with respect to the aftersets are the two FSS over K₁, as defined follows: ${\underline{Ω}}^{ψ} (e) (k_{1}) = {\begin{matrix} \land_{a \in k_{1} Ω (u)} ψ (a) if k_{1} Ω (u) \neq ϕ 0 if k_{1} Ω (u) = ϕ \end{matrix}$

and ${\bar{Ω}}^{ψ} (e) (k_{1}) = {\begin{matrix} \lor_{a \in k_{1} Ω (u)} ψ (a) if k_{1} Ω (u) \neq ϕ 0 if k_{1} Ω (u) = ϕ \end{matrix}$

And for a fsst, η of K₁, the lower approximation $(^{η} \underline{Ω}, H)$ and the upper approximation $(^{η} \bar{Ω}, H)$ of η with respect to the foresets are the two FSS over K₂, as defined follows: $^{η} \underline{Ω} (u) (k_{2}) = {\begin{matrix} \land_{a \in Ω (u) k_{2}} η (a) if Ω (u) k_{2} \neq ϕ 0 if Ω (u) k_{2} = ϕ \end{matrix}$

and $^{η} \bar{Ω} (u) (k_{2}) = {\begin{matrix} \lor_{a \in Ω (u) k_{2}} η (a) if Ω (u) k_{2} \neq ϕ 0 if Ω (u) k_{2} = ϕ \end{matrix}$ for all u ∈ H, where k₁Ω (u) = {k₂ ∈ K₂ : (k₁, k₂) ∈ Ω (u)} and is called the afterset of k₁ and Ω (u) k₂ ={ k₁ ∈ K₁ : (k₁, k₂) ∈ Ω (u) } and is called the foreset of k₂. Moreover, ${\underline{Ω}}^{ψ} : H \to ₣ (K_{1}),$ ${\bar{Ω}}^{ψ} : H \to ₣ (K_{1})$ , $^{η} \underline{Ω} : H \to ₣ (K_{2})$ and $^{η} \bar{Ω} : H \to ₣ (K_{2})$ for each ψ ∈ ₣(K₂) and η ∈ ₣(K₁) , respectively.

Theorem 1. [15] Let (Ω, H) and (Γ, H) be two soft binary relations from non-empty sets K₁ to K₂ and ψ₁, ψ₂ be non-empty fsst of K₂. Then the following hold:

(1) $ψ_{1} \leq ψ_{2} \Rightarrow {\underline{Ω}}^{ψ_{1}} (u) \leq {\underline{Ω}}^{ψ_{2}} (u)$ for all u∈ H ;

(2) $ψ_{1} \leq ψ_{2} \Rightarrow {\bar{Ω}}^{ψ_{1}} (u) \leq {\bar{Ω}}^{ψ_{2}} (u)$ for all u∈ H ;

(3) $({\underline{Ω}}^{ψ_{1}}, H) \cap ({\underline{Ω}}^{ψ_{2}}, H) = ({\underline{Ω}}^{ψ_{1} \cap ψ_{2}}, H);$

(4) $({\bar{Ω}}^{ψ_{1}}, H) \cap ({\bar{Ω}}^{ψ_{2}}, H) \supseteq ({\bar{Ω}}^{ψ_{1} \cap ψ_{2}}, H);$

(5) $({\underline{Ω}}^{ψ_{1}}, H) \cup ({\underline{Ω}}^{ψ_{2}}, H) \subseteq ({\underline{Ω}}^{ψ_{1} \cup ψ_{2}}, H);$

(6) $({\bar{Ω}}^{ψ_{1}}, H) \cup ({\bar{Ω}}^{ψ_{2}}, H) = ({\bar{Ω}}^{ψ_{1} \cup ψ_{2}}, H);$

(7) (Ω, H) ⊆ (Γ, H) implies $({\underline{Ω}}^{ψ_{1}}, H) \supseteq ({\underline{Γ}}^{ψ_{1}}, H);$

(8) (Ω, H) ⊆ (Γ, H) implies $({\bar{Ω}}^{ψ_{1}}, H) \subseteq ({\bar{Γ}}^{ψ_{1}}, H)$ .

Theorem 2. [15] Let (Ω, H) and (Γ, H) be two soft binary relations from non-empty sets K₁ to K₂ and ψ₁, ψ₂ be non-empty fsst of K₁. Then the following hold:

(1) ψ₁≤ ψ₂ ⇒ $^{ψ_{1}} \underline{Ω} (u) \leq$ $^{ψ_{2}} \underline{Ω} (u)$ for all u∈ H ;

(2) ψ₁≤ ψ₂ ⇒ $^{ψ_{1}} \bar{Ω} (u) \leq$ $^{ψ_{2}} \bar{Ω} (u)$ for all u∈ H ;

(3) $(^{ψ_{1}} \underline{Ω}, H) \cap (^{ψ_{2}} \underline{Ω}, H) = (^{ψ_{1} \cap ψ_{2}} \underline{Ω}, H);$

(4) $(^{ψ_{1}} \bar{Ω}, H) \cap (^{ψ_{2}} \bar{Ω}, H) \supseteq ({\bar{Ω}}^{ψ_{1} \cap ψ_{2}}, H);$

(5) $(^{ψ_{1}} \underline{Ω}, H) \cup (^{ψ_{2}} \underline{Ω}, H) \subseteq ({\underline{Ω}}^{ψ_{1} \cup ψ_{2}}, H);$

(6) $(^{ψ_{1}} \bar{Ω}, H) \cup (^{ψ_{2}} \bar{Ω}, H) = ({\bar{Ω}}^{ψ_{1} \cup ψ_{2}}, H);$

(7) (Ω, H) ⊆ (Γ, H) implies $(^{ψ_{1}} \underline{Ω}, H) \supseteq (^{ψ_{1}} \underline{Γ}, H);$

(8) (Ω, H) ⊆ (Γ, H) implies $(^{ψ_{1}} \bar{Ω}, H) \subseteq (^{ψ_{1}} \bar{Γ}, H)$ .

The following result follows from parts (7) and (8) of Theorem 1.

Theorem 3. Let (Ω, H) and (Γ, H) be soft binary relations from K₁ to K₂. If ψ is a fsst of K₂. Then

(1) $({(\bar{Ω \cap Γ})}^{ψ}, H) \subseteq ({\bar{Ω}}^{ψ}, H) \cap ({\bar{Γ}}^{ψ}, H)$ .

(2) $({(\underline{Ω \cap Γ})}^{ψ}, H) \supseteq ({\underline{Ω}}^{ψ}, H) \cup ({\underline{Γ}}^{ψ}, H)$ .

The following example shows that the equality does not hold in above Theorem 3.

Example 1. Let K₁ = {⊥ ₁, b, ⊤ ₁} and K₂ = {⊥ ₂, u, v, ⊤ ₂} be two complete lattices shown in Fig. 1 and Fig. 2 with the binary operations shown in Tables 1 and 2, respectively. Then K₁ and K₂ are quantales. Let H = {u_1,u₂} andΩ : H ⟶ P (K₁ × K₂) and Γ : H ⟶ P (K₁ × K₂) be defined by: $\begin{matrix} \begin{matrix} Ω (u_{1}) = {(⊥_{1}, ⊥_{2}), (b, u), (⊤_{1}, v), (b, ⊤_{2}), \\ (⊥_{1}, ⊤_{2}), (⊥_{1}, v), (b, v), (⊥_{1}, u),} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} Γ (u_{1}) = {(⊥_{1}, ⊥_{2}), (b, v), (⊤_{1}, u), (⊤_{1}, ⊤_{2}), \\ (⊥_{1}, v), (⊥_{1}, u), (b, u), (b, ⊥_{2}), (b, ⊤_{2})} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} (Ω \cap Γ) (u_{1}) = {(⊥_{1}, ⊥_{2}), (b, u), (b, ⊤_{2}), \\ (⊥_{1}, v), (b, v), (⊥_{1}, u)} \end{matrix} \end{matrix}$

Fig. 1

Illustration of K₁.

Fig. 2

Illustration of K₂.

Table 1

Binary operation cirle-1 subject of K₁

⊗₁	⊥₁	b	⊤₁
⊥₁	⊥₁	⊥₁	⊥₁
b	⊥₁	b	b
⊤₁	⊥₁	b	⊤₁

Table 2

Binary operation cirle-2 subject of K₂

⊗₂	⊥₂	u	v	⊤₂
⊥₂	⊥₂	u	v	⊤₂
u	⊥₂	u	v	⊤₂
v	⊥₂	u	v	⊤₂
⊤₂	⊥₂	u	v	⊤₂

The aftersets with respect to Ω (u₁) and Γ (u₁) are as follows,

$\begin{matrix} ⊥_{1} Ω (u_{1}) & = {⊥_{2}, ⊤_{2}, v, u} \underline{Ω} (u_{1}) & = {u, v, ⊤_{2}} ⊤_{1} Ω (u_{1}) & = {v} \end{matrix}$ $\begin{matrix} ⊥_{1} Ω (u_{1}) & = {⊥_{2}, v, u} b Ω (u_{1}) & = {⊥_{2}, u, ⊤_{2}, v} ⊤_{1} Ω (u_{1}) & = {⊤_{2}, u} \end{matrix}$ and

$\begin{matrix} ⊥_{1} (Ω \cap Γ) (u_{1}) & = {⊥_{2}, v, u} b (Ω \cap Γ) (u_{1}) & = {⊤_{2}, v, u} ⊤_{1} (Ω \cap Γ) (u_{1}) & = \emptyset \end{matrix}$

Define ψ₁ : K₂ ⟶ [0, 1] by, $ψ_{1} = \frac{0.6}{⊥_{2}} + \frac{0.4}{u} + \frac{0.1}{v} + \frac{0.3}{⊤_{2}}$

Then ψ₁ is a fsst of K₂, but ${\bar{Ω}}^{ψ_{1}} (u_{1}) = \frac{0.6}{⊥_{1}} + \frac{0.4}{b} + \frac{0.1}{⊤_{1}} {\bar{Γ}}^{ψ_{1}} (u_{1}) = \frac{0.6}{⊥_{1}} + \frac{0.6}{b} + \frac{0.4}{⊤_{1}} {\bar{(Ω \cap Γ)}}^{ψ_{1}} (u_{1}) = \frac{0.6}{⊥_{1}} + \frac{0, 4}{b} + \frac{0}{⊤_{1}}$

This shows that $\begin{matrix} {\bar{Ω}}^{ψ_{1}} (u_{1}) \cap {\bar{Γ}}^{ψ_{1}} (u_{1}) \neq & {\bar{(Ω \cap Γ)}}^{ψ_{1}} (u_{1}) \end{matrix}$

Now, define ψ₂ : K₂ ⟶ [0, 1] by $ψ_{2} = \frac{0.2}{⊥_{2}} + \frac{0.7}{u} + \frac{1}{v} + \frac{0}{⊤_{2}}$

Then ψ₂ is a fsst of K₂. However, $\begin{matrix} {\underline{Ω}}^{^{ψ_{2}}} (u_{1}) & = & \frac{0.7}{⊥_{1}} + \frac{0.7}{b} + \frac{1}{⊤_{1}} \\ {\underline{Γ}}^{ψ_{2}} (u_{1}) & = & \frac{0.7}{⊥_{1}} + \frac{0.7}{b} + \frac{0.7}{⊤_{1}} \\ {\underline{(Ω \cap Γ)}}^{ψ_{2}} (u_{1}) & = & \frac{0.7}{⊥_{1}} + \frac{0.7}{b} + \frac{0}{⊤_{1}} \end{matrix}$

This shows that ${\underline{Ω}}^{^{ψ_{2}}} (u_{1}) \cup {\underline{Γ}}^{ψ_{2}} (u_{1}) \neq {\underline{(Ω \cap Γ)}}^{ψ_{2}} (u_{1})$

The following result follows from parts (7) and (8) of Theorem 2.

Theorem 4. Let (Ω, H) and (Γ, H) be soft binary relations from non-empty sets K₁ to K₂. If η is a fuzzy subset of K₁, then

(1) $(^{η} (\bar{Ω \cap Γ}), H) \subseteq (^{η} \bar{Ω}, H) \cap (^{η} \bar{Γ}, H)$ ;

(2) $(^{η} (\underline{Ω \cap Γ}), H) \supseteq (^{η} \underline{Ω}, H) \cup (^{η} \underline{Γ}, H)$ .

The following example shows that the equality does not hold in above result.

Example 2. Consider the quantales in Example 1. Let H = {u_1,u₂} , Ω : H ⟶ P (K₁ × K₂) and Γ : H ⟶ P (K₁ × K₂) be soft binary relations defined by

Ω (u₁) = {(⊥ ₁, ⊥ ₂) , (b, u) , (⊤ ₁, v) , (b, ⊤ ₂) , (⊥ ₁, ⊤ ₂) , (⊥ ₁, v) , (b, v) , (⊥ ₁, u) , }

Γ (u₁) = {(⊥ ₁, ⊥ ₂) , (b, v) , (⊤ ₁, u) , (⊤ ₁, ⊤ ₂) , (⊥ ₁, v) , (⊥ ₁, u) , (b, u) , (b, ⊥ ₂) , (b, ⊤ ₂)}

(Ω ∩ Γ) (u₁) = {(⊥ ₁, ⊥ ₂) , (b, u) , (b, ⊤ ₂) , (⊥ ₁, v) , (b, v) , (⊥ ₁, u)}

The foresets with respect to Ω (u₁) and Γ (u₁) are as follows;

$\begin{matrix} Ω (u_{1}) ⊥_{2} & = {⊥_{1}} Ω (u_{1}) u & = {⊥_{1}, b} Ω (u_{1}) v & = {⊥_{1}, b, ⊤_{1}} Ω (u_{1}) ⊤_{2} & = {⊥_{1}, b} \end{matrix}$ $\begin{matrix} Γ (u_{1}) ⊥_{2} & = {⊥_{1}, b} Γ (u_{1}) u & = {⊥_{1}, b, ⊤_{1}} Γ (u_{1}) v & = {⊥_{1}, b} Γ (u_{1}) ⊤_{2} & = {b, ⊤_{1}} \end{matrix}$ and

$\begin{matrix} (Ω \cap Γ) (u_{1}) ⊥_{2} & = {⊥_{1}} (Ω \cap Γ) (u_{1}) u & = {⊥_{1}, b} (Ω \cap Γ) (u_{1}) v & = {⊥_{1}, b} (Ω \cap Γ) (u_{1}) ⊤_{2} & = {b} \end{matrix}$

Define η₁ : K₁ ⟶ [0, 1] by $η_{1} = \frac{0.5}{⊥_{1}} + \frac{0.2}{b} + \frac{1}{⊤_{1}}$

Then η₁ is a fsst of K₁. However, $\begin{array}{l} {\bar{Ω}}^{ψ_{1}} (u_{1}) = \frac{0.6}{⊥_{1}} + \frac{0.4}{b} + \frac{0.1}{⊤_{1}} \\ {\bar{Γ}}^{ψ_{1}} (u_{1}) = \frac{0.6}{⊥_{1}} + \frac{0.6}{b} + \frac{0.4}{⊤_{1}} \\ {\bar{(Ω \cap Γ)}}^{ψ_{1}} (u_{1}) = \frac{0.6}{⊥_{1}} + \frac{0, 4}{b} + \frac{0}{⊤_{1}} \end{array}$

This shows that $\begin{matrix} {\bar{Ω}}^{ψ_{1}} (u_{1}) \cap {\bar{Γ}}^{ψ_{1}} (u_{1}) \neq & {\bar{(Ω \cap Γ)}}^{ψ_{1}} (u_{1}) \end{matrix}$

Now define η₂ : K₁ ⟶ [0, 1] by $η_{2} = \frac{0.5}{⊥_{1}} + \frac{0.4}{b} + \frac{0.6}{⊤_{1}}$

Then η₂ is a fsst of K₁ but,

$\begin{array}{l} ^{η_{2}} \underline{Ω} (u_{1}) = \frac{0.5}{⊥_{2}} + \frac{0.5}{u %} + \frac{0.6}{v} + \frac{0.5}{⊤_{2}} \\ ^{η_{2}} \underline{Π} (u_{1}) = \frac{0.5}{⊥_{2}} + \frac{0.6}{u} + \frac{0.5}{v} + \frac{0.6}{⊤_{2}} \\ ^{η_{2}} \underline{(Ω \cap Π)} (u_{1}) = \frac{0.5}{⊥_{2}} + \frac{0.5}{u} + \frac{0.5}{v} + \frac{0.4}{⊤_{2}} \end{array}$

This show that, $^{η_{2}} \underline{Ω} (u_{1}) \cup^{η_{2}} \underline{Π} (u_{1}) \neq^{η_{2}} \underline{(Ω \cap Π)} (u_{1})$

4 Approximation of fuzzy substructures in quantale

In this section, we are considering soft compatible relations by taking two different quantales. By taking the fuzzy substructures of quantale K₂, their lower and upper approximations by aftersets, the resulting fuzzy substructures of K₁ are obtained. Moreover, for fuzzy substructures of K₁, their lower and upper approximations by foresets, the resulting fuzzy substructures of K₂ are obtained.

In the next, the following shortened form SBR, SCR, SCTR and FSSQ will be used for "soft binary relation from a quantale K₁ to a quantale K₂ ", "soft compatible relation from a quantale K₁ to a quantale K₂ ", "soft complete relation from a quantale K₁ to a quantale K₂ " and fuzzy soft subquantale, respectively.

Definition 15. Let (Ω, H) be a SBR and ψ be a non-empty fsst of K₂. Then ψ is said to be generalized upper (lower) rough fuzzy soft subquantale (ideal) ofK₁ with respect to the aftersets if the upper approximation $({\bar{Ω}}^{ψ}, H),$ $[({\underline{Ω}}^{ψ}, H)]$ is a fuzzy subquantale (fuzzy ideal) of K₁ for any nonempty fsst ψ of K₂. Likewise, let ψ be a non-empty fsst of K₁ and (Ω, H) be a SBR. Then ψ is said to be generalized upper (lower) fuzzy soft subquantale (ideal) ofK₂ with respect to the foresets, if the upper approximation $(^{ψ} \bar{Ω}, H),$ $[(^{ψ} \underline{Ω}, H)]$ is a fuzzy subquantale (fuzzy ideal) of K₂ for any nonempty fsst ψ of K₁.

Theorem 5. Let (Ω, H) be a SCR and ψ be a fsq of K₂. Then ψ is a generalized upper rough FSSQ of K₁ with respect to aftersets.

Proof. As ψ is a fsq of K₂, so we have ψ (∨ _i∈Ip_i) ≥ $\underset{i \in I}{Min}$ ψ (p_i) and ψ (p ⊗ ₂t) ≥ Min {ψ (p) , ψ (t)} ∀ p, t, p_i ∈ K₂. Since (Ω, H) is a SCR, so we have p₁Ω (u) ∨ p₂Ω (u) ∨ . . . ∨ p_iΩ (u) ⊆ (∨ _i∈Ip_i) Ω (u) for all u ∈ H and p_i ∈ K₁ (i ∈ I).

Let a_i ∈ K₁ for some i ∈ I. Then

$\begin{matrix} \underset{i \in I}{Min} {\bar{Ω}}^{ψ} (u) (a_{i}) = Min {{\bar{Ω}}^{ψ} (u) (a_{1}), {\bar{Ω}}^{ψ} (u) (a_{2}), . . ., {\bar{Ω}}^{ψ} (u) (a_{i})} & = Min {(\underset{a_{1} \in t_{1} Ω (u)}{\lor} ψ (t_{1})), (\underset{a_{2} \in t_{2} Ω (u)}{\lor} ψ (t_{2})), . . ., \\ (\underset{a_{i} \in t_{i} Ω (u)}{\lor} ψ (t_{i}))} & = \underset{t_{1} \in a_{1} Ω (u), t_{2} \in a_{2} Ω (u), . . ., t_{i} \in a_{i} Ω (u)}{\lor} Min [ψ (a_{1}), ψ (a_{2}), . . ., ψ (a_{i}] & = \underset{t_{1} \lor t_{2} \lor, . . ., \lor t_{i} \in a_{1} Ω (u) \lor a_{2} Ω (u) \lor, . . ., \lor \in a_{i} Ω (u)}{\lor} \underset{i \in I}{Min} [ψ (a_{i})] & \leq \underset{\lor_{i \in I} t_{i} \in (a_{1} \lor a_{2} \lor, . . ., \lor a_{i}) Ω (u)}{\lor} \underset{i \in I}{Min} [ψ (a_{i})] \\ = \underset{\lor_{i \in I} t_{i} \in \lor_{i \in I} a_{i} Ω (u)}{\lor} \underset{i \in I}{Min} [ψ (a_{i})] & \leq \underset{\lor_{i \in I} t_{i} \in \lor_{i \in I} a_{i} Ω (u)}{\lor} ψ (\lor_{i \in I} a_{i}) \\ = \underset{c \in \lor_{i \in I} a_{i} Ω (u)}{\lor} ψ (c) & = {\bar{Ω}}^{ψ} (u) (\lor_{i \in I} a_{i}) . \end{matrix}$

Hence ${\bar{Ω}}^{ψ} (u) (\lor_{i \in I} a_{i}) \geq$ $\underset{i \in I}{Min} {\bar{Ω}}^{ψ} (u) (a_{i})$ ∀ a_i ∈ K₁ and for all u ∈ H.

As (Ω, H) SCR, we have aΩ (u) ⊗ bΩ (u) ⊆ (a ⊗ b) Ω (u) for all a, b ∈ K₁ and for u ∈ H.

Consider a, b ∈ K₁ and

$\begin{matrix} Min {{\bar{Ω}}^{ψ} (u) (a), {\bar{Ω}}^{ψ} (u) (b)} = Min {\underset{y \in a Ω (u)}{\lor} ψ (y), & \underset{t \in b Ω (u)}{\lor} ψ (t)} & = \underset{y \in a Ω (u), t \in b Ω (u)}{\lor} Min {ψ (y), ψ (t)} & = \underset{y \otimes_{2} t \in a Ω (u) \otimes_{2} b Ω (u)}{\lor} Min {ψ (y), ψ (t)} & \leq \underset{y \otimes_{2} t \in a Ω (u) \otimes_{2} b Ω (u)}{\lor} ψ (y \otimes_{2} t) & \leq \underset{y \otimes_{2} t \in (a \otimes_{1} b) Ω (u)}{\lor} ψ (y \otimes_{2} t) & = \underset{c \in (a \otimes_{1} b) Ω (u)}{\lor} ψ (c) & = {\bar{Ω}}^{ψ} (ψ) (a \otimes_{1} b) . \end{matrix}$

Hence ${\bar{Ω}}^{ψ} (B) (a \otimes_{1} b) \geq Min {{\bar{Ω}}^{ψ} (u) (a), {\bar{Ω}}^{ψ} (u) (b)}$ ∀ a, b ∈ K₁. Thus ${\bar{Ω}}^{ψ} (u)$ is a FSSQ of K₁. □

The next example shows that the converse of Theorem 5 do not hold in general.

Example 3. Let K₁ = {⊥ ₁, c, d, ⊤ ₁} and K₂ = {⊥ ₂, p, q, r, ⊤ ₂} be two complete lattices shown in Fig. 3 and Fig. 4 with the binary operations shown in Tables 3 and 4, respectively. Then K₁ and K₂ are quantales. Let H = {u₁, u₂} and Ω : H ⟶ P (K₁ × K₂) be a SCR. $\begin{array}{l} Ω (u_{1}) = {\begin{matrix} (⊥_{1}, ⊥_{2}), (c, p), (d, q), (⊥_{1}, p), (⊥_{1}, q), (⊥_{1}, r), \\ (⊤_{1}, ⊤_{2}), (⊥_{1}, ⊤_{2}), (d, ⊤_{2}) (c, q), (⊤_{1}, ⊥_{2}), (d, ⊥_{2}), (c, ⊥_{2}) \end{matrix}} \\ Ω (u_{2}) = {(⊤_{1}, ⊥_{2}), (c, p), (d, q), (⊤_{1}, r), \\ (d, r), (c, r), (⊥_{1}, p), (⊥_{1}, q), (⊥_{1}, r)} . \end{array}$ $\begin{matrix} Ω (u_{2}) = {(⊤_{1}, ⊥_{2}), (c, p), (d, q), (⊤_{1}, r), \\ (d, r), (c, r), (⊥_{1}, p), (⊥_{1}, q), (⊥_{1}, r)} . \end{matrix}$

Fig. 3

Illustration of K₁.

Fig. 4

Illustration of K₂.

Table 3

Binary operation cirle-1 subject of K₁

⊗₁	⊥₁	c	d	⊤₁
⊥₁	⊥₁	⊥₁	⊥₁	⊥₁
c	⊥₁	c	⊥₁	c
d	⊥₁	⊥₁	d	d
⊤₁	⊥₁	c	d	⊤₁

Table 4

Binary operation cirle-2 subject of K₂

⊗₂	⊥₂	p	q	r	⊤₂
⊥₂	⊥₂	p	q	r	⊤₂
l	⊥₂	p	q	r	⊤₂
m	⊥₂	p	q	r	⊤₂
n	⊥₂	p	q	r	⊤₂
⊤₂	⊥₂	p	q	r	⊤₂

Then (Ω, H) is a SCR. Aftersets with respect to Ω (u₁) and Ω (u₂) are as follows; $\begin{matrix} ⊥_{1} Ω (u_{1}) = {⊥_{2}, p, q, r, ⊤_{2}}, c Ω (u_{1}) = {p, q, ⊥_{2}}, d Ω (u_{1}) = {q, ⊤_{2}, ⊥_{2}}, ⊤_{1} Ω (u_{1}) = {⊤_{2}, ⊥_{2}} . \end{matrix}$ $\begin{matrix} ⊥_{1} Ω (u_{2}) = {p, q, r}, c Ω (u_{2}) = {p, r}, \\ d Ω (u_{2}) = {q, r}, ⊤_{1} Ω (u_{2}) = {r, ⊥_{2}} . \end{matrix}$

Foresets with respect Ω (u₁) and Ω (u₂) are as follows, respectively; $\begin{matrix} Ω (u_{1}) ⊥_{2} = {c, d, ⊤_{1}}, Ω (u_{1}) p = {⊥_{1}, c}, \\ Ω (u_{1}) q = {c, d, ⊥_{1}}, Ω (u_{1}) r = {⊥_{1}}, \\ Ω (u_{1}) ⊤_{2} = {⊤_{1}, d, ⊥_{1}} \end{matrix}$ $\begin{matrix} Ω (u_{2}) ⊥_{2} = {⊤_{1}}, Ω (u_{2}) p = {⊥_{1}, c}, \\ Ω (u_{2}) q = {d, ⊥_{1}}, Ω (u_{2}) r = {⊤_{1}, d, c, \\ ⊥_{1}}, Ω (u_{2}) ⊤_{2} = ϕ \end{matrix}$

Let ψ : K₂ ⟶ [0, 1] be defined by, $ψ = \frac{1}{⊥_{2}} + \frac{0.5}{p} + \frac{0.4}{q} + \frac{0.2}{r} + \frac{0.8}{⊤_{2}}$

Then ψ is not a fsq of K₂. However, $\begin{matrix} {\bar{Ω}}^{ψ} (u_{1}) & = & \frac{1}{⊥_{1}} + \frac{1}{c} + \frac{1}{d} + \frac{1}{⊤_{1}} \\ {\bar{Ω}}^{ψ} (u_{2}) & = & \frac{0.5}{⊥_{1}} + \frac{0.5}{c} + \frac{0.4}{d} + \frac{1}{⊤_{1}} \end{matrix}$

The shows that ${\bar{Ω}}^{ψ} (u_{1})$ and ${\bar{Ω}}^{ψ} (u_{2})$ are fsq of K₁ but ψ is not a fsq of K₂. Hence ψ is generalized upper rough FSSQ of K₁with respect to aftersets.

Let η : K₁ ⟶ [0, 1] define by, $η = \frac{1}{⊥_{1}} + \frac{0.7}{c} + \frac{0.7}{d} + \frac{0.5}{⊤_{1}}$

Then η is not a fsq of K₁. However, $\begin{array}{l} ^{η} \bar{Ω} (u_{1}) = \frac{0.7}{⊥_{2}} + \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{1}{⊤_{2}} \\ ^{η} \bar{Ω} (u_{2}) = \frac{0.5}{⊥_{2}} + \frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{0}{⊤_{2}} \end{array}$

Thus above shows that $^{η} \bar{Ω} (u_{1})$ and $^{η} \bar{Ω} (u_{2})$ are fsq of K₂ . Hence η is generalized upper rough FSSQ of K₂ with respect to foresets.

Theorem 6. Let (Ω, H) be a SCTR and ψ be a fsq of K₂. Then ψ is a generalized lower rough FSSQ of K₁ with respect to aftersets.

Proof. Let ψ be afsq of K₂ and (Ω, H) SCTR. Then $ψ (\lor_{i \in I} c_{i}) \geq \underset{i \in I}{Min} ψ (c_{i})$ and ψ (p ⊗ ₂t) ≥ Min {ψ (p) , ψ (t)} ∀ p, t, c_i ∈ K₂ and m₁Ω (u) ∨ m₂Ω (u) ∨ , . . . , ∨ m_iΩ (u) = (∨ _i∈Im_i) Ω (u) for all m_i ∈ K₁ and u ∈ H.

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (\lor_{i \in I} m_{i}) = & \underset{c \in (\lor_{i \in I} m_{i}) Ω (u)}{\land} ψ (c) = & \underset{c \in m_{1} Ω (u) \lor m_{2} Ω (u) \lor, . . ., \lor m_{i} Ω (u)}{\land} ψ (c) . \end{matrix}$

Since c ∈ m₁Ω (u) ∨ m₂Ω (u) ∨ , . . . , ∨ m_iΩ (u), there exist c₁ ∈ m₁Ω (u), c₂ ∈ m₂Ω (u),..., c_i ∈ m_iΩ (u) be such that c = ∨ _i∈Ic_i.

Hence,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (\lor_{i \in I} m_{i}) = \underset{\lor_{i \in I} c_{i} \in m_{1} Ω (u) \lor m_{2} Ω (u) \lor, . . ., \lor m_{i} Ω (u)}{\land} ψ (\lor_{i \in I} c_{i}) \\ \geq \underset{\lor_{i \in I} c_{i} \in m_{1} Ω (u) \lor m_{2} Ω (u) \lor, . . ., \lor m_{i} Ω (u)}{\land} \underset{i \in I}{Min} ψ (c_{i}) & = \underset{c_{1} \in m_{1} Ω (u), c_{2} \in m_{2} Ω (u), . . ., c_{i} \in m_{i} Ω (u)}{\land} Min [ψ (c_{1}), ψ (c_{2}), . . ., ψ (c_{i}] \\ = Min {(\underset{c_{1} \in m_{1} Ω (u)}{\land} ψ (c_{1})), (\underset{c_{2} \in m_{2} Ω (u)}{\land} ψ (c_{2})), . . ., & (\underset{c_{i} \in m_{i} Ω (u)}{\land} ψ (c_{i}))} & = Min {{\underline{Ω}}^{ψ} (u) (m_{1}), {\underline{Ω}}^{ψ} (u) (m_{2}), . . ., {\underline{Ω}}^{ψ} (u) (m_{i})} & = \underset{i \in I}{Min} {\underline{Ω}}^{ψ} (u) (m_{i}) . \end{matrix}$

Thus we have, ${\underline{Ω}}^{ψ} (u) (\lor_{i \in I} m_{i}) \geq \underset{i \in I}{Min} {\underline{Ω}}^{ψ} (u) (m_{i})$ ∀ m_i ∈ K₁.

Since (Ω, H) SCTR, we have pΩ (u) ⊗ ₂qΩ (u) = (p ⊗ ₁q) Ω (u) for all p, q ∈ K₁ and for all u ∈ H.

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) = & \underset{c \in (p \otimes_{1} q) Ω (u)}{\land} ψ (c) {\overset{\land}{¯}}_{c \in p Ω (u) \otimes_{2} q Ω (u)} ψ (c) \end{matrix}$

As c ∈ pΩ (u) ⊗ ₂qΩ (u), we obtain x ∈ pΩ (u) and t ∈ qΩ (u) such that c = x ⊗ ₂t.

Hence,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) = \underset{x \otimes_{2} t \in p Ω (u) \otimes_{2} q Ω (u)}{\land} ψ (x \otimes_{2} t) \\ \geq \underset{x \otimes_{2} t \in p Ω (u) \otimes_{2} q Ω (u)}{\land} Min {ψ (x), ψ (t)} & = \underset{x \in p Ω (u), t \in q Ω (u)}{\land} Min {ψ (x), ψ (t)} & = Min {\underset{x \in p Ω (u)}{\land} ψ (x), \underset{t \in q Ω (u)}{\land} ψ (t)} & = Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q)} . \end{matrix}$

Hence ${\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) \geq Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q)}$ ∀ p, q ∈ K₁. Thus, ${\underline{Ω}}^{ψ} (u)$ is a FSSQ of K₁ ∀ u ∈ H. □

Moreover, the next example shows that the converse of Theorem 6 do not hold in general.

Example 4. Consider the quantales in Example 3. Let H = {u₁, u₂} and Ω : H ⟶ P (K₁ × K₂) be defined by, $\begin{matrix} Ω (u_{1}) = & {\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, p), (⊥_{1}, q), (⊥_{1}, r), (c, ⊥_{2}), (c, p), (c, q), (c, r) \\ (d, ⊥_{2}), (d, p), (d, q), (d, r), (⊤_{1}, ⊥_{2}), (⊤_{1}, p), (⊤_{1}, q), (⊤_{1}, r) \end{matrix}} \end{matrix}$ $\begin{matrix} Ω (u_{2}) = & & {\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, q), (⊥_{1}, r), (c, ⊥_{2}), (c, q), (c, r) \\ (d, ⊥_{2}), (d, q), (d, r), (⊤_{1}, ⊥_{2}), (⊤_{1}, q), (⊤_{1}, r) \end{matrix}} \end{matrix}$ Now, aftersets with respect to Ω (u₁) and Ω (u₂) are given below;

$\begin{matrix} ⊥_{1} Ω (u_{1}) = {⊥_{2}, p, q, r} & ⊥_{1} Ω (u_{2}) = {⊥_{2}, q, r} c Ω (u_{1}) = {⊥_{2}, p, q, r} & c Ω (u_{2}) = {⊥_{2}, q, r} \\ d Ω (u_{1}) = {⊥_{2}, p, q, r} & d Ω (u_{2}) = {⊥_{2}, q, r} \\ ⊤_{1} Ω (u_{1}) = {⊥_{2}, p, q, r} & ⊤_{1} Ω (u_{2}) = {⊥_{2}, q, r} \end{matrix}$

Then (Ω, H) is a SCTR with respect to aftersets. Define ψ : K₂ ⟶ [0, 1] by, $ψ = \frac{1}{⊥_{2}} + \frac{0.5}{p} + \frac{0.4}{q} + \frac{0.2}{r} + \frac{0.8}{⊤_{2}}$ Then ψ is not a fsq of K₂. But $\begin{matrix} {\underline{Ω}}^{ψ} (u_{1}) & = & \frac{0.2}{⊥_{1}} + \frac{0.2}{c} + \frac{0.2}{d} + \frac{0.2}{⊤_{1}} \\ {\underline{Ω}}^{ψ} (u_{2}) & = & \frac{0.2}{⊥_{1}} + \frac{0.2}{c} + \frac{0.2}{d} + \frac{0.2}{⊤_{1}} \end{matrix}$ This shows that ${\underline{Ω}}^{ψ} (u_{1})$ and ${\underline{Ω}}^{ψ} (u_{2})$ are fsq of K₁ . Hence ψ is generalized lower rough FSSQ of K₁with respect to aftersets.

Now define Ω : H ⟶ P (K₁ × K₂). Then $\begin{matrix} Ω (u_{1}) = & & {\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, p), (⊥_{1}, q), (⊥_{1}, r), (⊥_{1}, ⊤_{2}) \\ (c, ⊥_{2}) (c, p), (c, q), (c, r), (c, ⊤_{2}) \end{matrix}} \end{matrix}$ $\begin{matrix} Ω (u_{2}) = \\ {\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, p), (⊥_{1}, q), (⊥_{1}, r), (⊥_{1}, ⊤_{2}) \\ (d, ⊥_{2}) (d, p), (d, q), (d, r), (d, ⊤_{2}) \end{matrix}} \end{matrix}$

Now foresets in terms of Ω (u₁) and Ω (u₂) are as follows; $\begin{matrix} Ω (u_{1}) ⊥_{2} & = {⊥_{1}, c} Ω (u_{1}) p & = {⊥_{1}, c} Ω (u_{1}) q & = {⊥_{1}, c} Ω (u_{1}) r & = {⊥_{1}, c} Ω (u_{1}) ⊤_{2} & = {⊥_{1}, c} \end{matrix}$ $\begin{matrix} Ω (u_{2}) ⊥_{2} & = {⊥_{1}, d} Ω (u_{2}) p & = {⊥_{1}, d} Ω (u_{2}) q & = {⊥_{1}, d} Ω (u_{1}) r & = {⊥_{1}, d} Ω (u_{1}) ⊤_{2} & = {⊥_{1}, d} \end{matrix}$

Then (Ω, H) is a SCTR with respect to foresets. Define η : K₁ ⟶ [0, 1] by, $η = \frac{1}{⊥_{1}} + \frac{0.6}{c} + \frac{0.7}{d} + \frac{0.5}{⊤_{1}}$ Then η is not a fsq of K₁. The lower approximations with respect to foresets are as follows; $\begin{array}{l} ^{η} \underline{Ω} (u_{1}) = \frac{0.6}{⊥_{2}} + \frac{0.6}{p} + \frac{0.6}{q} + \frac{0.6}{r} + \frac{0.6}{⊤_{2}} \\ ^{η} \underline{Ω} (u_{2}) = \frac{0.7}{⊥_{2}} + \frac{0.7}{p} + \frac{0.7}{q} + \frac{0.7}{r} + \frac{0.7}{⊤_{2}} \end{array}$

Thus $^{η} \underline{Ω} (u_{1})$ and $^{η} \underline{Ω} (u_{2})$ are fsq of K₂ . Hence η is a generalized lower FSSQ of K₂ with respect to foresets.

Remark 1. Theorem 5 and Theorem 6 can be proved for foresets by considering fssts of K₁.

Proposition 1. Let (Ω, H) be a SCR. Let ψ be a fsst of K₂. Then for each α ∈ [0, 1], the following hold:

(1) $({\underline{Ω}}^{ψ} (u))_{α} = {\underline{Ω}}^{ψ_{α}} (u);$

(2) $({\bar{Ω}}^{ψ} (u))_{α} = {\bar{Ω}}^{ψ_{α}} (u);$

(3) $({\underline{Ω}}^{ψ} (u))_{α^{+}} = {\underline{Ω}}^{ψ_{α^{+}}} (u);$

(4) $({\bar{Ω}}^{ψ} (u))_{α^{+}} = {\bar{Ω}}^{ψ_{α^{+}}} (u)$ .

Proof. (1) Let $z \in ({\underline{Ω}}^{ψ} (u))_{α} \Leftrightarrow {\underline{Ω}}^{ψ} (u) (z) \geq α \Leftrightarrow \underset{a \in z Ω (u)}{⋀} ψ (a) \geq α$

⇔ψ (a) ≥ α $for all a∈ zΩ (u); ⇔ zΩ (u)⊆ ψα ⇔ z∈ Ω _ψα (u) . (2) Let z∈ (Ω ¯ψ (u))α ⇔ Ω ¯ψ (u) (z)≥ α ⇔ ⋁ a∈ zΩ (u) ψ (a)≥ α ⇔ ψ (a)≥ α$ for some a∈ zΩ (u) ;

⇔ zΩ (u)∩ ψ_α ≠ ∅ ⇔ $z \in {\bar{Ω}}^{ψ_{α}} (u)$ .

proof of (3) and (4) are similar to the proof of (1) and (2). □

Remark 2. The Proposition 1 also holds for foresets.

Theorem 7. Let ψ be a fsq of K₂ and (Ω, H) be a SCTR. Then ${\underline{Ω}}^{ψ} (u)$ $[{\bar{Ω}}^{ψ} (u)]$ is a FSSQ of K₁ with respect to afterset if and only if for each α ∈ [0, 1] , ${\underline{Ω}}^{ψ_{α}} (u)$ $[{\bar{Ω}}^{ψ_{α}} (u)$ where ψ_α ≠ ∅] is a subquantale of K₁ for all u ∈ H.

Proof. Let ${\underline{Ω}}^{ψ} (u)$ be a FSTQ of K₁ and $p_{i} \in {\underline{Ω}}^{ψ_{α}} (u)$ for some i ∈ I. Then ${\underline{Ω}}^{ψ} (u) (p_{i}) \geq α$ for all i ∈ I. But ${\underline{Ω}}^{ψ} (u)$ is a FSTQ, so ${\underline{Ω}}^{ψ} (u) (\lor_{i \in I} p_{i}) = \underset{i \in I}{Min} {{\underline{Ω}}^{ψ} (u) (p_{i})}$ ≥α. Hence $\lor_{i \in I} p_{i} \in {\underline{Ω}}^{ψ_{α}} (u)$ . Let p, q ∈ ${\underline{Ω}}^{ψ_{α}} (u)$ . Then ${\underline{Ω}}^{ψ} (u) (p) \geq α$ and ${\underline{Ω}}^{ψ} (u) (q) \geq α$ . But since ${\underline{Ω}}^{ψ} (u)$ be a FSTQ, we have ${\underline{Ω}}^{ψ} (u) (q \otimes_{1} p) \geq Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q) \geq α$ , we get $q \otimes_{1} p \in {\underline{Ω}}^{ψ_{α}} (u)$ . Hence, ${\underline{Ω}}^{ψ_{α}} (u)$ is subquantale of K₁.

Conversely, let ${\underline{Ω}}^{ψ_{α}} (u)$ be a subquantale of K₁ and $m_{i} \in {\underline{Ω}}^{ψ_{α}} (u)$ for all u ∈ H.

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (\lor_{i \in I} m_{i}) = \underset{c \in (\lor_{i \in I} m_{i}) Ω (u)}{⋀} ψ (c) \\ quad = \underset{c \in m_{1} Ω (u) \lor m_{2} Ω (u) \lor, . . . ., \lor m_{i} Ω (u)}{⋀} ψ (c) \end{matrix}$ Since (Ω, A) be a SCTR and c ∈ m₁Ω (u) ∨ m₂Ω (u) ∨ , . . . . , ∨ m_iΩ (u) , then there exist c₁ ∈ m₁Ω (u) and c₂ ∈ m₂Ω (u) , . . . . , c_i ∈ m_iΩ (u) such that c = ∨ _i∈Ic_i. Hence

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (\lor_{i \in I} m_{i}) = \underset{\lor_{i \in I} c_{i} \in m_{1} Ω (u) \lor m_{2} Ω (u) \lor, . . . ., \lor m_{i} Ω (u)}{⋀} ψ (\lor_{i \in I} c_{i}) \\ \geq \underset{\lor_{i \in I} c_{i} \in m_{1} Ω (u) \lor m_{2} Ω (u) \lor, . . . ., \lor m_{i} Ω (u)}{⋀} \underset{i \in I}{Min} {ψ (c_{i})} \end{matrix}$ $\begin{matrix} = \underset{c_{1} \in m_{1} Ω (u), c_{2} \in m_{2} Ω (u), . . . ., c_{i} \in m_{i} Ω (u)}{⋀} Min {ψ (c_{1}), ψ (c_{2}), . . . . (c_{i})} \\ = Min {(\underset{c_{1} \in m_{1} Ω (u)}{⋀} ψ (c_{1})), (\underset{c_{2} \in m_{2} Ω (u)}{⋀} ψ (c_{2})) & , . . ., (\underset{c_{i} \in m_{i} Ω (u)}{⋀} ψ (c_{i}))} & = Min {{\underline{Ω}}^{ψ} (u) (m_{1}), {\underline{Ω}}^{ψ} (u) (m_{2}), . . . . ., {\underline{Ω}}^{ψ} (u) (m_{i})} & = \underset{i \in I}{Min} {\underline{Ω}}^{ψ} (u) (m_{i}) \end{matrix}$

So, ${\underline{Ω}}^{ψ} (u) (\lor_{i \in I} m_{i}) \geq$ $\underset{i \in I}{Min} {\underline{Ω}}^{ψ} (u) (m_{i})$ ∀ m_i ∈ K₁. Let $p, q \in {\underline{Ω}}^{ψ_{α}} (u)$ ∀ p, q ∈ K₁. Then $p \otimes_{1} q \in {\underline{Ω}}^{ψ_{α}} (u)$ . Hence, ${\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) \geq α$ . If either ${\underline{Ω}}^{ψ} (u) (p) \geq α$ or ${\underline{Ω}}^{ψ} (u) (q) \geq α$ . In both the cases, $Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q)}$ ≥α. Let $Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q)}$ = α. So ${\underline{Ω}}^{ψ} (u) (p \otimes_{1} q)$ ≥ $Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q)}$ . Hence, ${\underline{Ω}}^{ψ} (u)$ is a FSSQ of K₁. □

Theorem 8. Let (Ω, A) be a SCTR and ψ be a fid of K₂. Then ψ is a generalized lower rough FSTI of K₁ with respect to aftersets.

Proof. Let ψ be a fid of K₂ and (Ω, A) be a SCTR. Then ψ (p ∨ q) = Min {ψ (p) , ψ (q)} and ψ (p ⊗ ₂q) ≥ Max {ψ (p) , ψ (q)} ∀ p, q ∈ K₂. Also, (z₁ ∨ z₂) Ω (u) = z₁Ω (u) ∨ z₂Ω (u) ∀ z₁, z₂ ∈ K₁.

Therefore,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z_{1} \lor z_{2}) = & \underset{c \in (z_{1} \lor z_{2}) Ω (u)}{⋀} ψ (c) = & \underset{c \in z_{1} Ω (u) \lor z_{2} Ω (u)}{⋀} ψ (c) \end{matrix}$

Since c ∈ z₁Ω (u) ∨ z₂Ω (u), there exist c₁ ∈ z₁Ω (u) and c₂ ∈ z₂Ω (u) such that c = c₁ ∨ c₂.

Hence,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z_{1} \lor z_{2}) = \underset{c_{1} \lor c_{2} \in z_{1} Ω (u) \lor z_{2} Ω (u)}{⋀} ψ (c_{1} \lor c_{2}) & = \underset{c_{1} \in z_{1} Ω (u), c_{2} \in z_{2} Ω (u)}{⋀} Min {ψ (c_{1}), ψ (c_{2})} & = Min {\underset{c_{1} \in z_{1} Ω (u)}{⋀} ψ (c_{1}), \underset{c_{2} \in z_{2} Ω (u)}{⋀} ψ (c_{2})} & = Min {{\underline{Ω}}^{ψ} (u) (z_{1}), {\underline{Ω}}^{ψ} (u) (z_{2})} . \end{matrix}$

Hence ${\underline{Ω}}^{ψ} (u) (z_{1} \lor z_{2}) = Min {{\underline{Ω}}^{ψ} (u) (z_{1}),$ ${\underline{Ω}}^{ψ} (u) (z_{2})}$ ∀z₁, z₂ ∈ K₁.

As (Ω, A) be a SCTR so (z₁ ⊗ ₁z₂) Ω (u) = z₁Ω (u) ⊗ ₂z₂Ω (u) ∀ z₁, z₂ ∈ K₁.

Thus, we have,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z_{1} \otimes_{1} z_{2}) = & \underset{c \in (z_{1} \otimes_{1} z_{2}) Ω (u)}{⋀} ψ (c) = & \underset{c \in z_{1} Ω (u) \otimes_{2} z_{2} Ω (u)}{⋀} ψ (c) . \end{matrix}$

Now since c ∈ z₁Ω (u) ⊗ ₂z₂Ω (u) so there exist c₁ ∈ z₁Ω (u), c₂ ∈ z₂Ω (u) such that u = c₁ ⊗ ₂c₂.

Thus,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z_{1} \otimes_{1} z_{2}) = \underset{c_{1} \otimes_{2} c_{2} \in z_{1} Ω (u) \otimes_{2} z_{2} Ω (u)}{⋀} ψ (c_{1} \otimes_{2} c_{2}) & \geq \underset{c_{1} \otimes_{2} c_{2} \in z_{1} Ω (u) \otimes_{2} z_{2} Ω (u)}{⋀} Max {ψ (c_{1}), ψ (c_{2})} & = \underset{c_{1} \in z_{1} Ω (u), c_{2} \in z_{2} Ω (u)}{⋀} Max {ψ (c_{1}), ψ (c_{2})} & = Max {\underset{c_{1} \in z_{1} Ω (u)}{⋀} ψ (c_{1}), \underset{c_{2} \in z_{2} Ω (u)}{⋀} ψ (c_{2})} \\ = Max {{\underline{Ω}}^{ψ} (u) (z_{1}), {\underline{Ω}}^{ψ} (u) (z_{2})} . \end{matrix}$

Hence, ${\underline{Ω}}^{ψ} (u) (z_{1} \otimes_{1} z_{2}) \geq Max {{\underline{Ω}}^{ψ} (u) (z_{1}), {\underline{Ω}}^{ψ} (u) (z_{2})}$ ∀z₁, z₂ ∈ K₁. Thus, ${\underline{Ω}}^{ψ} (u)$ is a FSTI of K₁. □

Likewise, we show that the converse of Theorem 8 do not hold in general.

Example 5. Consider the quantales in Example 1. Let H = {u_1,u₂} and Ω : H ⟶ P (K₁ × K₂) be, $\begin{matrix} Ω (u_{1}) = & & {\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, u), (⊥_{1}, v), (⊥_{1}, ⊤_{2}), (b, ⊥_{2}), (b, u) \\ (b, v), (b, ⊤_{2}), (⊤_{1}, ⊥_{2}), (⊤_{1}, u), (⊤_{1}, v), (⊤_{1}, ⊤_{2}) \end{matrix}} \end{matrix}$ $\begin{matrix} Ω (u_{2}) = & & {\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, v), (⊥_{1}, ⊤_{2}), (b, ⊥_{2}) \\ (b, v), (b, ⊤_{2}), (⊤_{1}, ⊥_{2}), (⊤_{1}, v), (⊤_{1}, ⊤_{2}) \end{matrix}} \end{matrix}$

Now,the aftersets with respect to Ω (u₁) and Ω (u₂) are calculated as follows;

$\begin{matrix} ⊥_{1} Ω (u_{1}) = {⊥_{2}, u, v, ⊤_{2}} & ⊥_{1} Ω (u_{2}) = {⊥_{2}, v, ⊤_{2}} \\ b Γ (u_{1}) = {⊥_{2}, u, v, ⊤_{2}} & b Γ (u_{2}) = {⊥_{2}, v, ⊤_{2}} ⊤_{1} Ω (u_{1}) = {⊥_{2}, u, v, ⊤_{2}} & ⊤_{1} Ω (u_{2}) = {⊥_{2}, v, ⊤_{2}} \end{matrix}$

Then (Ω, H) is a SCTR with respect to aftersets. Define ψ₁ : K₂ ⟶ [0, 1] by, $ψ_{1} = \frac{0.5}{⊥_{2}} + \frac{0.3}{u} + \frac{0.7}{v} + \frac{0.4}{⊤_{2}}$

Then ψ₁ is not a fid of K₂. But, $\begin{matrix} \underline{Ω} ψ_{1} (u_{1}) & = & \frac{0.3}{⊥_{1}} + \frac{0.3}{b} + \frac{0.3}{⊤_{1}} \\ {\underline{Ω}}^{ψ_{2}} (u_{2}) & = & \frac{0.4}{⊥_{1}} + \frac{0.4}{b} + \frac{0.4}{⊤_{1}} \end{matrix}$ Thus ${\underline{Ω}}^{ψ_{1}} (u_{1})$ and ${\underline{Ω}}^{ψ_{1}} (u_{2})$ are fid of K₁ . Hence ψ₁ is generalized lower rough FSTI of K₁ with respect to aftersets by SCTR.

The following Theorem has similar proof as reported in the proof of Theorem 8.

Theorem 9. Let (Ω, A) be a SCR and ψ be a fid of K₂. Then ψ is a generalized upper rough FSTI of K₂ with respect to aftersets.

Remark 3. Theorems 8 and 9 can be obtained for foresets by considering fssts of K₁.

Theorem 10. Let ψ be a fid of K₂ and (Ω, H) be a SCTR. Then ${\underline{Ω}}^{ψ} (u)$ $[{\bar{Ω}}^{ψ} (u)]$ is a FSTI of K₁ with respect to afterset if and only if for each α ∈ [0, 1], ${\underline{Ω}}^{ψ_{α}} (u)$ $[{\bar{Ω}}^{ψ_{α}} (u)$ where ψ_α ≠ ∅] is an ideal of K₁ for all u ∈ H.

Proof. Let ${\underline{Ω}}^{ψ} (u)$ be a FSTI of K₁. Let $p, q \in {\underline{Ω}}^{ψ_{α}} (u)$ . Then ${\underline{Ω}}^{ψ} (u) (p) \geq α$ and ${\underline{Ω}}^{ψ} (u) (q) \geq α$ . But ${\underline{Ω}}^{ψ} (u)$ is a FSTI, so ${\underline{Ω}}^{ψ} (u) (p \lor q) = Min {{\underline{Ω}}^{ψ} (u) (p), {\underline{Ω}}^{ψ} (u) (q)}$ ≥α. Hence $p \lor q \in {\underline{Ω}}^{ψ_{α}} (u)$ . Let $q \in {\underline{Ω}}^{ψ_{α}} (u)$ and p ∈ K₁ such that p ≤ q. Then ${\underline{Ω}}^{ψ} (u) (p) \geq {\underline{Ω}}^{ψ} (u) (q) \geq α$ . Thus $p \in {\underline{Ω}}^{ψ_{α}} (u)$ . Let $p \in {\underline{Ω}}^{ψ_{α}} (u)$ and ∀ z ∈ K₁. Then ${\underline{Ω}}^{ψ} (u) (z \otimes_{1} p) \geq {\underline{Ω}}^{ψ} (u) (z) \lor {\underline{Ω}}^{ψ} (u) (p) = {\underline{Ω}}^{ψ} (u) (p) \geq α$ , we get $z \otimes_{1} p \in {\underline{Ω}}^{ψ_{α}} (u)$ . Similarly, $p \otimes_{1} z \in {\underline{Ω}}^{ψ_{α}} (u)$ . Hence, ${\underline{Ω}}^{ψ_{α}} (u)$ is an ideal of K₁.

Conversely assume that ${\underline{Ω}}^{ψ_{α}} (u)$ is an ideal of K₁. Let α = $Min {\underline{Ω} (g) (z_{1}), \underline{Ω} (g) (z_{2})}$ ∈ $rang ({\underline{Ω}}^{ψ} (u))$ for any z₁, z₂ ∈ K₁. Then ${\underline{Ω}}^{ψ} (u) (z_{1}) \geq α$ and ${\underline{Ω}}^{ψ} (u) (z_{2}) \geq α$ . That is $z_{1} \in {\underline{Ω}}^{ψ_{α}} (u)$ and $z_{2} \in {\underline{Ω}}^{ψ_{α}} (u)$ . Hence, $z_{1} \lor z_{2} \in {\underline{Ω}}^{ψ_{α}} (u)$ .

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z_{1} \lor z_{2}) = & \underset{c \in (z_{1} \lor z_{2}) Ω (u)}{⋀} ψ (c) = & \underset{c \in z_{1} Ω (u) \lor z_{2} Ω (u)}{⋀} ψ (c) \end{matrix}$

Since (Ω, A) be a SCTR and c ∈ z₁Ω (u) ∨ z₂Ω (u) then there exist a₁ ∈ z₁Ω (u) and a₂ ∈ z₂Ω (u) such that c = a₁ ∨ a₂.

Hence we have,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z_{1} \lor z_{2}) = \underset{a_{1} \lor a_{2} \in z_{1} Ω (u) \lor z_{2} Ω (u)}{⋀} ψ (a_{1} \lor a_{2}) & = \underset{a_{1} \lor a_{2} \in z_{1} Ω (u) \lor z_{2} Ω (u)}{⋀} Min {ψ (a_{1}), ψ (a_{2})} & = \underset{a_{1} \in z_{1} Ω (u), a_{2} \in z_{2} Ω (u)}{⋀} Min {ψ (a_{1}), ψ (a_{2})} \end{matrix}$

$\begin{matrix} = Min {\underset{a_{1} \in z_{1} Ω (u)}{⋀} ψ (a_{1}), \underset{a_{2} \in z_{2} Ω (u)}{⋀} ψ (a_{2})} & = Min {{\underline{Ω}}^{ψ} (u) (z_{1}), {\underline{Ω}}^{ψ} (u) (z_{2})} \end{matrix}$

So, ${\underline{Ω}}^{ψ} (u) (z_{1} \lor z_{2}) = Min {$ ${\underline{Ω}}^{ψ} (u) (z_{1}), {\underline{Ω}}^{ψ} (u) (z_{2})}$ ∀ z₁, z₂ ∈ K₁.

Now for $z \in {\underline{Ω}}^{ψ_{α}} (u)$ and ∀ y ∈ K₁, we obtain $z \otimes_{1} y \in {\underline{Ω}}^{ψ_{α}} (u)$ and $y \otimes_{1} z \in {\underline{Ω}}^{ψ_{α}} (u)$ . Hence, ${\underline{Ω}}^{ψ} (u) (z \otimes_{1} y) \geq α$ and ${\underline{Ω}}^{ψ} (u) (z) \geq α$ . If either ${\underline{Ω}}^{ψ} (u) (y) \geq α$ or ${\underline{Ω}}^{ψ} (u) (y) < α$ . In both the cases, $Max {{\underline{Ω}}^{ψ} (u) (x), {\underline{Ω}}^{ψ} (u) (y)}$ ≥α. We suppose $Max {{\underline{Ω}}^{ψ} (u) (x), {\underline{Ω}}^{ψ} (u) (y)}$ = α. So ${\underline{Ω}}^{ψ} (u) (x \otimes_{1} y)$ ≥ $Max {{\underline{Ω}}^{ψ} (u) (x), {\underline{Ω}}^{ψ} (u) (y)}$ . Hence, ${\underline{Ω}}^{ψ} (u)$ is a fid of K₁. □

Theorem 11. Let (Ω, H) be a SCTR and ψ be a fpid of K₂. Then ψ is a generalized lower rough FSTPI of K₁ with respect to aftersets.

Proof. Let ψ be a fpid of K₂. Then ψ (p ⊗ ₂q) = ψ (p) or ψ (p ⊗ ₂q) = ψ (q) ∀ p, q∈ K₂. Since ψ is a fpid of K₂ so it is fid of K₂, so by Theorem 8, ${\underline{Ω}}^{ψ} (u)$ is a FSTI of K₁.

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) = & \underset{c \in (p \otimes_{1} q) Ω (u)}{⋀} ψ (c) = & \underset{c \in p Ω (u) \otimes_{2} q Ω (u)}{⋀} ψ (c) \end{matrix}$

Since (Ω, A) be a SCTR, therefore for c ∈ pΩ (u) ⊗ ₂qΩ (u) there exist b ∈ pΩ (u) and d ∈ qΩ (u) such that c = b ⊗ ₂d.

Hence,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) = & \underset{b \otimes_{2} d \in p Ω (u) \otimes_{2} p Ω (u)}{⋀} ψ (b \otimes_{2} d) \overset{⋀}{¯} b \in p Ω (u), d \in q Ω (u) ψ (b \otimes_{2} d) \overset{⋀}{¯} b \in p Ω (u), d \in q Ω (u) [ψ (b) or ψ (d)] = & \underset{b \in p Ω (u)}{⋀} ψ (b) or \underset{d \in q Ω (u)}{⋀} ψ (d) {\underline{\overset{Ω}{¯}}}^{ψ} (u) (x_{1}) or {\underline{Ω}}^{ψ} (u) (y_{1}) \end{matrix}$

Thus, ${\underline{Ω}}^{ψ} (u) (p \otimes_{1} q) = {\underline{Ω}}^{ψ} (u) (p)$ or ${\underline{Ω}}^{ψ} (u) (p \otimes q) = {\underline{Ω}}^{ψ} (u) (q)$ ∀ p, q ∈ K₁. Hence ${\underline{Ω}}^{ψ} (u)$ is a FSTPI of K₁ for all u ∈ H. □

Example 6. Consider the quantales in Example 1. Let H = {u_1,u₂} and Ω : H ⟶ P (K₁ × K₂) be defined by Ω (u₁) = ${\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, u), (⊥_{1}, v), (⊥_{1}, ⊤_{2}), (b, ⊥_{2}), (b, u) \\ (b, v), (b, ⊤_{2}), (⊤_{1}, ⊥_{2}), (⊤_{1}, u), (⊤_{1}, v), (⊤_{1}, ⊤_{2}) \end{matrix}}$ and Ω (u₂) = ${\begin{matrix} (⊥_{1}, ⊥_{2}), (⊥_{1}, v), (⊥_{1}, ⊤^{'}), (b, ⊥_{2}) \\ (b, v), (b, ⊤_{2}), (⊤_{1}, ⊥_{2}), (⊤_{1}, v), (⊤_{1}, ⊤_{2}) \end{matrix}}$ .

Now, aftersets in terms of Ω (u₁) and Ω (u₂) are below;

$\begin{matrix} ⊥_{1} Ω (u_{1}) = {⊥_{2}, u, v, ⊤_{2}} & ⊥_{1} Ω (u_{2}) = {⊥_{2}, v, ⊤_{2}} \\ b Γ (u_{1}) = {⊥_{2}, u, v, ⊤_{2}} & b Ω (u_{2}) = {⊥_{2}, v, ⊤_{2}} ⊤_{1} Ω (u_{1}) = {⊥_{2}, u, v, ⊤_{2}} & ⊤_{1} Ω (u_{2}) = {⊥_{2}, v, ⊤_{2}} \end{matrix}$

It is easy to confirm that (Ω, H) is a SCTR with respect to aftersets. Define ψ₁ : K₂ ⟶ [0, 1] by, $ψ_{1} = \frac{0.9}{⊥^{'}} + \frac{0.6}{u} + \frac{0.7}{v} + \frac{0.6}{⊤^{'}}$ Then ψ₁ is not a fpid of K₂ but, $\begin{matrix} {\bar{Ω}}^{ψ_{1}} (u_{1}) & = & \frac{0.9}{⊥} + \frac{0.9}{b} + \frac{0.9}{⊤} \\ {\bar{Ω}}^{ψ_{1}} (u_{2}) & = & \frac{0.9}{⊥} + \frac{0.9}{b} + \frac{0.9}{⊤} \end{matrix}$ This show that ${\bar{Ω}}^{ψ_{1}} (u_{1})$ and ${\bar{Ω}}^{ψ_{1}} (u_{2})$ are fpid of K₁ . Hence ψ₁ is generalized upper FSTPI of K₁with respect to aftersets. Define ψ₂ : K₂ ⟶ [0, 1] by, $ψ_{2} = \frac{0.5}{⊥^{'}} + \frac{0.2}{u} + \frac{0.3}{v} + \frac{0.2}{⊤^{'}}$ Then ψ₂ is not a fpid of K₂ but, $\begin{matrix} {\underline{Ω}}^{^{ψ_{2}}} (u_{1}) & = & \frac{0.2}{⊥} + \frac{0.2}{b} + \frac{0.2}{⊤} \\ {\underline{Ω}}^{^{ψ_{2}}} (u_{2}) & = & \frac{0.2}{⊥} + \frac{0.2}{b} + \frac{0.2}{⊤} \end{matrix}$ It is concluded from above that ${\underline{Ω}}^{^{ψ_{2}}} (u_{1})$ and ${\underline{Ω}}^{^{ψ_{2}}} (u_{2})$ are fpid of K₁ . Hence ψ₂ is generalized lower FSTPI of K₁ with respect to aftersets.

The Theorem 12 has similar proof as the proof of Theorem 11.

Theorem 12. Let ψ be a fpid of K₂ and (Ω, A) be a SCTR. Then ψ is a generalized upper rough FSTPI of K₁ with respect to afterset.

Theorem 13. Let (Ω, H) be a SCR and ψ be a fpid of K₂. Then ${\underline{Ω}}^{ψ} (e)$ $[{\bar{Ω}}^{ψ} (e)]$ is a FSTPI of K₁ with respect to afterset if and only if ${\underline{Ω}}^{ψ_{α}} (e)$ $[{\bar{Ω}}^{ψ_{α}} (e)$ where ψ_α ≠ ∅] for each α ∈ [0, 1] is a pid of K₁ for all u ∈ H.

Proof. Let ψ be a fpid of K₂. Then ψ (a ⊗ ₂c) = ψ (a) or ψ (a ⊗ ₂c) = ψ (c) ∀ a, c ∈ K₂. Suppose ${\underline{Ω}}^{ψ} (u)$ is a FSTPI of K₁, then ${\underline{Ω}}^{ψ} (u)$ is a FSTI of K₁. By Theorem 10, ${\underline{Ω}}^{ψ_{α}} (u)$ is an ideal of K₁. To show that ${\underline{Ω}}^{ψ_{α}} (u)$ is a pid for all α ∈ [0, 1], we have to show that for a⊗ ₁c ∈ ${\underline{Ω}}^{ψ_{α}} (u)$ implies that $a \in {\underline{Ω}}^{ψ_{α}} (u)$ or $c \in {\underline{Ω}}^{ψ_{α}} (u)$ . Let $a \otimes_{1} c \in {\underline{Ω}}^{ψ_{α}} (u)$ . Then ${\underline{Ω}}^{ψ} (u) (a) = {\underline{Ω}}^{ψ} (u) (a \otimes_{1} c) \geq α$ or ${\underline{Ω}}^{ψ} (u) (a \otimes_{1} c) = {\underline{Ω}}^{ψ} (u) (c) \geq α$ . Thus, $a \in {\underline{Ω}}^{ψ_{α}} (u)$ or $c \in {\underline{Ω}}^{ψ_{α}} (u)$ . Hence ${\underline{Ω}}^{ψ_{α}} (u)$ is a pid of K₁.

Conversely, let ${\underline{Ω}}^{ψ_{α}} (e)$ be a pid of K₁. Then ${\underline{Ω}}^{ψ_{α}} (e)$ is an ideal of K₁. By Theorem 10, ${\underline{Ω}}^{ψ} (e)$ is a FSTI of K₁.

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (x_{2} \otimes_{1} y_{2}) = & \underset{d \in (x_{2} \otimes_{1} y_{2}) Ω (u)}{⋀} ψ (d) = & \underset{d \in x_{2} Ω (u) \otimes_{2} y_{2} Ω (u)}{⋀} ψ (d) \end{matrix}$

Since (Ω, A) be a SCTR, so we have a ∈ xΩ (u) , c ∈ yΩ (u) such that d = a ⊗ ₂c.

Hence,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (x_{2} \otimes_{1} y_{2}) = & \underset{a \otimes_{2} c \in x_{2} Ω (u) \otimes_{2} y_{2} Ω (u)}{⋀} ψ (a \otimes_{2} c) = & \underset{a \in x_{2} Ω (u), c \in y_{2} Ω (u)}{⋀} ψ (a \otimes_{2} c) = & \underset{a \in x_{2} Ω (u), c \in y_{2} Ω (u)}{⋀} [ψ (a) or ψ (c)] = & \underset{a \in x_{2} Ω (u)}{⋀} ψ (a) or \underset{c \in y_{2} Ω (u)}{⋀} ψ (c) {\overline{\underset{Ω}{¯}}}^{ψ} (u) (x_{2}) or {\underline{Ω}}^{ψ} (u) (y_{2}) \end{matrix}$

Therefore ${\underline{Ω}}^{ψ} (u) (x_{2} \otimes_{1} y_{2}) = {\underline{Ω}}^{ψ} (u) (x_{2})$ or $\underline{Ω} (u) (x_{2} *_{1} y_{2}) = {\underline{Ω}}^{ψ} (u) (y_{2})$ ∀ x₂, y₂ ∈ K₁. Hence ${\underline{Ω}}^{ψ} (u)$ is a FSTPI of K₁. □

Theorem 14. Let ψ be a fspid of K₂ and (Ω, H) be a SCTR. Then ψ is a generalized lower rough FSTSPI of K₁ with respect to aftersets.

Proof. Let ψ be a fspid of K₂. Then ψ (p²) = ψ (p) ∀ p ∈ K₂ and thus ψ is a fid of K₂. By Theorem 10, ${\underline{Ω}}^{ψ} (u)$ is a FSI of K₁ for all u ∈ H.

Consider,

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (y) = & \underset{p \in y Ω (u)}{⋀} ψ (p) {\overset{⋀}{¯}}_{p \in y Ω (u)} ψ (p^{2}) \\ = & \underset{p \otimes_{2} p \in y Ω (u) \otimes_{2} y Ω (u)}{⋀} ψ (p^{2}) \end{matrix}$

$\begin{matrix} = & \underset{p \otimes_{2} p \in (y \otimes_{1} y) Ω (u)}{⋀} ψ (p^{2}) {\overset{⋀}{¯}}_{p^{2} \in y^{2} Ω (u)} ψ (p^{2}) = & {\underline{Ω}}^{ψ} (u) (y^{2}) \end{matrix}$

Thus ${\underline{Ω}}^{ψ} (u) (y^{2}) = {\underline{Ω}}^{ψ} (u) (y)$ ∀ y ∈ K₁. Therefore ${\underline{Ω}}^{ψ} (u)$ is a FSTSPI of K₁ for all u ∈ H. □

Proposition 2. Let ψ be a fspid of K₂ and (Ω, H) be a SCTR. Then ψ is a generalized upper rough FSTSPI of K₁ with respect to aftersets.

Theorem 15. Let (Ω, H) be a SCTR and ψ be a fspid of K₂. Then ${\underline{Ω}}^{ψ} (u)$ $({\bar{Ω}}^{ψ} (u))$ is a FSTSPI of K₁ with respect to afterset if and only if for each α ∈ [0, 1], ${\underline{Ω}}^{ψ_{α}} (u)$ $[{\bar{Ω}}^{ψ_{α}} (u)$ where ψ_α ≠ ∅] is a spid of K₁ for all u ∈ H.

Proof. Let ${\underline{Ω}}^{ψ} (u)$ is a FSTSPI of K₁. Then ${\underline{Ω}}^{ψ} (u)$ is a FSTI of K₁. By Theorem 10, ${\bar{Ω}}^{ψ_{α}} (u)$ is an ideal of K₁. Let $p \otimes_{1} p \in {\bar{Ω}}^{ψ_{α}} (u)$ . Then ${\underline{Ω}}^{ψ} (u) (p) = {\underline{Ω}}^{ψ} (u) (p \otimes_{1} p) \geq α$ . Thus, we have $p \in {\bar{Ω}}^{ψ_{α}} (u)$ . Hence ${\bar{Ω}}^{ψ_{α}} (u)$ is a spid of K₁.

Conversely, let ${\bar{Ω}}^{ψ_{α}} (u)$ is a spid of K₁. Then ${\bar{Ω}}^{ψ_{α}} (u)$ is an ideal of K₁. By Theorem 10, ${\underline{Ω}}^{ψ} (u)$ is a FSTI.

We have to show that ${\underline{Ω}}^{ψ} (u) (z \otimes_{1} z) = {\underline{Ω}}^{ψ} (u) (z)$ ∀ z ∈ K₁. As (Ω, H) be a SCTR and ψ is a fspid of K₂. Then

$\begin{matrix} {\underline{Ω}}^{ψ} (u) (z) = & \underset{p \in z Ω (u)}{⋀} ψ (p) {\overset{⋀}{¯}}_{p \in z Ω (u)} ψ (p^{2}) = & \underset{p \otimes_{2} p \in z Ω (u) \otimes_{2} z Ω (u)}{⋀} ψ (p^{2}) {\overset{⋀}{¯}}_{p \otimes_{2} p \in (z \otimes_{1} z) Ω (u)} ψ (p^{2}) {\overset{⋀}{¯}}_{p^{2} \in (z \otimes_{1} z) Ω (u)} ψ (p^{2}) = & {\underline{Ω}}^{ψ} (u) (z \otimes_{1} z) \end{matrix}$

Thus ${\underline{Ω}}^{ψ} (u) (z \otimes_{1} z) = {\underline{Ω}}^{ψ} (u) (z)$ ∀ z ∈ K₁. Hence ${\underline{Ω}}^{ψ} (u)$ is a FSTSPI of K₁ for all u ∈ H. □

5 Homomorphic images of generalized rough fuzzy soft substructures

This section includes the relations between the upper (lower) generalized rough fuzzy soft substructures of quantales and the upper (lower) approximations of their images under weak quantale homomorphism. Some notions related to establish such results are first explained.

Definition 16. [30] Let (K₁, ⊗ ₁) and (K₂, ⊗ ₂) be two quantales. A map ξ : K₁ ⟶ K₂ is called a weak quantale homomorphism (WQH) if

(1) ξ (p⊗ ₁q) = ξ (p) ⊗ ₂ξ (q) ;

(2) ξ (p ∨ q) = ξ (p) ∨ ξ (p) for all p, q ∈ K.

A weak quantale homomorphism ξ : K₁ ⟶ K₂ is called an epimorphism if ξ is on to K₂ and ξ is called a monomorphism if ξ is one-one. If ξ is bijective, then it is called an isomorphism. It is clear that if p ≤ q, then ξ (p) ≤ ξ (q). That is ξ is order-preserving. In the following the term WQH will be utilized for weak quantale homomorphism.

Lemma 1. Let ξ : K₁ → K₂ be a surjective WQH and (Ω₂, H) be a SBR on K₂. Set Ω₁ (u) ={ (s, t) ∈ K₁ × K₁ : (ξ (s) , ξ (t)) ∈ Ω₂ (u) } for all u ∈ H. Then the following hold:

(1) (Ω₁, H) is soft compatible if (Ω₂, H) is soft compatible.

(2) (Ω₁, H) is SCTR with respect to aftersets (with respect to foresets) if (Ω₂, H) is SCTR with respect to aftersets (with respect to foresets) and ξ is one one.

(3) $ξ ({\bar{Ω}}_{1}^{C} (u))$ = ${\bar{Ω}}_{2}^{ξ (C)} (u)$ for C ⊆ K₁ and for all u ∈ H.

(4) $ξ ({\underline{Ω}}_{1}^{C} (u))$ $\subseteq {\underline{Ω}}_{2}^{ξ (C)} (u)$ for all u ∈ H and if ξ is one one, then $ξ ({\underline{Ω}}_{1}^{C} (u))$ = ${\underline{Ω}}_{2}^{ξ (C)} (u)$ for all u ∈ H.

(5) Let ξ : K₁ ⟶ K₂ be one-one. Then $ξ (x) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ $\Leftrightarrow x \in ({\bar{Ω}}_{1}^{C} (u))$ for all u ∈ H.

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

(3) Let $z \in ξ ({\bar{Ω}}_{1}^{C}) (u))$ for some z ∈ K₂. Then there exists a ∈ K₁ such that $a \in {\bar{Ω}}_{1}^{C} (u)$ and ξ (a) = z. Thus there exists x ∈ aΩ₁ (u) ∩ C such that (a, x) ∈ Ω₁ (u) and x ∈ C. This shows that (ξ (a) , ξ (x)) ∈ Ω₂ (u) ⇒ ξ (x) ∈ ξ (a) Ω₂ (u). Moreover, ξ (x) ∈ ξ (C). Thus $ξ (a) Ω_{2} (u) \cap ξ (C) \neq \emptyset \Rightarrow z = ξ (a) \in {\bar{Ω}}_{2}^{ξ (C)}$ . That is $ξ ({\bar{Ω}}_{1}^{C} (u))$ $\subseteq {\bar{Ω}}_{2}^{ξ (C)} (u)$ . Conversely, let $w \in {\bar{Ω}}_{2}^{ξ (C)} (u)$ . Then wΩ₂ (u)∩ ξ (C) ≠ ∅. So there exists a ∈ wΩ₂ (u) ∩ ξ (C) such that (w, a) ∈ Ω₂ (u) and a ∈ ξ (C). Since ξ is onto so there exist x ∈ C and s ∈ K₁ such that a = ξ (x) and w = ξ (s). Thus (ξ (s) , ξ (x)) = (w, a) ∈ Ω₂ (u) ⇒ (s, x) ∈ Ω₁ (u). This implies x ∈ sΩ₁ (u) ∩ C, so we have $s \in {\bar{Ω}}_{1}^{C} (u)$ . That is $w = ξ (s) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Thus ${\bar{Ω}}_{2}^{ξ (C)} (u)$ ⊆ $ξ ({\bar{Ω}}_{1}^{C} (u))$ . Hence $ξ ({\bar{Ω}}_{1}^{C} (u)) = {\bar{Ω}}_{2}^{ξ (C)} (u)$ for all u ∈ H.

(4) Let $b \in ξ ({\underline{Ω}}_{1}^{C} (u))$ for all u ∈ H. Then there exists $a \in {\underline{Ω}}_{1}^{C} (u)$ such that aΩ₁ (u) ⊆ C and ξ (a) = b. Let p ∈ bΩ₂ (u). Then there exist q ∈ K₁ such that ξ (q) = p and ξ (q) ∈ ξ (a) Ω₂ (u) , i.e., (ξ (a) , ξ (q)) ∈ Ω₂ (u). Hence (a, q) ∈ Ω₁ (u). That is q ∈ aΩ₁ (u) ⊆ C and so ξ (q) ∈ ξ (C) and bΩ₂ (u) ⊆ ξ (C). This shows that $b \in {\underline{Ω}}_{2}^{ξ (C)} (u)$ . So we have $ξ ({\underline{Ω}}_{1}^{C} (u))$ $\subseteq {\underline{Ω}}_{2}^{ξ (C)} (u)$ for all u ∈ H. Now let $b \in {\underline{Ω}}_{2}^{ξ (C)} (u)$ . Then there exists a unique a ∈ K₁ such that ξ (a) = b and ξ (a) Ω₂ (u) ⊆ ξ (C). Let q ∈ aΩ₁ (u) , i.e., (a, q) ∈ Ω₁ (u). Then (ξ (a) , ξ (q) ∈ Ω₂ (u). That is ξ (q) ∈ ξ (a) Ω₂ (u) ⊆ ξ (C) , and so q ∈ C. Thus, aΩ₁ (u) ⊆ C, which gives $a \in {\underline{Ω}}_{1}^{C} (u)$ . Then $b = ξ (a) \in ξ ({\underline{Ω}}_{1}^{C} (u)),$ and so ${\underline{Ω}}_{2}^{ξ (C)} (u) \subseteq ξ ({\underline{Ω}}_{1}^{C} (u))$ for all u ∈ H.

(5) Let $p \in {\bar{Ω}}_{1}^{C} (u)$ for all u ∈ H. Then $ξ (p) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ for all u ∈ H. Conversely suppose that $ξ (w) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Then there exists $p^{'} \in {\bar{Ω}}_{1}^{C} (u)$ such that ξ (p) = ξ (p′). Since ξ is ono-one, we get $p = p^{'} \in {\bar{Ω}}_{1}^{C} (u)$ . □

Theorem 16. Let ξ be a surjective WQH and (Ω₂, H) be a SCR with respect to aftersets on K₂. Set $Ω_{1} (u) = {(s, t) \in K_{1} \times K_{1} : (ξ (s), ξ (t)) \in Ω_{2} (u)}$ for all u ∈ H. Then for all ∅ ≠ C ⊆ K₁ and u ∈ H, the following hold;

(1) ${\bar{Ω}}_{1}^{C} (u)$ is an ideal of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂ for all u ∈ H.

(2) ${\bar{Ω}}_{1}^{C} (u)$ is a pid of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is a pid of K₂ for all u ∈ H.

(3) ${\bar{Ω}}_{1}^{C} (u)$ is a spid of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is a spid of K₂ for all u ∈ H.

(4) ${\bar{Ω}}_{1}^{C} (u)$ is a subquantale of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is a subquantale of K₂ for all u ∈ H.

Proof. (1) Let ${\bar{Ω}}_{1}^{C} (u)$ be an ideal of K₁ for all u ∈ H. It is to show that ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂ for all u ∈ H. By Lemma 1(3), we have $ξ ({\bar{Ω}}_{1}^{C} (u)) = {\bar{Ω}}_{2}^{ξ (C)} (u)$ for all C ⊆ K₁ and for all u ∈ H.

(i) Let $z, y \in ξ ({\bar{Ω}}_{1}^{C} (u))$ and for all u ∈ H. Then $p, q \in {\bar{Ω}}_{1}^{C} (u)$ be such that ξ (p) = z, ξ (q) = y. Since ${\bar{Ω}}_{1}^{C} (u)$ is an ideal and ξ is WQH, we have $z \lor y = ξ (p) \lor ξ (q) = ξ (p \lor q) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ .

(ii) Let $p \leq q \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Then there exist w₁ ∈ K₁ and $w_{2} \in {\bar{Ω}}_{1}^{C} (u)$ such that p = ξ (w₁) and q = ξ (w₂). Since ξ (w₁) ≤ ξ (w₂) , we have $ξ (w_{1} \lor w_{2}) = ξ (w_{1}) \lor ξ (w_{2}) = ξ (w_{2}) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . By Lemma 1(5), we have $w_{1} \lor w_{2} \in {\bar{Ω}}_{1}^{C} (u)$ . Since ${\bar{Ω}}_{1}^{C} (u)$ is an ideal and w₁ ≤ w₁ ∨ w₂, , we have w₁ $\in {\bar{Ω}}_{1}^{C} (u)$ and p $= ξ (w_{1}) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ .

(iii) Let y = ξ (a) ∈ K₂ and w = ξ (b) ∈ $ξ ({\bar{Ω}}_{1}^{C} (u))$ . Then $b \in {\bar{Ω}}_{1}^{C} (u)$ by Lemma 1(5). Since ${\bar{Ω}}_{1}^{C} (u)$ is an ideal, we have $a \otimes_{1} b \in {\bar{Ω}}_{1}^{C} (u)$ and thus y ⊗ ₂w = ξ (a) ⊗ ₂ξ (b) = ξ (a ⊗ ₁b) ∈ $ξ ({\bar{Ω}}_{1}^{C} (u))$ . Similar argument shows that $w \otimes_{2} y \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Thus $ξ ({\bar{Ω}}_{1}^{C} (u))$ = ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂ from (i)-(iii) for all u ∈ H.

Conversely, let $ξ ({\bar{Ω}}_{1}^{C} (u))$ = ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂ for all u ∈ H.

(i) Let p₁, p₂ $\in {\bar{Ω}}_{1}^{C} (u)$ for all u ∈ H. Then ξ (p₁) , ξ (p₂) $\in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Since $ξ ({\bar{Ω}}_{1}^{C} (u))$ is an ideal, so ξ (p₁ ∨ p₂) = ξ (p₁) ∨ ξ (p₂) $\in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Then by Lemma 1(5), we have $p_{1} \lor p_{2} \in {\bar{Ω}}_{1}^{C} (u)$ .

(ii) Let $p_{1} \leq p_{2} \in {\bar{Ω}}_{1}^{C} (u)$ . Then ξ (p₁) ≤ ξ (p₂) $\in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Since, $ξ ({\bar{Ω}}_{1}^{C} (u))$ is an ideal, we have ξ (p₁) $\in ξ ({\bar{Ω}}_{1}^{C} (u))$ and thus p₁ ∈ ${\bar{Ω}}_{1}^{C} (u)$ by Lemma 1(5).

(iii) Let z ∈ K₁ and $p \in {\bar{Ω}}_{1}^{C} (u)$ . Then $ξ (p) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ and ξ (z) ∈ K₂. Since $ξ ({\bar{Ω}}_{1}^{C} (u))$ is an ideal of K₂, we have $ξ (p \otimes_{1} z) = ξ (p) \otimes_{2} ξ (z) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ and thus $p \otimes_{1} z \in {\bar{Ω}}_{1}^{C} (u)$ by Lemma 1(5). The similar argument shows that $z \otimes_{1} p \in {\bar{Ω}}_{1}^{C} (u)$ . Thus, ${\bar{Ω}}_{1}^{C} (u)$ is an ideal of K₁ from (i)-(iii).

(2) First, we show that ${\bar{Ω}}_{1}^{C} (u) \neq K_{1} \Leftrightarrow ξ ($ ${\bar{Ω}}_{1}^{C} (u)) \neq K_{2}$ for all u ∈ H, that is ${\bar{Ω}}_{1}^{C} (u) = K_{1} \Leftrightarrow ξ ($ ${\bar{Ω}}_{1}^{C} (u)) = K_{2}$ . Suppose that ${\bar{Ω}}_{1}^{C} (u) = K_{1}$ . Since ξ is surjective, we have ξ ( ${\bar{Ω}}_{1}^{C} (u)) = ξ (K_{1}) = K_{2}$ . Conversely, assume that $ξ ({\bar{Ω}}_{1}^{C} (u)) = K_{2}$ . For each z ∈ K₁, we have ξ (z) ∈ ξ (K₁) = K₂ = ξ ( ${\bar{Ω}}_{1}^{C} (u))$ . Then by Lemma 1(5), we have $z \in {\bar{Ω}}_{1}^{C} (u)$ and thus ${\bar{Ω}}_{1}^{C} (u) = K_{1}$ . Assume that ${\bar{Ω}}_{1}^{C} (u)$ is a pid of K₁ for all u ∈ H. Then ${\bar{Ω}}_{1}^{C} (u)$ is an ideal of K₁ and ${\bar{Ω}}_{1}^{C} (u) \neq K_{1}$ . By (1), ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂. It is also known that $ξ ({\bar{Ω}}_{1}^{C} (u)) = {\bar{Ω}}_{2}^{ξ (C)} (u) \neq K_{2}$ . Now suppose w₁, w₂ ∈ K₂ and $w_{1} \otimes_{2} w_{2} \in {\bar{Ω}}_{2}^{ξ (C)} (u)$ . Since ξ is surjective, there exist z₁, z₂ ∈ K₁ such that ξ (z₁) = w₁, ξ (z₂) = w₂. Then $ξ (z_{1} \otimes_{1} z_{2}) = ξ (z_{1}) \otimes_{2} ξ (z_{2}) = w_{1} \otimes_{2} w_{2} \in {\bar{Ω}}_{2}^{ξ (C)} (u)$ . By Lemma 1(5), we have $z_{1} \otimes_{1} z_{2} \in {\bar{Ω}}_{1}^{C} (u)$ . Since, ${\bar{Ω}}_{1}^{C} (u)$ is pid, we have $z_{1} \in {\bar{Ω}}_{1}^{C} (u)$ or $z_{2} \in {\bar{Ω}}_{1}^{C} (u)$ and thus $w_{1} \in {\bar{Ω}}_{2}^{ξ (C)} (u) = ξ ({\bar{Ω}}_{1}^{C} (u))$ or $w_{2} \in {\bar{Ω}}_{2}^{ξ (C)} (u) = ξ ({\bar{Ω}}_{1}^{C} (u))$ . So, ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is a pid of K₂.

Conversely, assume that ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is a pid of K₂. Then ${\bar{Ω}}_{2}^{ξ (C)} (u) = ξ ({\bar{Ω}}_{1}^{C} (u)) \neq K_{2}$ and thus ${\bar{Ω}}_{1}^{C} (u) \neq K_{1}$ . Since ${\bar{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂. By (1), ${\bar{Ω}}_{1}^{C} (u)$ is an ideal of K₁. Now suppose z₁, z₂ ∈ K₁ and $z_{1}, z_{2} \in {\bar{Ω}}_{1}^{C} (u)$ . Then $ξ (z_{1}) \otimes_{2} ξ (z_{2}) = ξ (z_{1} \otimes_{1} z_{2}) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . Since ${\bar{Ω}}_{2}^{ξ (C)} (u) = ξ ({\bar{Ω}}_{1}^{C} (u))$ is pid, we have $ξ (z_{1}) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ or $ξ (z_{2}) \in ξ ({\bar{Ω}}_{1}^{C} (u))$ . By Lemma, 1(5), we have $z_{1} \in {\bar{Ω}}_{1}^{C} (u)$ or $z_{2} \in {\bar{Ω}}_{1}^{C} (u)$ . Thus ${\bar{Ω}}_{1}^{C} (u)$ is a pid of K₁.

The proof of (3) and (4) are similar to the proof of (1) and (2). □

The next Proposition has similar proof as reported in Theorem 16.

Proposition 3. Let ξ be a surjective WQH and (Ω₂, H) be a SCR on K₂. Set $Ω_{1} (u) = {(s, t) \in K_{1} \times K_{1} : (ξ (s), ξ (t)) \in Ω_{2} (u)}$ for all u ∈ H. Then for all ∅ ≠ C ⊆ K₁ and u ∈ H, we have

(1) ${\underline{Ω}}_{1}^{C} (u)$ is an ideal of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (C)} (u)$ is an ideal of K₂ for all u ∈ H;

(2) ${\underline{Ω}}_{1}^{C} (u)$ is a pid of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (C)} (u)$ is a pid of K₂ for all u ∈ H;

(3) ${\underline{Ω}}_{1}^{C} (u)$ is a spid of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (C)} (u)$ is a spid of K₂ for all u ∈ H;

(4) ${\underline{Ω}}_{1}^{C} (u)$ is a subquantale of K₂ if and only if ${\underline{Ω}}_{2}^{ξ (C)} (u)$ is a subquantale of K₂ for all u ∈ H.

Remark 4. Theorem 16 and Proposition 3 can be obtained for foresets and have similar proof.

The relation between fuzzy algebraic substructures of a quantale K₁ and fuzzy algebraic substructures of a quantale K₂ are described in the following.

Theorem 17. Let ξ : K₁ ⟶ K₂ be a surjective WQH and (Ω₂, A) be a SCR on K₂ and ψ be a fsst of K₁. Set $Ω_{1} (u) = {(k, t) \in K_{1} \times K_{1} : (ξ (s), ξ (t)) \in Ω_{2} (e)}$ for all u ∈ H . Then following hold;

(1) ${\bar{Ω}}_{1}^{ψ} (u)$ is a fid of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fid of K₂ ;

(2) ${\bar{Ω}}_{1}^{ψ} (u)$ is a fpid of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fpid of K₂ ;

(3) ${\bar{Ω}}_{1}^{ψ} (u)$ is a fspid of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fspid of K₂ ;

(4) ${\bar{Ω}}_{1}^{ψ} (u)$ is a fsq of K₁ if and only if ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fsq of K₂.

Where, $ξ (ψ) (y) = {\begin{matrix} \underset{x \in ξ^{- 1} (y)}{⋁} ψ (x), & if ξ (y) \neq \emptyset \forall y \in K_{2} 0, & otherwise \end{matrix}$

That is, ξ (ψ) is the standard Zadeh image of the fsst ψ under the mapping ξ.

Proof. (1) Note that (ξ (ψ)) _{α
⁺} = ξ (ψ_{α
⁺}) for each α ∈ [0, 1]. Also $({\bar{Ω}}_{1}^{ψ} (u))_{α^{+}} \neq \emptyset$ if and only if ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} (u) \neq \emptyset$ .

Let ${\bar{Ω}}_{1}^{ψ} (u)$ be a fid of K₁. Then ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} \neq \emptyset$ if ${\bar{Ω}}_{1}^{ψ_{α^{+}}} (u) \neq \emptyset$ for all α ∈ (0, 1]. By Theorem 10, we have ${\bar{Ω}}_{1}^{ψ_{α^{+}}} (u)$ is an ideal of K₁. Also by using Proposition 1, we obtain $({\bar{Ω}}^{ψ} (u))_{α^{+}}$ is an ideal of K₁. Now, by Theorem 16(1) and Proposition 1, we have ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} (u) = ({\bar{Ω}}_{2}^{ξ (ψ)} (u))_{α^{+}} = {\bar{Ω}}_{2}^{ξ (ψ_{α^{+}})} (u)$ is an ideal of K₂. Thus, by Theorem 10, we have ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fid of K₂.

Conversely, suppose ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fid of K₂. We have ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} (u) = ({\bar{Ω}}_{2}^{ξ (ψ)} (u))_{α^{+}} = {\bar{Ω}}_{2}^{ξ (ψ_{α^{+}})} (u)$ is an ideal of K₂ by utilizing Theorem 10 and Proposition 1. Thus, ${\bar{Ω}}^{ψ_{α^{+}}} (u)$ is an ideal of K₁ from Theorem 16(1). Hence ${\bar{Ω}}_{1}^{ψ} (u)$ is a fid of K₁ by Theorem 10.

(2) Let ${\bar{Ω}}_{1}^{ψ} (u)$ be a fpid of K₁. Now for ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} \neq \emptyset$ , then ${\bar{Ω}}_{1}^{ψ_{α^{+}}} (u) \neq \emptyset$ for each α ∈ [0, 1]. Since ${\bar{Ω}}_{1}^{ψ} (u)$ is a fpid of K₁, then by Theorem 13 and Proposition 1, we have $({\bar{Ω}}_{1}^{ψ} (u))_{α^{+}} = {\bar{Ω}}_{1}^{ψ_{α^{+}}} (u)$ is a pid of K₁. Hence ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} (u) = ({\bar{Ω}}_{2}^{ξ (ψ)} (u))_{α^{+}} = {\bar{Ω}}_{2}^{ξ (ψ_{α^{+}})} (u)$ is a pid of K₂, by Theorem 16(2). Thus, by Theorem 13, we have ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fpid of K₂.

Conversely, suppose ${\bar{Ω}}_{2}^{ξ (ψ)} (u)$ is a fpid of K₁. By Theorem 13, we have ${\bar{Ω}}_{2}^{(ξ (ψ))_{α^{+}}} (u) = ({\bar{Ω}}_{2}^{ξ (ψ)} (u))_{α^{+}} = {\bar{Ω}}_{2}^{ξ (ψ_{α^{+}})} (u)$ is a pid of K₂. Thus from Theorem 16(2), ${\bar{Ω}}_{1}^{ψ_{α^{+}}} (u)$ is a pid of K₁. Hence ${\bar{Ω}}_{1}^{ψ} (u)$ is a fpid of K₁ by Theorem 13.

The similar proof of (3) and (4) can be obtained from (1) and (2). □

The similar proof of next Proposition can be obtained from Theorem 17.

Proposition 4. Let (Ω₂, H) be a SCR with respect to aftersets on K₂ and ψ be a fsst of K₂. Let ξ : K₁ ⟶ K₂ be a surjective WQH. Set $Ω_{1} (u) = {(s, t) \in K_{1} \times K_{1} : (ξ (s), ξ (t)) \in Ω_{2} (u)}$ for all u ∈ H . Then following hold;

(1) ${\underline{Ω}}_{1}^{ψ} (u)$ is a fid of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (ψ)} (u)$ is a fid of K₂ ;

(2) ${\underline{Ω}}_{1}^{ψ} (u)$ is a fpid of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (ψ)} (u)$ is a fpid of K₂ ;

(3) ${\underline{Ω}}_{1}^{ψ} (u)$ is a fsq of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (ψ)} (u)$ is a fsq of K₂.

(4) ${\underline{Ω}}_{1}^{ψ} (u)$ is a fsq of K₁ if and only if ${\underline{Ω}}_{2}^{ξ (ψ)} (u)$ is a fsq of K₂.

Remark 5. Theorem 17 and Proposition 4 can be obtained for foresets and have similar proof.

6 Conclusion

This paper extend and develop the notion of "rough approximation" defined by Pawlak and "rough fuzzy set" by Dubois and Prade. In this paper, we are motivated to establish the relations between fuzzy substructures and quantales with the help of soft compatible relations. This is done with the help of afterset and foreset. Moreover, we present soft complete relation to find out the lower and upper approximation of ideals, prime ideals, semi-prime ideals and subquantales of quantales with the help of afterset and foreset. We also present the relation between two quantales and observed that if ψ is a fuzzy substructure of K₂ then, its upper and lower approximation with the help of afterset are fuzzy substructure of K₂ by using compatible relation. Similarly, upper and lower approximation with the help of foresets are fuzzy substructure of K₁ by using compatible relation, if we are considering ψ as a fuzzy substructure of K₁.

From all motivation, one can proceed the above mentioned ideas to others algebraic structures, like rings and semirings.

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