# Soft Substructures in Quantales and Their Approximations Based on Soft Relations

Authors: Huan Zhou; Saqib Mazher Qurashi; Muti Ur Rehman; Muhammad Shabir; Rani Sumaira Kanwal; Ahmed Mostafa Khalil
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

The aim of this research article is to derive a new relation between rough sets and soft sets with an algebraic structure quantale by using soft binary relations. The aftersets and foresets are utilized to define lower approximation and upper approximation of soft subsets of quantales. As a consequence of this new relation, different characterization of rough soft substructures of quantales is obtained. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Moreover, this concept generalizes the concept of rough soft substructures of other structures. Furthermore, the algebraic relations between the upper (lower) approximation of soft substructures of quantales and the upper (lower) approximation of their homomorphic images with the help of soft quantales homomorphism are examined. In comparison with the different type of approximations in different type of algebraic structures, it is concluded that this new study is much better.


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## Body

## 1. Introduction

Quantale theory was proposed by Mulvey[1]. It is based on defining an algebraic structure on a complete lattice. Since quantale was defined on a complete lattice, there must be a correlation between linear logic and quantale theory which was studied by Yetter, in his study. He presented a new class of models for linear intuitionistic logic [2]. In recent years, quantale is applied in vast research areas, such as algebraic theory [3], rough set theory [4–7], topological theory [8], theoretical computer science [9], and linear logic [10].In 1982, Pawlak developed the famous rough set theory [11], which is a mathematization of inadequate knowledge. The rough set deals with the categorization and investigation of inadequate information and knowledge. After Pawlak’s work, Zhu [12] provided some new views on the rough set theory. In [13], Ali et al. studied some properties of generalized rough sets. Nowadays, rough sets are applied in many different areas, such as cognitive sciences, machine learning, pattern recognition, and process control.There are many problems that arise in different fields such as engineering, economics, and social sciences in which data have some sort of uncertainty. Well-known mathematical tools have so many limitations because these tools are introduced for particular circumstances. There are many theories to overcome uncertainty such as fuzzy set theory, probability theory, rough sets, and vague sets, but these are limited due to its design.In 1998, Molodtsov present the idea of soft set theory, which is a mathematical tool to overcome the adversities affecting the above theories [14]. Many authors like Maji et al. present different operations on soft sets and try to consolidate the algebraic aspects of soft sets [15]. A new and different idea of operations was presented by Ali et al. [16]. Many soft algebraic structures such as soft modules [17], soft groups [18], soft rings [19], and soft ordered semigroups [20] were studied. The basic theme and purpose of soft sets are to create the idea of parametrization, and this idea has been utilized to find soft binary relation (SBR) which is a parameterized collection of binary relations on a universe under consideration. This puts forward the consideration for complicated objects that may be perceived from different points of view. In [21–23], Feng et al. presented the relationship between soft, rough, and fuzzy sets and produced rough soft sets, soft rough sets, and soft-rough fuzzy sets.By using aftersets and foresets notions associated with SBR, a new approximation space is widely utilized these days. By using generalized approximation space based on SBR, different soft substructures in semigroups were approximated by Kanawal and Shabir [24]. Motivated by the idea in [24], soft substructures in quantales are defined, and the aftersets and foresets are employed to construct the lower approximation and upper approximation of soft substructures. Since we are dealing with the approximation of soft subsets of quantale, further soft substructures are employed for further characterization.There are several authors who introduced rough sets theory in algebraic structures and soft algebraic structures. Iwinski analyzes algebraic properties of rough sets [25]. Qurashi and Shabir present the idea of roughness in Q- module [5]. Idea of the generalized rough quantales (subquantales) was presented by Xiao and Li [6]. Rough prime (semiprime and primary) ideals in quantales were investigated by Yang and Xu [7]. Fuzzy ideals (prime, semiprime, and primary) in quantales were introduced by Luo and Wang [4]. Generalized roughness of fuzzy substructures in quantale is studied by Qurashi et al. [26]. In [27], Yamak et al. proposed the idea of set-valued mappings as the basis of the generalized upper (lower) approximations of a ring with the help of ideals. Rough prime bi Γ-hyper ideals of Γ-semihypergroups were proposed by Yaqoob et al. [28, 29]. Rough substructures of semigroups were studied by Kuroki [30].The following scheme is designed for the rest of the paper. Some essential explanations related to quantales, its substructures, soft substructures, and their corresponding sequels are connected in Section2. Notion of approximations of soft sets over quantale generated by soft binary relations is discussed in Section 3. In Section 4, by using these ideas, generalized soft substructures are defined and investigated further fundamental algebraic characteristics of these phenomena. Additionally, we extend this study to define the relationship between homomorphic images and their approximation by soft binary relation in Section 5.


## 2. Preliminaries

LetΘ be a nonempty finite set called the universe set and Ψ be an E.R (equivalence relation) over Θ. Let qΨ denotes the equivalence class of the relation containing q. Any definable set in Θ would be written as finite union of equivalence classes of Θ. Let R⊆Θ in general R is not a definable set in Θ. However, the set R can be approximated by two definable sets in Θ. The first one is called Ψ-lower approximation (Ψ−Lappr) of R, and the second is called Ψ-upper approximation (Ψ−Uappr). They are defined as follows:(1)Ψ¯R=q∈Θ:qΨ⊆R.(2)Ψ¯R=q∈Θ:qΨ∩R≠∅.TheΨ−Lappr of R in Θ is the greatest definable in Θ contained in R. The Ψ−Uappr of R in Θ is the least definable set in Θ containing R. For any nonempty subset R in Θ, ΨR=Ψ¯R,Ψ¯R is called rough set with respect to Ψ or simply a Ψ-rough subset of PΘ×PΘ if Ψ¯R≠Ψ¯R, where PΘ denotes the set of all subsets of Θ.Definition 1 (see [31]).
LetΘ be a complete lattice. Define an associative binary relation ∘ on Θ satisfying(3)l∘∨i∈Iwi=∨i∈Il∘wiand∨i∈Ili∘w=∨i∈Ili∘w,∀l,w,li,wi∈Θ. Then, Θ,∘ is called quantale.
LetT1,T2,TI⊂Θ, i∈I. We define some notions as follows:(4)T1°T2=t1°t2:t1∈T1,t2∈T2;T1∨T2=t1∨t2:t1∈T1,t2∈T2;∨i∈ITi=∨i∈Iti:ti∈Ti.
Throughout the paper, quantales are denoted byΘ1 and Θ2.
Let∅≠W⊆Θ. Then, W is called a subquantale of Θ if the following holds:(1)
w1°w2∈W, ∀w1,w2∈W.(2)
∨i∈Iwi,∈W, ∀wi,∈W.
That is,Θ closed under ∘ and arbitrary supremum.Definition 2 (see [32]).
LetΘ be a quantale, ∅≠I⊆Θ is called left (right) ideal if the following satisfied:(1)
u,v∈I implies u∨v∈1(2)
p∈Θ, u∈I such that p≤u implies p∈I(3)
q∈Θ and u∈I implies q°u∈Iu°q∈I
A nonempty subsetI⊆Θ is called ideal of Θ if it is left as well as right ideal.Example 1.
LetΘ=0,p,q,r,1 complete lattices are shown in Figure 1. We define ∘ be the associative binary operation on Θ as shown in Table 1.
Then,Θ is a quantale. Then, 0, 0,p, 0,q, 0,p,q,r, and Θ are all I of quantale Θ.Figure 1
Illustration ofΘ.Table 1
Binary operation subject toΘ
∘10pqr100pqr1p0pqr1q0pqr1r0pqr110pqr1Definition 3 (see [32]).
Let∅≠I⊆Θ be an ideal. I is called prime ideal if, ∀u,v∈Θ, u°v∈I⇒u∈I or v∈I. I is called semiprime I if, ∀u∈Θ, u°u∈I⇒u∈II is called primary I if, ∀u,v∈Θ, u°v∈I and u∉I implies vn∈I for some n∈Ν.Definition 4 (see [14]).
A pairΨ,C is called a soft set over Θ if Ψ:C⟶PΘ where C is a subset of E (the set of parameters).Definition 5 (see [16]).
LetF,C1 and H,C2 be two soft sets over Θ. Then, F,C1 soft subset H,C2 if the following conditions are fulfilled:(1)
C1⊆C2(2)
Fc⊆Hc, ∀c∈C1Definition 6 (see [33]).
LetΨ,C be a soft set over Θ×Θ, that is, Ψ:C⟶PΘ×Θ. Then, Ψ,C is called a soft binary relation (SBR) over Θ×Θ. A SBR over Θ1×Θ2 is a soft set Ψ,C over Θ1×Θ2. That is, Ψ:C⟶PΘ1×Θ2.Definition 7.
LetΨ,C be a soft set over quantale Θ. Then,(1)
Ψ,C is called soft subquantale over Θ iff Ψc is a subquantale of Θ, ∀c∈C(2)
Ψ,C is called soft ideal over Θ iff Ψc is an ideal of Θ, ∀c∈C(3)
Ψ,C is called soft prime ideal over Θ iff Ψc is a prime ideal of Θ, ∀c∈C(4)
Ψ,C is called soft semiprime ideal over Θ iff Ψc is a semiprime ideal of Θ, ∀c∈C(5)
Ψ,C is called soft primary ideal over Θ iff Ψc is a primary ideal of Θ, ∀c∈C


## 3. Approximation of Soft Sets over Quantale by Soft Binary Relation

In this section, we present some important aspects regarding to the approximation of soft sets in quantaleΘ by SBR. We utilized aftersets and foresets to approximate soft sets.Definition 8 (see [34]).
LetΨ,C be a SBR over Θ1×Θ2, where C⊆E (parametric set). Then, Ψ:C⟶PΘ1×Θ2. For a soft set F,C over Θ2, the LapprΨ¯F,C and UapprΨ¯F,C of F,C w.r.t the afterset are essentially two soft sets over Θ1, which is defined as(5)Ψ¯Fc=q1∈Θ1:∅≠q1Ψc⊆Fc,(6)Ψ¯Fc=q1∈Θ1:q1Ψc∩Fc≠∅,∀,c∈C.
And for a soft setH,C over Θ1, the LapprΨ¯H,C and UapprΨ¯H,C of H,C w.r.t the foreset are actually two soft sets over Θ2, which is defined as(7)Ψ¯Hc=q2∈Θ2:∅≠Ψcq2⊆Hc,Ψ¯Hc=q2∈Θ2:Ψcq2∩Hc≠∅.
For allc∈C, where q1Ψc=q2∈Θ2:q1,q2∈Ψc is called the afterset of q1 and Ψcq2=q1∈Θ1:q1,q2∈Ψc is called the foreset of q2.Remark 1.
(1)
For each soft setF,C over Θ2, Ψ¯F:C⟶PΘ1 and Ψ¯F:C⟶PΘ1(2)
For each soft setH,C over Θ1, Ψ¯H:C⟶PΘ2 and Ψ¯H:C⟶PΘ2Definition 9.
LetΨ,C be a SBR over Θ1×Θ2, that is, Ψ:C⟶PΘ1×Θ2. Then, Ψ,C is called soft compatible relation (SCPR) if for all p,r,ji∈Θ1 and q,s,ki∈Θ2i∈I, we have(1)
p,q,r,s∈Ψc⇒p∘1r,q∘2s∈Ψc(2)
ji,ki∈Ψc⇒∨i∈Iji,∨i∈Iki∈Ψcfor every c∈C.Definition 10.
A SCPRΨ,C over Θ1×Θ2 is called soft complete relation (SCTR) with respect to the afterset if, for all p,r,∈Θ1, we have(1)
pΨc∨rΨc=p∨rΨc(2)
pΨc∘2rΨc=p∘1rΨcfor all c∈C.
A SCPRΨ,C is called ∨-complete w.r.t the aftersets if it satisfies only condition (1). A SCPR Ψ,C is called °-complete w.r.t the aftersets if it satisfies only condition (2).
A SCPRΛ,C over Θ1×Θ2 is called soft complete relation (SCTR) with respect to the foreset if for all q,s∈Θ2, and we have(1)
Λcq∨Λcs=Λcq∨s(2)
Λcq∘1Λcs=Λcq∘2sfor all c∈C.
A SCPRΛ,C is called ∨-complete w.r.t the foresets if it satisfies only condition (1).
A SCPRΛ,C is called °-complete w.r.t the foresets if it satisfies only condition (2).Theorem 1.
LetΨ,C be a SCPR with respect to the afterset over Θ1×Θ2. Then, for any two soft sets F1,C and F2,C over Θ2, we have(1)
Ψ¯F1,C∘1Ψ¯F2,C⊆Ψ¯F1∘2F2,C(2)
Ψ¯F1,C∨Ψ¯F2,C⊆Ψ¯F1∨F2,CProof.
For arbitraryc∈C, let x∈Ψ¯F1c∘1Ψ¯F2c. Then, x=y1∘1y2 for some y1∈Ψ¯F1c and y2∈Ψ¯F2c. This implies that y1Ψc∩F1c≠∅ and y2Ψc∩F2c≠∅, so there exist elements l,m∈Θ2 such that l∈y1Ψc∩F1c and m∈y2Ψc∩F2c. Thus, l∈y1Ψc,m∈y2Ψc,l∈F1c and m∈F2c. So y1,l∈Ψc and y2,m∈Ψc imply y1∘1y2,l∘2m∈Ψc; that is, l∘2m∈y1∘1y2Ψc. Also, l∘2m∈F1c∘2F2c; therefore, l∘2m∈y1∘1y2Ψc∩F1c∘2F2c. This shows that x=y1∘1y2∈Ψ¯F1∘2F2c.
Now, for arbitraryc∈C, let x∈Ψ¯F1c∨Ψ¯F2c. Then, x=y1∨y2 for some y1∈Ψ¯F1c and y2∈Ψ¯F2c. This implies that y1Ψc∩F1c≠∅ and y2Ψc∩F2c≠∅, so there exist elements l,m∈Θ2 such that l∈y1Ψc∩F1c and m∈y2Ψc∩F2c. Thus, l∈y1Ψc,m∈y2Ψc,l∈F1c, and m∈F2c. So y1,l∈Ψc and y2,m∈Ψc imply y1∨y2,l∨m∈Ψc; that is, l∨m∈y1∨y2Ψc. Also, l∨m∈F1c∨F2c; therefore, l∨m∈y1∨y2Ψc∩F1c∨F2c. This shows that x=y1∨y2∈Ψ¯F1∨F2c.Theorem 2.
LetΨ,C be a SCPR with respect to the foreset over Θ1×Θ2. Then, for any two soft sets L1,C and L2,C over Θ1, we have(1)
Ψ¯L1,C∘2Ψ¯L2,C⊆Ψ¯L1∘1L2,C(2)
Ψ¯L1,C∨Ψ¯L2,C⊆Ψ¯L1∨L2,CProof.
The proof is simple.Theorem 3.
LetΨ,C be a SCTR w.r.t the afterset over Θ1×Θ2. Then, for any two soft sets F1,C and F2,C over Θ2, we have(1)
Ψ¯F1,C∘1Ψ¯F2,C⊆Ψ¯F1∘2F2,C(2)
Ψ¯F1,C∨Ψ¯F2,C⊆Ψ¯F1∨F2,CProof.
For arbitraryc∈C, if at least one of ΨF1c and ΨF2c is empty, then (1) is obvious. Now, for arbitrary c∈C, consider that ΨF1c≠∅ and ΨF2c≠∅. Then, ΨF1c°1ΨF2c≠∅. So, let x∈ΨF1c°1ΨF2c. Then, x=y°11y2 for some y1∈ΨF1c and y2∈ΨF2c. This implies that ∅≠y1Ψc⊆F1c and ∅≠y2Ψc⊆F2c. As y1°1y2Ψc=y1Ψc°2Ψc⊆F1c°2F2c. This shows that x=y1°1y2∈ΨF1F2c. Hence, (1) is proved.
For arbitraryc∈C, if at least one of ΨF1c and ΨF2c is empty, then (2) is obvious. Now, for arbitrary c∈C, consider that ΨF1c≠∅ and ΨF2c≠∅. Then, ΨF1c∨ΨF2c≠∅. So, let x∈ΨF1c∨ΨF2c. Then, x=y1∨y2 for some y1∈ΨF1c and y2∈ΨF2c. This implies that ∅≠y1Ψc⊆F1c and ∅≠y2Ψc⊆F2c. As y1∨y2Ψc=y1Ψc∨y2Ψc⊆F1c∨F2c. This shows that x=y1∨y2∈ΨF1∨F2c. Hence, (2) is proved.Theorem 4.
LetΨ,C be a SCTR with respect to the foreset over Θ1×Θ2. Then, for any two soft sets L1,C and L2,C over Θ1, we have(1)
Ψ¯L1,C∘2Ψ¯L2,C⊆Ψ¯L1∘1L2,C(2)
Ψ¯L1,C∨Ψ¯L2,C⊆Ψ¯L1∨L2,CProof.
The proof is obvious.


## 4. Approximation of Soft Substructures in Quantales

In this section, we consider two quantalesΘ1 and Θ2 and approximate different soft substructures of quantales by using different SBR over Θ1×Θ2. We will show that Uappr of a soft substructure of quantales by using SCPR is again a soft substructure of quantales and provide counter examples to support the argument that the converse is not true. Also, we will show that Lappr of a soft substructure of quantales by using SCTR is again a soft substructure of quantales and provide a counter example to support the argument that the converse is not true.Throughout this section, we considerΨ,C to be the SBR over Θ1×Θ2 and xΨc≠∅ for all x∈Θ1, c∈C, and Ψcy≠∅ for all y∈Θ2, c∈C unless otherwise specified.Definition 11.
LetΨ,C be a SBR over Θ1×Θ2 and F,C be a soft set over Θ2. If Uappr. Ψ¯F,C is a soft subquantale of Θ1, then F,C is called generalized upper soft (GUpS) subquantale of Θ1 w.r.t the aftersets. If UapprΨ¯F,C is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ1, then F,C is called GUpS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ1 w.r.t the aftersets.Definition 12.
LetΨ,C be a SBR over Θ1×Θ2 and L,C be a soft set over Θ1. If UapprΨ¯L,C is a soft subquantale of Θ2, then L,C is called generalized upper soft (GUpS) subquantale of Θ2 w.r.t the foresets. If UapprΨ¯L,C is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ2, then L,C is called GUpS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ2 w.r.t the foresets.Theorem 5.
LetΨ,C be a SCPR over Θ1×Θ2. If F,C is a soft subquantale of Θ2, then F,C is a GUpS subquantale of Θ1 w.r.t the aftersets.Proof.
Suppose thatF,C is a soft subquantale, then ∅≠Ψ¯Fc for any c∈C. Let pi∈Ψ¯Fc, i∈I. Then, piΨc∩Fc≠∅. So, there exists qi∈piΨc∩Fc. Thus, qi∈piΨc and qi∈Fc since Ψ,C is a SCPR. Therefore, pi,qi∈Ψc, i∈I implies ∨i∈Ipi,∨i∈Iqi∈Ψc. This implies that ∨i∈Iqi∈∨i∈IpiΨc. Also, ∨i∈Iqi∈Fc (as F,C is a soft subquantale). So, ∨i∈Iqi∈∨i∈Ipi∈Ψc∩Fc. Hence, ∨i∈Ipi∈Ψ¯Fc.
Letp1,p2∈Ψ¯Fc. Then, p1Ψc∩Fc≠∅ and p2Ψ¯c∩Fc≠∅. So, there exists q1∈p1Ψc∩Fc and q2∈p2Ψc∩Fc. Thus, q1∈p1Ψc, q1∈Fc, q2∈p2Ψc, and q2∈Fc since Ψ,C is a SCPR. Therefore, p1,q1,p2,q2∈Ψc implies p1°1q1,p2°2q2∈Ψc. This implies that q1°2q2∈p1°p2Ψc. Also, q1°2q2∈c (as F,C is a soft subquantale). So, q1°2q2∈p1°p2Ψc∩Fc. Hence, p1°1q1∈Ψ¯Fc. This completes the proof.
With the same arguments, the next Theorem6 can be achieved.Theorem 6.
LetΨ,C be a SCPR over Θ1×Θ2. If L,C is a soft subquantale of Θ1, then L,C is a GUpS subquantale of Θ2 w.r.t the foresets.Theorem 7.
LetΨ,C be a soft ∨-complete relation over Θ1×Θ2 w.r.t the aftersets. If F,C is a soft left (right) ideal of Θ2, then F,C is a GUpS left (right) ideal of Θ1 w.r.t the aftersets.Proof.
Suppose thatF,C is a soft left ideal of Θ2, then ∅≠Ψ¯Fc for any c∈C. Let u1,u2∈Ψ¯Fc. Then, u1Ψc∩Fc≠∅ and u2Ψc∩Fc≠∅. So, there exists v1∈u1Ψc∩Fc and v2∈u2Ψc∩Fc. Thus, v1∈u1Ψc, v1∈Fc, v2∈u2Ψc, and v2∈Fc since Ψ,C is a SCPR. Therefore, u1∨u2,v1∨v2∈Ψc; that is, v1∨v2∈u1∨u2Ψc. Also, v1∨v2∈Fc (as F,C is a soft left ideal). So, v1∨v2∈u1∨u2Ψc∩Fc. Hence, u1∨u2∈Ψ¯Fc.
Now, letu1,u2∈Θ1 such that u1≤u2 and u2∈Ψ¯Fc. So, u1∨u2=u2∈Ψ¯Fc. Since u2∈Ψ¯Fc, so there exist v2∈u2Ψc∩Fc. Thus, v2∈u2Ψc and v2∈Fc. Since Ψ,C is a soft ∨-complete relation, therefore, v2∈u2Ψc=u1∨u2Ψc=u1Ψc∨u2Ψc. This implies that v2=s∨t, for some s∈u1Ψc and t∈u2Ψc. Thus, s≤v2 and v2∈Fc imply s∈Fc (as Fc is ideal). So, s∈u1Ψc∩Fc. Hence, u1∈Ψ¯Fc.
Letp,x∈Θ1 and x∈Ψ¯Fc. Then, xΨc∩Fc≠∅. So, there exist q∈xΨc∩Fc. Thus, q∈xΨc and q∈Fc. Since F,C is a soft left ideal so, y∘2q∈Fc for any y∈pΨc⊆Θ2. This implies that p,y∈Ψc. So, p∘1x,y∘2q∈Ψc; that is, y∘2q∈p∘1xΨc. So, y∘2q∈p∘1xΨc∩Fc. Hence, p∘1x∈Ψ¯Fc. Similarly, we can show that x∘1p∈Ψ¯Fc.Theorem 8.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If F,C is a soft prime ideal of Θ2, then F,C is a GUpS prime ideal of Θ1 w.r.t the aftersets.Proof.
Assume thatF,C is a soft prime ideal of Θ2, then ∅≠Ψ¯Fc for any c∈C. Then, by Theorem 5, F,C is generalized upper soft ideal of Θ1. Let p1,p2∈Θ1 such that p1∘1p2∈Ψ¯Fc. Then, p1∘1p2Ψc∩Fc≠∅. So, there exist q∈p1∘1p2Ψc∩Fc. This implies that q∈p1∘1p2Ψc and q∈Fc. Since Ψ,C is a SCTR, q∈p1∘1p2Ψc=p1Ψc∘2p2Ψc. Thus, q=c∘2d for some c∈p1Ψc and d∈p2Ψc. Thus, c∘2d∈Fc and F,C is a soft prime ideal of Θ2 so, c∈Fc or d∈Fc. Thus, c∈p1Ψc∩Fc or d∈p2Ψc∩Fc. Hence, p1∈Ψ¯Fc or p2∈Ψ¯Fc.Theorem 9.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If F,C is a soft semiprime ideal of Θ2, then F,C is a GUpS semiprime ideal of Θ1 w.r.t the aftersets.Proof.
Assume thatF,C is a soft semiprime ideal of Θ2, then ∅≠Ψ¯Fc for any c∈C. Then, by Theorem 5, F,C is generalized upper soft ideal of Θ1. Let p1∈Θ1 such that p1∘1p1∈Ψ¯Fc. Then, p1∘1p1Ψc∩Fc≠∅. So, there exist q∈p1∘1p1Ψc∩Fc. This implies that q∈p1∘1p1Ψc and q∈Fc. Since Ψ,C is a SCTR, q∈p1∘1p1Ψc=p1Ψc∘2p1Ψc. Thus, q=c∘2c for some c∈p1Ψc. Thus, c∘2c∈Fc and F,C is a soft semiprime ideal of Θ2 so, c∈Fc. Thus, c∈p1Ψc∩Fc. Hence, p1∈Ψ¯Fc.Theorem 10.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If F,C is a soft primary ideal of Θ2, then F,C is a GUpS primary ideal of Θ1 w.r.t the aftersets.Proof.
Assume thatF,C is a soft primary ideal of Θ2, then ∅≠Ψ¯Fc for any c∈C. Then, by Theorem 5, F,C is generalized upper soft ideal of Θ1. Let p1,p2∈Θ1 such that p1∘1p2∈Ψ¯Fc and p1∉Ψ¯Fc. Then, p1∘1p2Ψc∩Fc≠∅. So, there exist q∈p1∘1p2Ψc∩Fc. This implies that q∈p1∘1p2Ψc and q∈Fc. Since Ψ,C is a SCTR, q∈p1∘1p2Ψc=p1Ψc∘2p2Ψc. Thus, q=c∘2d for some c∈p1Ψc and d∈p2Ψc. Thus, c∘2d∈Fc and F,C is a soft primary ideal of Θ2 so dn∈Fc for some n∈ℕ. Also, dn∈p2nΨc for n∈ℕ. Thus, dn∈p2nΨc∩Fc. Hence, p2n∈Ψ¯Fc.Remark 2.
In general, the converse of the above theorem is not true. We will present examples to justify our claim as follows.Example 2.
LetΘ1=0,p,q,1 and Θ2=0′,s′,q′,p′,1′,r′ be two complete lattices described in Figures 2 and 3, respectively.
We define∘1 and ∘2 the associative binary operation on Θ1 and Θ2, respectively, as shown in Tables 2 and 3. Then, and are quantales.(1)
LetC=c1,c2 and define SBR Ψ,C over Θ1×Θ2 by the rule(8)Ψc1=0,q′,0,0′,p,s′,0,s′,q,p′,p,r′,0,1′,0,p′,p,1′,q,s′,q,1′,p,q′,q,r′,1,r′,1,s′,1,1′,0,r′,Ψc2=0,r′,0,0′,0,p′,0,1′,0,s′,p,r′,0,q′,p,s′,p,1′,p,q′.Then,Ψ,C is SCPR. The aftersets with respect to Ψc1 and Ψc2 are given as follows:(9)0Ψc1=0′,s′,p′,q′,r′,1′,0Ψc2=0′,s′,p′,r′,1′,q′,pΨc1=r′,s′,1′,q′,pΨc2=r′,s′,1′,q′,qΨc2=∅,1Ψc1=r′,s′,1′1Ψc2=∅.Define soft setF,C over Θ2 by the rule(10)Fc1=r′,s′,Fc2=q′,r′.Then,F,C is not a soft subquantale of Θ2. But Ψ¯Fc1=0,p,q,1 and Ψ¯Fc2=0,p are subquantale of Θ1. So F,C is a GUpSSΘ of Θ1 w.r.t the aftersets.Foresets with respect toΨc1 and Ψc2 are given as follows:(11)Ψc10′=0,Ψc20′=0,Ψc1p′=0,q,Ψc2p′=0,Ψc1q′=0,p,Ψc2q′=0,p,Ψc1r′=0,p,q,1,Ψc2r′=0,p,Ψc1s′=0,p,q,1,Ψc2s′=0,p,Ψc11′=0,p,q,1,Ψc21′=0,p.Define soft setL,C over Θ1 by the rule(12)Lc1=p,q,Lc2=0,p,q.Then,L,C is not a soft subquantale of Θ1. But Ψ¯Lc1=p′,q′,r′,s′,1′ and Ψ¯Lc2=0′,p′,q′,r′,s′,1′ are subquantale of Θ2. So, L,C is a GUpS subquantale of Θ2 w.r.t the foresets.(2)
Now, letC=c1,c2 and define SBR Ψ,C over Θ1×Θ2 by the rule(13)Ψc1=q,0′,p,0′,q,p′,0,p′,1,0′,1,p′,p,p′,0,0′.(14)Ψc2=p,q′,0,s′,0,q′,q,s′,1,s′,1,q′,q,q′,p,s′.
Aftersets with respect toΨc1 and Ψc2 are given as follows:(15)0Ψc1=0′,p′,0Ψc2=s′,q′,pΨc1=0′,p′,pΨc2=s′,q′,qΨc1=0′,p′,qΨc2=s′,q′,1Ψc1=0′,p′,1Ψc2=s′,q′.
Then,Ψ,C is ∨-complete relation over Θ1×Θ2 w.r.t the aftersets. Define soft set F,C over Θ2 by the rule(16)Fc1=s′,p′,Fc2=s′,r′.
Then,F,C is not a soft ideal of Θ2. But Ψ¯Fc1=0,p,q,1 and Ψ¯Fc2=0,p,q,1 are ideal of Θ1. So, F,C is a GUpS ideal of Θ1 w.r.t the aftersets.
Now, define SBRΨ,C over Θ1×Θ2 by the rule(17)Ψc1=0,q′,p,q′,0,0′,p,0′,p,s′,p,r′,0,1′,0,r′,p,p′,0,s′,p,1′,0,p′(18)Ψc2=0,0′,q,r′,q,q′,0,s′,0,1′,q,0′,0,r′,q,p′,0,p′,q,s′,q,1′,0,q′
Foresets with respect toΨc1 and Ψc2 are given as follows:(19)Ψc10′=p,0,Ψc20′=q,0,Ψc1p′=p,0,Ψc2p′=0,q,Ψc1q′=p,0,Ψc2q′=q,0,Ψc1r′=0,p,Ψc2r′=q,0,Ψc1s′=0,p,Ψc2s′=q,0,Ψc11′=0,p,Ψc21′=0,q.
Then,Ψ,C is soft ∨-complete relation over Θ1×Θ2 w.r.t the foresets. Define soft set L,C over Θ1 by the rule(20)Lc1=p,q,Lc2=0,p,q.
Then,L,C is not a soft ideal of Θ1. But Ψ¯Lc1=0′,p′,q′,r′,s′,1′ and Ψ¯Lc2=0′,p′,q′,r′,s′,1′ are ideal of Θ2. So, L,C is a GUpS ideal of Θ2 w.r.t the foresets.
Similar examples can be presented to justify that converse of Theorems11 to 13 is not true.Figure 2
Illustration ofΘ1.Figure 3
Illustration ofΘ2.Table 2
Binary operation subject toΘ1.
O10pq100000p0p0pq00qq10pq1Table 3
Binary operation subject toΘ2.
O20′s′p′q′r′1′0′0′s′p′q′r′1′s′0′s′p′q′r′1′p′0′s′p′q′r′1′q′0′s′p′q′r′1′r′0′s′p′q′r′1′1′0′s′p′q′r′1′Definition 13.
LetΨ,C be a SBR over Θ1×Θ2. Consider the soft set M,C over Θ2, if LapprΨ¯M,C is a soft subquantale of Θ1, then M,C is called generalized lower soft (GLWS) subquantale of Θ1 w.r.t the aftersets. If LapprΨ¯M,C is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ1, then M,C is called GUpS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ1 w.r.t the aftersets.Definition 14.
LetΨ,C be a SBR over Θ1×Θ2. Consider the soft set L,C over Θ1, if LapprΨ¯L,C is a soft subquantale of Θ2, then L,C is called GLWS subquantale of Θ2 w.r.t the foresets. If LapprΨ¯L,C is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ2, then L,C is called GUpS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ2 w.r.t the foresets.Theorem 11.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If M,C is a soft subquantale of Θ2, then M,C is a GLWS subquantale of Θ1 w.r.t the aftersets.Proof.
Suppose thatM,C is a soft subquantale of Θ2 and Ψ¯Mc≠∅ for any c∈C. Let ui∈Ψ¯Mc,i∈I. Then, uiΨc⊆Mc. Since Ψ,C is a SCTR, therefore, ∨i∈IuiΨc=∨i∈IuiΨc⊆Mc. Hence, ∨i∈Iui∈Ψ¯Mc.
Now, letu1,u2∈Ψ¯Mc. Then, u1Ψc⊆Mc and u2Ψc⊆Mc. Since Ψ,C is a SCTR and M,C is a soft subquantale, therefore, u1Ψc∘2u2Ψc⊆Mc∘2Mc implies u1∘1u2Ψc⊆Mc. Hence, u1∘1u2∈Ψ¯Mc.
With the same arguments, next Theorem12 can be achieved.Theorem 12.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the foresets. If L,C is a soft subquantale of Θ1, then L,C is a GLWS subquantale of Θ2 w.r.t the foresets.Theorem 13.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If M,C is a soft ideal of Θ2, then M,C is a GLWS ideal of Θ1 w.r.t the aftersets.Proof.
Suppose thatM,C is a soft ideal of Θ2 and Ψ¯Mc≠∅ for any c∈C. Let u1,u2∈Ψ¯Mc. Then, u1Ψc⊆Mc and u2Ψc⊆Mc. Since Ψ,C is a SCTR and M,C is a soft ideal of Θ2 so u1Ψc∨u2Ψc=u1∨u2Ψc⊆Mc∨Mc; that is, u1∨u2Ψc⊆Mc Hence, u1∨u2∈Ψ¯Mc.
Now, letu1,u2∈Θ1 such that u1≤u2 and u2∈Ψ¯Mc. So, u1∨u2=u2∈Ψ¯Mc. Let, v1∈u1Ψc and v2∈u2Ψc⊆Mc. So, v1∨v2∈u1∨u2Ψc, that is, v1∨v2∈u2Ψc⊆Mc. Since Mc is ideal so v1≤v1∨v2∈Mc implies v1∈Mc. Thus, u1Ψc⊆Mc. Hence, u1∈Ψ¯Mc.
Now, letu,y∈Θ1 and y∈Ψ¯Mc. Then, ∅≠yΨc⊆Mc. Consider v1∈u∘1yΨc since Ψ,C is a SCTR so v1∈uΨc∘2yΨc. Thus, v1=c∘2d for some c∈uΨc and d∈yΨc. But yΨc⊆Mc so d∈Mc and M,C is a soft ideal of Θ2; therefore, c∘2d∈Mc, that is, v1∈Mc. Thus, u∘1yΨc⊆Mc. Hence, u∘1y∈Ψ¯Mc. Similarly, we can show that y∘1u∈Ψ¯Mc.Theorem 14.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If M,C is a soft prime ideal of Θ2, then M,C is a GLWS prime ideal of Θ1 w.r.t the aftersets.Proof.
Assume thatM,C is a soft prime ideal of Θ2 and Ψ¯Mc≠∅ for any c∈C. Then, by Theorem 4.19, M,C is GLWS ideal of Θ1. Let u1,u2∈Θ1 such that u1∘1u2∈Ψ¯Mc. Then, u1∘1u2Ψc⊆Mc. Consider v∈u1∘1u2Ψc⊆Mc. Since Ψ,C is a SCTR, v∈u1∘1u2Ψc=u1Ψc∘2u2Ψc. Thus, v=c∘2d for some c∈u1Ψc and d∈u2Ψc. This implies that v=c∘2d∈Mc. As M,C is a soft prime ideal so, c∈Mc or d∈Mc. Thus, c∈u1Ψc⊆Mc or d∈u2Ψc⊆Mc. Hence, u1∈Ψ¯Mc or u2∈Ψ¯Mc.Theorem 15.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If M,C is a soft semiprime ideal of Θ2, then M,C is a GLWS semiprime ideal of Θ1 w.r.t the aftersets.Proof.
Assume thatM,C is a soft semiprime ideal of Θ2 and Ψ¯Mc≠∅ for any c∈C. Then, by Theorem 14,M,C is GLWS ideal of Θ1. Let u∈Θ1 such that u∘1u∈Ψ¯Mc. Then, u∘1uΨc⊆Mc. Let v∈uΨc. As Ψ,C is a SCTR so v∘2v∈u∘1uΨc⊆Mc. Since M,C is a soft semiprime ideal, v∘2v∈Mc implies v∈Mc. Thus, uΨc⊆Mc. Hence, u∈Ψ¯Mc.Theorem 16.
LetΨ,C be a SCTR over Θ1×Θ2 w.r.t the aftersets. If M,C is a soft primary ideal of Θ2, then M,C is a GLWS primary ideal of Θ1 w.r.t the aftersets.Proof.
Suppose thatM,C is a soft primary ideal of Θ2 and ∅≠Ψ¯Mc for any c∈C. Then, by Theorem 4.19., M,C is a GLWS ideal of Θ1. Let u1,u2∈Θ1 such that u1∘1u2∈Ψ¯Mc and u1∉Ψ¯Mc. Then, u1∘1u2Ψc⊆Mc. Let v∈u1∘1u2Ψc. Since Ψ,C is a SCTR, v∈u1Ψc∘2u2Ψc. Thus, v=c∘2d for some c∈u1Ψc and d∈u2Ψc. Thus, dn∈u2nΨc for some n∈ℕ. Also, c∘2d∈Mc. As M,C is a soft primary ideal, c∉Mc and dn∈Mc. Thus, u2nΨc⊆Mc. Hence, u2n∈Ψ¯Mc for some n∈ℕ.Remark 3.
One can find examples like Example2 to show that converse of Theorems 11 to 16 is not true.


## 5. Relationship between Soft Quantale Homomorphism and Their Approximation

In this section, we define soft weak quantale homomorphism (SWQH), and then, we established the relationship between homomorphic images and their approximation by SBR.Definition 15 (see [4]).
A functionη:Θ1⟶Θ2 is called weak quantale homomorphism (WQH) if ηp∘1q=ηp∘2ηq and ηp∨q=ηp∨ηq, where Θ1,∘1 and Θ2,∘2 are quantales. If η is one-one, then η is monomorphism. If η is onto, then η is called epimorphism, and if η is bijective, then η is called isomorphism between Θ1,∘1 and Θ2,∘2.Definition 16.
LetH,C1 be a soft quantale over Θ1 and F,C2 be a soft quantale over Θ2. Then, H,C1 is said to soft weak homomorphic to F,C2 if there exist ordered pair of functions η,ζ satisfies the following(1)
η:Θ1⟶Θ2 is onto WQH, that is, ηp∘1q=ηp∘2ηq and ηp∨q=ηp∨ηq(2)
ζ:C1⟶C2 is surjective(3)
ηHc1=Fζc1, ∀c1∈C1
The ordered pairη,ζ of functions is SWQH. If in ordered pair η,ζ both η and ζ are one-to-one functions, then H,C1 is said to soft weak isomorphic to F,C2 and η,ζ is called SWQI.Lemma 1.
LetH,C1 be soft weak homomorphic to F,C2 with SWQH η,ζ. Let Ψ2,C3 be a SBR over Θ2 and H1,C1′⊆H,C1. Define Ψ1c3=x,y∈Θ1×Θ1:ηx,ηy∈Ψ2c3 be a SBR over Θ1. Then, the following holds:(1)
Ψ1,C3 is SCPR if Ψ2,C3 is SCPR(2)
Ifη,ζ is SWQI and Ψ2,C3 is SCPR w.r.t the aftersets (w.r.t the foresets), then Ψ1,C3 is SCPR w.r.t the aftersets (w.r.t the foresets)(3)
ηΨ¯1H1c3=Ψ¯2ηH1c3(4)
ηΨ¯1H1c3⊆Ψ¯2ηH1c3 and if η,ζ is SWQI, then ηΨ¯1H1c3=Ψ¯2ηH1c3(5)
Letη,ζ be a SWQI. Then, ηx∈ηΨ¯1H1c3⇔x∈Ψ¯1H1c3 and ηx∈ηΨ¯1H1c3⇔x∈Ψ¯1H1c3Proof.
(1)
and (2) are obvious(3)
SupposeH1,C1′⊆H,C1 and for any c3∈C3, z∈ηΨ¯1H1c3 for some z∈Θ2. Then, there exist a∈Θ1 such that a∈Ψ¯1H1c3 and ηa=z. Thus, x∈aΨ1c3∩H1c1′. So, a,x∈Ψ1c3 and x∈H1c1′. Thus, ηa,ηx∈Ψ2c3, that is, ηx∈ηaΨ2c3. Also, ηx∈ηH1c1′. So, ηaΨ2c3∩ηH1c1′≠∅. This implies that ηa∈Ψ¯2ηH1c3. Hence, ηΨ¯1H1c3⊆Ψ¯2ηH1c3.Now, letw∈Ψ¯2ηH1c3. Then, wΨ2c3∩ηH1c1′≠∅. This implies that y∈wΨ2c3∩ηH1c1′. Thus, y∈wΨ2c3 and y∈ηH1c1′. This implies that there exists x∈H1c1′⊆Θ1 and x1∈Θ1 such that ηx=y and ηx1=w. So, w,y=ηx1,ηx∈Ψ2c3. This implies that x1,x∈Ψ1c3. So, x∈x1Ψ1c3∩H1c1′. Thus, x1∈Ψ¯1H1c3. So, w=ηx1∈ηΨ¯1H1c3. Hence, Ψ¯2ηH1c3⊆ηΨ¯1H1c3. Consequently, ηΨ¯1H1c3=Ψ¯2ηH1c3.(4)
SupposeH1,C1′⊆H,C1 and for any c3∈C3, z∈ηΨ¯1H1c3 for some z∈Θ2. Then, there exist a∈Θ1 such that a∈Ψ¯1H1c3 and ηa=z. Thus, aΨ1c3⊆H1c1′. Let x∈zΨ2c3. Then, there exist y∈Θ1 such that ηy=x. So, ηy∈ηaΨ2c3, that is, ηa,ηy∈Ψ2c3. So, a,y∈Ψ1c3, that is, y∈aΨ1c3⊆H1c1′. Thus, ηy∈ηH1c1′. So, ηaΨ2c3⊆ηH1c1′. Thus, z=ηa∈Ψ¯2ηH1c3. Hence, ηΨ¯1H1c3⊆Ψ¯2ηH1c3.Now, letz∈Ψ¯2ηH1c3. Then, there exist unique a∈Θ1 such that ηa=z and ηaΨ2c3⊆ηH1c1′. Let x∈aΨ1c3, that is, a,x∈Ψ1c3. Then, ηa,ηx∈Ψ2c3. Then, ηx∈ηaΨ2c3⊆ηH1c1′. So, ηx∈ηH1c1′. This implies that x∈H1c1′. So, aΨ1c3⊆H1c1′. Then, a∈Ψ¯1H1c3. So, z=ηa∈ηΨ¯1H1c3. Hence, Ψ¯2ηH1c3⊆ηΨ¯1H1c3. Consequently, ηΨ¯1H1c3=Ψ¯2ηH1c3.(6)
Letx∈Ψ¯1H1c3 for any c3∈C3. Then, ηx∈ηΨ¯1H1c3. Conversely, suppose that ηx∈ηΨ¯1H1c3. As η is bijection so x∈Ψ¯1H1c3. Similarly, we can show that ηx∈ηΨ¯1H1c3⇔x∈Ψ¯1H1c3.Remark 4.
With a similar technique, Lemma1 can be proved but for the foresets.Theorem 17.
LetH,C1 be soft weak isomorphic to F,C2 with SWQI η,ζ. Let Ψ2,C3 be a SCPR over Θ2 and H1,C1′⊆H,C1. Define Ψ1c3=x,y∈Θ1×Θ1:ηx,ηy∈Ψ2c3 for any c3∈C3. Then, the following holds:(1)
Ψ¯1H1c3 is an ideal of Θ1 iff Ψ¯2ηH1c3 is an ideal of Θ2 for all c3∈C3(2)
Ψ¯1H1c3 is a subquantale of Θ1 iff Ψ¯2ηH1c3 is a subquantale of Θ2 for all c3∈C3(3)
Ψ¯1H1c3 is a prime ideal of Θ1 iff Ψ¯2ηH1c3 is a prime ideal of Θ2 for all c3∈C3(4)
Ψ¯1H1c3 is a semiprime ideal of Θ1 iff Ψ¯2ηH1c3 is a semiprime ideal of Θ2 for all c3∈C3(5)
Ψ¯1H1c3 is a primary ideal of Θ1 iff Ψ¯2ηH1c3 is a primary ideal of Θ2 for all c3∈C3Proof.
(1)
LetΨ¯1H1c3 be an ideal of Θ1 for any c3∈C3. We will show that Ψ¯2ηH1c3 is an ideal of Θ2. By Lemma 1 (3), we have ηΨ¯1H1c3=Ψ¯2ηH1c3.
Letp,q∈ηΨ¯1H1c3. Then, there exist u,v∈Ψ¯1H1c3 such that ηu=p and ηv=q. Since Ψ¯1H1c3 is ideal and η,ζ is SWQI so p∨q=ηu∨ηv=ηu∨v∈ηΨ¯1H1c3.
Now, letp,q∈Θ2 such that p≤q and q∈ηΨ¯1H1c3. Then, there exist u∈Θ1 and v∈Ψ¯1H1c3 such that ηu=p and ηv=q. So, ηu≤ηv implies ηu∨v=ηu∨ηv=ηv∈ηΨ¯1H1c3. This implies that u∨v=v∈Ψ¯1H1c3. This implies that u≤v and Ψ¯1H1c3 are ideal so u∈Ψ¯1H1c3. Thus, ηu=p∈ηΨ¯1H1c3.
Finally, letp∈Θ2 and q∈ηΨ¯1H1c3. Then, there exist u∈Θ1 and v∈Ψ¯1H1c3 such that ηu=p and ηv=q. Since Ψ¯1H1c3 ideal, u∘1v∈Ψ¯1H1c3. Thus, ηu∘1v=ηu∘2ηv=p∘2q∈ηΨ¯1H1c3. Similarly, q∘2p∈ηΨ¯1H1c3. Hence, Ψ¯2ηH1c3 is ideal of Θ2.
Conversely, suppose thatΨ¯2ηH1c3 = ηΨ¯1H1c3 be an ideal of Θ2 for any c3∈C3. We will show that Ψ¯1H1c3 is ideal of Θ1.
Letu,v∈Ψ¯1H1c3. Then, ηu,ηv∈ηΨ¯1H1c3. Since ηΨ¯1H1c3 is ideal so ηu∨v=ηu∨ηv∈ηΨ¯1H1c3. Then, by Lemma 5.2(5), u∨v∈Ψ¯1H1c3.
Now, letu,v∈Θ1 such that u≤v and v∈Ψ¯1H1c3. Then, u∨v=v∈Ψ¯1H1c3. Thus, ηu∨v=ηu∨ηv=ηv∈ηΨ¯1H1c3. This implies that ηu≤ηv. Since ηΨ¯1H1c3 is ideal ηu∈ηΨ¯1H1c3. Then, by Lemma 5.2(5), u∈Ψ¯1H1c3. Finally, let u∈Θ1 and v∈Ψ¯1H1c3. Then, ηu∈Θ2 and ηv∈ηΨ¯1H1c3. Since ηΨ¯1H1c3 is ideal, ηu∘2ηv∈ηΨ¯1H1c3, that is, ηu∘1v∈ηΨ¯1H1c3. Thus, u∘1v∈Ψ¯1H1c3. In a similar way, we can show that v∘1u∈Ψ¯1H1c3. This completes the proof.
The proof of (2)–(5) is similar to the proof of (1).Remark 5.
Theorem17with a similar technique can be proved but for the foresets.
With the same arguments, the next Theorem18 can be achieved.Theorem 18.
LetH,C1 be soft weak isomorphic to F,C2 with SWQI η,ζ. Let Ψ2,C3 be a SCTR over Θ2 and H1,C1′⊆H,C1. Define Ψ1c3=x,y∈Θ1×Θ1:ηx,ηy∈Ψ2c3 for any c3∈C3. Then, the following holds:(1)
Ψ¯1H1c3 is an ideal of Θ1 iff Ψ¯2ηH1c3 is an ideal of Θ2 for all c3∈C3(2)
Ψ¯1H1c3 is a subquantale of Θ1 iff Ψ¯2ηH1c3 is a subquantale of Θ2 for all c3∈C3(3)
Ψ¯1H1c3 is a prime ideal of Θ1 iff Ψ¯2ηH1c3 is a prime ideal of Θ2 for all c3∈C3(4)
Ψ¯1H1c3 is a semiprime ideal of Θ1 iff Ψ¯2ηH1c3 is a semiprime ideal of Θ2 for all c3∈C3(5)
Ψ¯1H1c3 is a primary ideal of Θ1 iff Ψ¯2ηH1c3 is a primary ideal of Θ2 for all c3∈C3


## 6. Comparison

Yang and Xu [7] introduced rough approximations in quantale which is a kind of partially ordered algebraic structure with an associative binary operation. The main idea of work in [7] is based on equivalence relation equipped with congruence relation in quantale. In fact, the generalization of Pawlak’s space is discussed in [7]. Further approximation of fuzzy substructures of quantale in crisp atmospheric space was discussed in [4]. Sometimes, it is difficult to find out an equivalence relation and then congruence while finding rough substructures in quantale. To remove this hurdle, soft binary relations are utilized in this paper. Since suitable soft binary relations are easy to find out, it is an easy approach to apply soft rough properties to approach different characterizations of soft rough structures in quantale with the help of aftersets and foresets.


## 7. Conclusion

The new combined effect of an algebraic structure quantale with rough and soft sets is presented by using soft binary relation, in this paper. The soft substructures of quantales like soft subquantale and soft ideal are discussed. The approximation w.r.t aftersets and foresets of these substructures by SBR which is an extended notion of Pawlak’s rough approximation space are presented. The more generalized version of approximation space implied from SBR overΘ1×Θ2 is employed. This new relation over Θ1×Θ2 enables us to use the concept of aftersets and foresets to express the lower and upper approximation. Important results regarding to the approximation of soft substructures of quantales under SBR with some essential algebraic conditions such as compatible and complete relations are introduced. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Crux of these results is that whenever we approximate a soft algebraic structure of quantale, corresponding upper and lower approximations, are again the same kind of soft algebraic structure. Furthermore, we presented the soft quantale homomorphism and established the relationship of soft homomorphic images with their approximation under SBR.In future, one can use this work and generalize it to different soft algebraic structures such as soft quantale modules, soft hypergroups, soft hyperquantales, and soft hyperrings. One can take motivation from our generalized approximation space and define new approximation spaces.



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