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.

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 Section 2. 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

Let Θ be a nonempty finite set called the universe set and Ψ be an E.R (equivalence relation) over Θ. Let [q]_Ψ 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 − L_appr) of R, and the second is called Ψ-upper approximation (Ψ − U_appr). They are defined as follows:

(1)

(2)

The Ψ − L_appr of R in Θ is the greatest definable in Θ contained in R. The Ψ − U_appr of R in Θ is the least definable set in Θ containing R. For any nonempty subset R in Θ, is called rough set with respect to Ψ or simply a Ψ-rough subset of P(Θ) × P(Θ) if , where P(Θ) denotes the set of all subsets of Θ.

Definition 1 (see [31].)Let Θ be a complete lattice. Define an associative binary relation ∘ on Θ satisfying

(3)

∀ l, w, l_i, w_i ∈ Θ. Then, (Θ, ∘) is called quantale.

Let T₁, T₂, T_I ⊂ Θ, i ∈ I. We define some notions as follows:

(4)

Throughout the paper, quantales are denoted by Θ₁ 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∈Iw_i, ∈W, ∀w_i, ∈W.

That is, Θ closed under ∘ and arbitrary supremum.

Definition 2 (see [32].)Let Θ 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 ∈ I(u^°q ∈ I)

A nonempty subset I⊆Θ is called ideal of Θ if it is left as well as right ideal.

Example 1. Let Θ={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.

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

Illustration of Θ.

Table 1. Binary operation subject to Θ

∘₁	p	q	r	1
0	p	q	r	1
p	p	q	r	1
q	p	q	r	1
r	p	q	r	1
1	p	q	r	1

Definition 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 pair (Ψ, C) is called a soft set over Θ if Ψ : C⟶P(Θ) where C is a subset of E (the set of parameters).

Definition 5 (see [16].)Let (F, 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)
F(c)⊆H(c), ∀ c ∈ C₁

Definition 6 (see [33].)Let (Ψ, C) be a soft set over Θ × Θ, that is, Ψ : C⟶P(Θ × 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⟶P(Θ₁ × Θ₂).

Definition 7. Let (Ψ, C) be a soft set over quantale Θ. Then,

(1)
(Ψ, C) is called soft subquantale over Θ iff Ψ(c) is a subquantale of Θ, ∀ c ∈ C
(2)
(Ψ, C) is called soft ideal over Θ iff Ψ(c) is an ideal of Θ, ∀ c ∈ C
(3)
(Ψ, C) is called soft prime ideal over Θ iff Ψ(c) is a prime ideal of Θ, ∀ c ∈ C
(4)
(Ψ, C) is called soft semiprime ideal over Θ iff Ψ(c) is a semiprime ideal of Θ, ∀ c ∈ C
(5)
(Ψ, C) is called soft primary ideal over Θ iff Ψ(c) 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 quantale Θ by SBR. We utilized aftersets and foresets to approximate soft sets.

Definition 8 (see [34].)Let (Ψ, C) be a SBR over Θ₁ × Θ₂, where C⊆E (parametric set). Then, Ψ : C⟶P(Θ₁ × Θ₂). For a soft set (F, C) over Θ₂, the L_appr and U_appr of (F, C) w.r.t the afterset are essentially two soft sets over Θ₁, which is defined as

(5)

(6)

And for a soft set (H, C) over Θ₁, the L_appr and U_appr of (H, C) w.r.t the foreset are actually two soft sets over Θ₂, which is defined as

(7)

For all c ∈ C, where q₁Ψ(c)={q₂ ∈ Θ₂ : (q₁, q₂) ∈ Ψ(c)} is called the afterset of q₁ and Ψ(c)q₂={q₁ ∈ Θ₁ : (q₁, q₂) ∈ Ψ(c)} is called the foreset of q₂.

Remark 1.

(1)
For each soft set (F, C) over Θ₂, and
(2)
For each soft set (H, C) over Θ₁, and

Definition 9. Let (Ψ, C) be a SBR over Θ₁ × Θ₂, that is, Ψ : C⟶P(Θ₁ × Θ₂). Then, (Ψ, C) is called soft compatible relation (SCPR) if for all p, r, j_i ∈ Θ₁ and q, s, k_i ∈ Θ₂(i ∈ I), we have

(1)
(p, q), (r, s) ∈ Ψ(c)⇒(p∘₁r, q∘₂s)∈Ψ(c)
(2)
(j_i, k_i) ∈ Ψ(c)⇒(∨_i∈Ij_i, ∨_i∈Ik_i)∈Ψ(c)

for every c ∈ C.

Definition 10. A SCPR (Ψ, C) over Θ₁ × Θ₂ is called soft complete relation (SCTR) with respect to the afterset if, for all p, r, ∈ Θ₁, we have

(1)
pΨ(c)∨rΨ(c)=(p∨r)Ψ(c)
(2)
pΨ(c)∘₂rΨ(c)=(p∘₁r)Ψ(c)

for all c ∈ C.

A SCPR (Ψ, 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 SCPR (Λ, 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)
Λ(c)q∨Λ(c)s=Λ(c)(q∨s)
(2)
Λ(c)q∘₁Λ(c)s=Λ(c)(q∘₂s)

for all c ∈ C.

A SCPR (Λ, C) is called ∨-complete w.r.t the foresets if it satisfies only condition (1).

A SCPR (Λ, C) is called °-complete w.r.t the foresets if it satisfies only condition (2).

Theorem 1. Let (Ψ, 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

Proof. For arbitrary c ∈ C, let Then, x=y₁∘₁y₂ for some and . This implies that y₁Ψ(c)∩F₁(c) ≠ ∅ and y₂Ψ(c)∩F₂(c) ≠ ∅, so there exist elements l, m ∈ Θ₂ such that l ∈ y₁Ψ(c)∩F₁(c) and m ∈ y₂Ψ(c)∩F₂(c). Thus, l ∈ y₁Ψ(c), m ∈ y₂Ψ(c), l ∈ F₁(c) and m ∈ F₂(c). So (y₁, l) ∈ Ψ(c) and (y₂, m) ∈ Ψ(c) imply (y₁∘₁y₂, l∘₂m) ∈ Ψ(c); that is, (l∘₂m) ∈ (y₁∘₁y₂)Ψ(c). Also, l∘₂m∈F₁(c)∘₂F₂(c); therefore, l∘₂m ∈ y₁∘₁y₂Ψ(c)∩F₁(c)∘₂F₂(c). This shows that

Now, for arbitrary c ∈ C, let Then, x=y₁∨y₂ for some and . This implies that y₁Ψ(c)∩F₁(c) ≠ ∅ and y₂Ψ(c)∩F₂(c) ≠ ∅, so there exist elements l, m ∈ Θ₂ such that l ∈ y₁Ψ(c)∩F₁(c) and m ∈ y₂Ψ(c)∩F₂(c). Thus, l ∈ y₁Ψ(c), m ∈ y₂Ψ(c), l ∈ F₁(c), and m ∈ F₂(c). So (y₁, l) ∈ Ψ(c) and (y₂, m) ∈ Ψ(c) imply (y₁∨y₂, l∨m) ∈ Ψ(c); that is, (l∨m) ∈ (y₁∨y₂)Ψ(c). Also, l∨m ∈ F₁(c)∨F₂(c); therefore, l∨m ∈ y₁∨y₂Ψ(c)∩F₁(c)∨F₂(c). This shows that

Theorem 2. Let (Ψ, 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

Proof. The proof is simple.

Theorem 3. Let (Ψ, 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

Proof. For arbitrary c ∈ C, if at least one of and is empty, then (1) is obvious. Now, for arbitrary c ∈ C, consider that and . Then, . So, let . Then, x=y₁°₁y₂ for some and . This implies that ∅≠y₁Ψ(c)⊆F₁(c) and ∅≠y₂Ψ(c)⊆F₂(c). As (y₁°₁y₂)Ψ(c)=y₁Ψ(c)°₂Ψ(c)⊆F₁(c)°₂F₂(c). This shows that . Hence, (1) is proved.

For arbitrary c ∈ C, if at least one of and is empty, then (2) is obvious. Now, for arbitrary c ∈ C, consider that and . Then, . So, let . Then, x=y₁∨y₂ for some and . This implies that ∅≠y₁Ψ(c)⊆F₁(c) and ∅≠y₂Ψ(c)⊆F₂(c). As (y₁∨y₂)Ψ(c)=y₁Ψ(c)∨y₂Ψ(c)⊆F₁(c)∨F₂(c). This shows that . Hence, (2) is proved.

Theorem 4. Let (Ψ, 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

Proof. The proof is obvious.

4. Approximation of Soft Substructures in Quantales

In this section, we consider two quantales Θ₁ and Θ₂ and approximate different soft substructures of quantales by using different SBR over Θ₁ × Θ₂. We will show that U_appr 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 L_appr 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 consider (Ψ, C) to be the SBR over Θ₁ × Θ₂ and xΨ(c) ≠ ∅ for all x ∈ Θ₁, c ∈ C, and Ψ(c)y ≠ ∅ for all y ∈ Θ₂, c ∈ C unless otherwise specified.

Definition 11. Let (Ψ, C) be a SBR over Θ₁ × Θ₂ and (F, C) be a soft set over Θ₂. If U_appr. is a soft subquantale of Θ₁, then (F, C) is called generalized upper soft (GU_pS) subquantale of Θ₁ w.r.t the aftersets. If U_appr is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₁, then (F, C) is called GU_pS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₁ w.r.t the aftersets.

Definition 12. Let (Ψ, C) be a SBR over Θ₁ × Θ₂ and (L, C) be a soft set over Θ₁. If U_appr is a soft subquantale of Θ₂, then (L, C) is called generalized upper soft (GU_pS) subquantale of Θ₂ w.r.t the foresets. If U_appr is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₂, then (L, C) is called GU_pS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₂ w.r.t the foresets.

Theorem 5. Let (Ψ, C) be a SCPR over Θ₁ × Θ₂. If (F, C) is a soft subquantale of Θ₂, then (F, C) is a GU_pS subquantale of Θ₁ w.r.t the aftersets.

Proof. Suppose that (F, C) is a soft subquantale, then for any c ∈ C. Let , i ∈ I. Then, p_iΨ(c)∩F(c) ≠ ∅. So, there exists q_i ∈ p_iΨ(c)∩F(c). Thus, q_i ∈ p_iΨ(c) and q_i ∈ F(c) since (Ψ, C) is a SCPR. Therefore, (p_i, q_i) ∈ Ψ(c), i ∈ I implies (∨_i∈Ip_i, ∨_i∈Iq_i) ∈ Ψ(c). This implies that ∨_i∈Iq_i ∈ ∨_i∈Ip_iΨ(c). Also, ∨_i∈Iq_i ∈ F(c) (as (F, C) is a soft subquantale). So, ∨_i∈Iq_i ∈ ∨_i∈Ip_i ∈ Ψ(c)∩F(c). Hence, .

Let . Then, p₁Ψ(c)∩F(c) ≠ ∅ and . So, there exists q₁ ∈ p₁Ψ(c)∩F(c) and q₂ ∈ p₂Ψ(c)∩F(c). Thus, q₁ ∈ p₁Ψ(c), q₁ ∈ F(c), q₂ ∈ p₂Ψ(c), and q₂ ∈ F(c) since (Ψ, C) is a SCPR. Therefore, (p₁, q₁), (p₂, q₂) ∈ Ψ(c) implies (p₁°₁q₁), (p₂°₂q₂) ∈ Ψ(c). This implies that q₁°₂q₂ ∈ p₁°p₂Ψ(c). Also, q₁°₂q₂ ∈ (c) (as (F, C) is a soft subquantale). So, q₁°₂q₂ ∈ p₁°p₂Ψ(c)∩F(c). Hence, . This completes the proof.

With the same arguments, the next Theorem 6 can be achieved.

Theorem 6. Let (Ψ, C) be a SCPR over Θ₁ × Θ₂. If (L, C) is a soft subquantale of Θ₁, then (L, C) is a GU_pS subquantale of Θ₂ w.r.t the foresets.

Theorem 7. Let (Ψ, 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 GU_pS left (right) ideal of Θ₁ w.r.t the aftersets.

Proof. Suppose that (F, C) is a soft left ideal of Θ₂, then for any c ∈ C. Let . Then, u₁Ψ(c)∩F(c) ≠ ∅ and u₂Ψ(c)∩F(c) ≠ ∅. So, there exists v₁ ∈ u₁Ψ(c)∩F(c) and v₂ ∈ u₂Ψ(c)∩F(c). Thus, v₁ ∈ u₁Ψ(c), v₁ ∈ F(c), v₂ ∈ u₂Ψ(c), and v₂ ∈ F(c) since (Ψ, C) is a SCPR. Therefore, (u₁∨u₂, v₁∨v₂) ∈ Ψ(c); that is, v₁∨v₂ ∈ (u₁∨u₂)Ψ(c). Also, v₁∨v₂ ∈ F(c) (as (F, C) is a soft left ideal). So, v₁∨v₂ ∈ (u₁∨u₂)Ψ(c)∩F(c). Hence, .

Now, let u₁, u₂ ∈ Θ₁ such that u₁ ≤ u₂ and . So, . Since , so there exist v₂ ∈ u₂Ψ(c)∩F(c). Thus, v₂ ∈ u₂Ψ(c) and v₂ ∈ F(c). Since (Ψ, C) is a soft ∨-complete relation, therefore, v₂ ∈ u₂Ψ(c)=u₁∨u₂Ψ(c)=u₁Ψ(c)∨u₂Ψ(c). This implies that v₂=s∨t, for some s ∈ u₁Ψ(c) and t ∈ u₂Ψ(c). Thus, s ≤ v₂ and v₂ ∈ F(c) imply s ∈ F(c) (as F(c) is ideal). So, s ∈ u₁Ψ(c)∩F(c). Hence, .

Let p, x ∈ Θ₁ and . Then, xΨ(c)∩F(c) ≠ ∅. So, there exist q ∈ xΨ(c)∩F(c). Thus, q ∈ xΨ(c) and q ∈ F(c). Since (F, C) is a soft left ideal so, y∘₂q ∈ F(c) for any y ∈ pΨ(c)⊆Θ₂. This implies that (p, y) ∈ Ψ(c). So, (p∘₁x, y∘₂q) ∈ Ψ(c); that is, y∘₂q ∈ p∘₁xΨ(c). So, y∘₂q ∈ p∘₁xΨ(c)∩F(c). Hence, . Similarly, we can show that .

Theorem 8. Let (Ψ, 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 GU_pS prime ideal of Θ₁ w.r.t the aftersets.

Proof. Assume that (F, C) is a soft prime ideal of Θ₂, then for any c ∈ C. Then, by Theorem 5, (F, C) is generalized upper soft ideal of Θ₁. Let p₁, p₂ ∈ Θ₁ such that . Then, (p₁∘₁p₂)Ψ(c)∩F(c) ≠ ∅. So, there exist q ∈ (p₁∘₁p₂)Ψ(c)∩F(c). This implies that q ∈ (p₁∘₁p₂)Ψ(c) and q ∈ F(c). Since (Ψ, C) is a SCTR, q ∈ (p₁∘₁p₂)Ψ(c)=p₁Ψ(c)∘₂p₂Ψ(c). Thus, q=c∘₂d for some c ∈ p₁Ψ(c) and d ∈ p₂Ψ(c). Thus, c∘₂d ∈ F(c) and (F, C) is a soft prime ideal of Θ₂ so, c ∈ F(c) or d ∈ F(c). Thus, c ∈ p₁Ψ(c)∩F(c) or d ∈ p₂Ψ(c)∩F(c). Hence, or .

Theorem 9. Let (Ψ, 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 GU_pS semiprime ideal of Θ₁ w.r.t the aftersets.

Proof. Assume that (F, C) is a soft semiprime ideal of Θ₂, then for any c ∈ C. Then, by Theorem 5, (F, C) is generalized upper soft ideal of Θ₁. Let p₁ ∈ Θ₁ such that . Then, (p₁∘₁p₁)Ψ(c)∩F(c) ≠ ∅. So, there exist q ∈ (p₁∘₁p₁)Ψ(c)∩F(c). This implies that q ∈ (p₁∘₁p₁)Ψ(c) and q ∈ F(c). Since (Ψ, C) is a SCTR, q ∈ (p₁∘₁p₁)Ψ(c)=p₁Ψ(c)∘₂ p₁Ψ(c). Thus, q=c ∘₂ c for some c ∈ p₁Ψ(c). Thus, c∘₂c ∈ F(c) and (F, C) is a soft semiprime ideal of Θ₂ so, c ∈ F(c). Thus, c ∈ p₁Ψ(c)∩F(c). Hence, .

Theorem 10. Let (Ψ, 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 GU_pS primary ideal of Θ₁ w.r.t the aftersets.

Proof. Assume that (F, C) is a soft primary ideal of Θ₂, then 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∈ and . Then, (p₁∘₁p₂)Ψ(c)∩F(c) ≠ ∅. So, there exist q ∈ (p₁∘₁p₂)Ψ(c)∩F(c). This implies that q ∈ (p₁∘₁p₂)Ψ(c) and q ∈ F(c). Since (Ψ, C) is a SCTR, q ∈ (p₁∘₁p₂)Ψ(c)=p₁Ψ(c)∘₂p₂Ψ(c). Thus, q=c∘₂d for some c ∈ p₁Ψ(c) and d ∈ p₂Ψ(c). Thus, c∘₂d ∈ F(c) and (F, C) is a soft primary ideal of Θ₂ so dⁿ ∈ F(c) for some n ∈ ℕ. Also, for n ∈ ℕ. Thus, . Hence, .

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. Let Θ₁={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)
Let C={c₁, c₂} and define SBR (Ψ, C) over Θ₁ × Θ₂ by the rule
(8)
Then, (Ψ, C) is SCPR. The aftersets with respect to Ψ(c₁) and Ψ(c₂) are given as follows:
(9)
Define soft set (F, C) over Θ₂ by the rule
(10)
Then, (F, C) is not a soft subquantale of Θ₂. But and are subquantale of Θ₁. So (F, C) is a GU_pSS_Θ of Θ₁ w.r.t the aftersets.
Foresets with respect to Ψ(c₁) and Ψ(c₂) are given as follows:
(11)
Define soft set (L, C) over Θ₁ by the rule
(12)
Then, (L, C) is not a soft subquantale of Θ₁. But and are subquantale of Θ₂. So, (L, C) is a GU_pS subquantale of Θ₂ w.r.t the foresets.
(2)
Now, let C={c₁, c₂} and define SBR (Ψ, C) over Θ₁ × Θ₂ by the rule

(13)

(14)

Aftersets with respect to Ψ(c₁) and Ψ(c₂) are given as follows:

(15)

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)

Then, (F, C) is not a soft ideal of Θ₂. But and are ideal of Θ₁. So, (F, C) is a GU_pS ideal of Θ₁ w.r.t the aftersets.

Now, define SBR (Ψ, C) over Θ₁ × Θ₂ by the rule

(17)

(18)

Foresets with respect to Ψ(c₁) and Ψ(c₂) are given as follows:

(19)

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)

Then, (L, C) is not a soft ideal of Θ₁. But and are ideal of Θ₂. So, (L, C) is a GU_pS ideal of Θ₂ w.r.t the foresets.

Similar examples can be presented to justify that converse of Theorems 11 to 13 is not true.

Table 2. Binary operation subject to Θ₁.

O₁	p	q	1
0	0	0	0
p	p	0	p
q	0	q	q
1	p	q	1

Table 3. Binary operation subject to Θ₂.

O₂	0^′	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. Let (Ψ, C) be a SBR over Θ₁ × Θ₂. Consider the soft set (M, C) over Θ₂, if L_appr is a soft subquantale of Θ₁, then (M, C) is called generalized lower soft (GL_WS) subquantale of Θ₁ w.r.t the aftersets. If L_appr is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₁, then (M, C) is called GU_pS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₁ w.r.t the aftersets.

Definition 14. Let (Ψ, C) be a SBR over Θ₁ × Θ₂. Consider the soft set (L, C) over Θ₁, if L_appr is a soft subquantale of Θ₂, then (L, C) is called GL_WS subquantale of Θ₂ w.r.t the foresets. If L_appr is a soft ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₂, then (L, C) is called GU_pS ideal (prime ideal, semiprime ideal, and primary ideal) of Θ₂ w.r.t the foresets.

Theorem 11. Let (Ψ, 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 GL_WS subquantale of Θ₁ w.r.t the aftersets.

Proof. Suppose that (M, C) is a soft subquantale of Θ₂ and for any c ∈ C. Let . Then, u_iΨ(c)⊆M(c). Since (Ψ, C) is a SCTR, therefore, ∨_i∈I(u_iΨ(c))=(∨_i∈Iu_i)Ψ(c)⊆M(c). Hence, .

Now, let . Then, u₁Ψ(c)⊆M(c) and u₂Ψ(c)⊆M(c). Since (Ψ, C) is a SCTR and (M, C) is a soft subquantale, therefore, u₁Ψ(c)∘₂u₂Ψ(c)⊆M(c)∘₂M(c) implies (u₁∘₁u₂)Ψ(c)⊆M(c). Hence, .

With the same arguments, next Theorem 12 can be achieved.

Theorem 12. Let (Ψ, 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 GL_WS subquantale of Θ₂ w.r.t the foresets.

Theorem 13. Let (Ψ, 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 GL_WS ideal of Θ₁ w.r.t the aftersets.

Proof. Suppose that (M, C) is a soft ideal of Θ₂ and for any c ∈ C. Let . Then, u₁Ψ(c)⊆M(c) and u₂Ψ(c)⊆M(c). Since (Ψ, C) is a SCTR and (M, C) is a soft ideal of Θ₂ so u₁Ψ(c)∨u₂Ψ(c)=(u₁∨u₂)Ψ(c)⊆M(c)∨M(c); that is, (u₁∨u₂)Ψ(c)⊆M(c) Hence, .

Now, let u₁, u₂ ∈ Θ₁ such that u₁ ≤ u₂ and . So, . Let, v₁ ∈ u₁Ψ(c) and v₂ ∈ u₂Ψ(c)⊆M(c). So, v₁∨v₂ ∈ (u₁∨u₂)Ψ(c), that is, v₁∨v₂ ∈ u₂Ψ(c)⊆M(c). Since M(c) is ideal so v₁ ≤ v₁∨v₂ ∈ M(c) implies v₁ ∈ M(c). Thus, u₁Ψ(c)⊆M(c). Hence, .

Now, let u, y ∈ Θ₁ and . Then, ∅≠yΨ(c)⊆M(c). Consider v₁∈(u∘₁y)Ψ(c) since (Ψ, C) is a SCTR so v₁ ∈ u Ψ(c)∘₂yΨ(c). Thus, v₁=c∘₂d for some c ∈ uΨ(c) and d ∈ yΨ(c). But yΨ(c)⊆M(c) so d ∈ M(c) and (M, C) is a soft ideal of Θ₂; therefore, c∘₂d ∈ M(c), that is, v₁ ∈ M(c). Thus, (u∘₁y)Ψ(c)⊆M(c). Hence, . Similarly, we can show that .

Theorem 14. Let (Ψ, 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 GL_WS prime ideal of Θ₁ w.r.t the aftersets.

Proof. Assume that (M, C) is a soft prime ideal of Θ₂ and for any c ∈ C. Then, by Theorem 4.19, (M, C) is GL_WS ideal of Θ₁. Let u₁, u₂ ∈ Θ₁ such that . Then, (u₁∘₁u₂)Ψ(c)⊆M(c). Consider v ∈ (u₁∘₁u₂)Ψ(c)⊆M(c). Since (Ψ, C) is a SCTR, v ∈ (u₁∘₁u₂)Ψ(c)=u₁Ψ(c)∘₂u₂Ψ(c). Thus, v=c∘₂d for some c ∈ u₁Ψ(c) and d ∈ u₂Ψ(c). This implies that v=c∘₂d ∈ M(c). As (M, C) is a soft prime ideal so, c ∈ M(c) or d ∈ M(c). Thus, c ∈ u₁Ψ(c)⊆M(c) or d ∈ u₂Ψ(c)⊆M(c). Hence, or .

Theorem 15. Let (Ψ, 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 GL_WS semiprime ideal of Θ₁ w.r.t the aftersets.

Proof. Assume that (M, C) is a soft semiprime ideal of Θ₂ and for any c ∈ C. Then, by Theorem 14,(M, C) is GL_WS ideal of Θ₁. Let u ∈ Θ₁ such that . Then, (u∘₁u)Ψ(c)⊆M(c). Let v ∈ uΨ(c). As (Ψ, C) is a SCTR so v∘₂v ∈ (u∘₁u)Ψ(c)⊆M(c). Since (M, C) is a soft semiprime ideal, v∘₂v ∈ M(c) implies v ∈ M(c). Thus, uΨ(c)⊆M(c). Hence, .

Theorem 16. Let (Ψ, 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 GL_WS primary ideal of Θ₁ w.r.t the aftersets.

Proof. Suppose that (M, C) is a soft primary ideal of Θ₂ and for any c ∈ C. Then, by Theorem 4.19., (M, C) is a GL_WS ideal of Θ₁. Let u₁, u₂ ∈ Θ₁ such that and . Then, (u₁∘₁u₂)Ψ(c)⊆M(c). Let v ∈ (u₁∘₁u₂)Ψ(c). Since (Ψ, C) is a SCTR, v ∈ u₁Ψ(c)∘₂u₂Ψ(c). Thus, v=c∘₂d for some c ∈ u₁Ψ(c) and d ∈ u₂Ψ(c). Thus, for some n ∈ ℕ. Also, c∘₂d ∈ M(c). As (M, C) is a soft primary ideal, c ∉ M(c) and dⁿ ∈ M(c). Thus, . Hence, for some n ∈ ℕ.

Remark 3. One can find examples like Example 2 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 function η : Θ₁⟶Θ₂ is called weak quantale homomorphism (WQH) if η(p∘₁q)=η(p)∘₂η(q) and η(p∨q)=η(p)∨η(q), 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. Let (H, 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, η(p∘₁q)=η(p)∘₂η(q) and η(p∨q)=η(p)∨η(q)
(2)
ζ : C₁⟶C₂ is surjective
(3)
η(H(c₁))=F(ζ(c₁)), ∀c₁ ∈ C₁

The ordered pair (η, 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. Let (H, C₁) be soft weak homomorphic to (F, C₂) with SWQH (η, ζ). Let (Ψ₂, C₃) be a SBR over Θ₂ and . Define Ψ₁(c₃)={}(x, y)∈Θ₁ × Θ₁:(η(x), η(y)) ∈ Ψ₂(c₃){} be a SBR over Θ₁. Then, the following holds:

(1)
(Ψ₁, C₃) is SCPR if (Ψ₂, C₃) is SCPR
(2)
If (η, 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)
(4)
⊆ and if (η, ζ) is SWQI, then
(5)
Let (η, ζ) be a SWQI. Then, and ⇔

Proof.

(1)
and (2) are obvious
(3)
Suppose and for any c₃ ∈ C₃, for some z ∈ Θ₂. Then, there exist a ∈ Θ₁ such that and η(a)=z. Thus, . So, (a, x) ∈ Ψ₁(c₃) and . Thus, (η(a), η(x)) ∈ Ψ₂(c₃), that is, η(x) ∈ η(a)Ψ₂(c₃). Also, . So, η(a)Ψ₂(c₃)∩≠∅. This implies that . Hence, .
Now, let . Then, wΨ₂(c₃)∩. This implies that . Thus, y ∈ wΨ₂(c₃) and . This implies that there exists and x₁ ∈ Θ₁ such that η(x)=y and η(x₁)=w. So, (w, y)=(η(x₁), η(x))∈Ψ₂(c₃). This implies that (x₁, x) ∈ Ψ₁(c₃). So, . Thus, . So, . Hence, . Consequently, .
(4)
Suppose and for any c₃ ∈ C₃, for some z ∈ Θ₂. Then, there exist a ∈ Θ₁ such that and η(a)=z. Thus, . Let x ∈ zΨ₂(c₃). Then, there exist y ∈ Θ₁ such that η(y)=x. So, η(y) ∈ η(a)Ψ₂(c₃), that is, (η(a), η(y)) ∈ Ψ₂(c₃). So, (a, y) ∈ Ψ₁(c₃), that is, y ∈ aΨ₁(c₃). Thus, η(y)∈. So, η(a)Ψ₂(c₃)⊆. Thus, . Hence, .
Now, let . Then, there exist unique a ∈ Θ₁ such that η(a)=z and . Let x ∈ aΨ₁(c₃), that is, (a, x) ∈ Ψ₁(c₃). Then, (η(a), η(x)) ∈ Ψ₂(c₃). Then, . So, . This implies that . So, . Then, . So, . Hence, . Consequently, .
(6)
Let for any c₃ ∈ C₃. Then, . Conversely, suppose that . As η is bijection so . Similarly, we can show that ⇔.

Remark 4. With a similar technique, Lemma 1 can be proved but for the foresets.

Theorem 17. Let (H, C₁) be soft weak isomorphic to (F, C₂) with SWQI (η, ζ). Let (Ψ₂, C₃) be a SCPR over Θ₂ and . Define Ψ₁(c₃)={}(x, y)∈Θ₁ × Θ₁:(η(x), η(y))∈Ψ₂(c₃){} for any c₃ ∈ C₃. Then, the following holds:

(1)
is an ideal of Θ₁ iff is an ideal of Θ₂ for all c₃ ∈ C₃
(2)
is a subquantale of Θ₁ iff is a subquantale of Θ₂ for all c₃ ∈ C₃
(3)
is a prime ideal of Θ₁ iff is a prime ideal of Θ₂ for all c₃ ∈ C₃
(4)
is a semiprime ideal of Θ₁ iff is a semiprime ideal of Θ₂ for all c₃ ∈ C₃
(5)
is a primary ideal of Θ₁ iff is a primary ideal of Θ₂ for all c₃ ∈ C₃

Proof.

(1)
Let be an ideal of Θ₁ for any c₃ ∈ C₃. We will show that is an ideal of Θ₂. By Lemma 1 (3), we have

Let . Then, there exist such that η(u)=p and η(v)=q. Since is ideal and (η, ζ) is SWQI so p∨q=η(u)∨η(v)=η(u∨v)∈.

Now, let p, q ∈ Θ₂ such that p ≤ q and . Then, there exist u ∈ Θ₁ and such that η(u)=p and η(v)=q. So, η(u) ≤ η(v) implies η(u∨v)=η(u). This implies that . This implies that u ≤ v and are ideal so . Thus, .

Finally, let p ∈ Θ₂ and . Then, there exist u ∈ Θ₁ and such that η(u)=p and η(v)=q. Since ideal, . Thus, η(u∘₁v). Similarly, . Hence, is ideal of Θ₂.

Conversely, suppose that = be an ideal of Θ₂ for any c₃ ∈ C₃. We will show that is ideal of Θ₁.

Let . Then, η(u), η(v)∈. Since is ideal so . Then, by Lemma 5.2(5), .

Now, let u, v ∈ Θ₁ such that u ≤ v and . Then, . Thus, η(u∨v)=η(u)∨η(v)=η(v)∈. This implies that η(u) ≤ η(v). Since is ideal . Then, by Lemma 5.2(5), . Finally, let u ∈ Θ₁ and . Then, η(u) ∈ Θ₂ and . Since is ideal, , that is, . Thus, u∘₁v∈. In a similar way, we can show that . This completes the proof.

The proof of (2)–(5) is similar to the proof of (1).

Remark 5. Theorem 17with a similar technique can be proved but for the foresets.

With the same arguments, the next Theorem 18 can be achieved.

Theorem 18. Let (H, C₁) be soft weak isomorphic to (F, C₂) with SWQI (η, ζ). Let (Ψ₂, C₃) be a SCTR over Θ₂ and . Define Ψ₁(c₃)={}(x, y)∈Θ₁ × Θ₁:()η(x), η()y∈Ψ₂(c₃){} for any c₃ ∈ C₃. Then, the following holds:

(1)
is an ideal of Θ₁ iff is an ideal of Θ₂ for all c₃ ∈ C₃
(2)
is a subquantale of Θ₁ iff is a subquantale of Θ₂ for all c₃ ∈ C₃
(3)
is a prime ideal of Θ₁ iff is a prime ideal of Θ₂ for all c₃ ∈ C₃
(4)
is a semiprime ideal of Θ₁ iff is a semiprime ideal of Θ₂ for all c₃ ∈ C₃
(5)
is a primary ideal of Θ₁ iff 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 over Θ₁ × 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Open Research

Data Availability

No data were used to support this study.

References

Citing Literature

All articles

Soft Substructures in Quantales and Their Approximations Based on Soft Relations

Abstract

1. Introduction

2. Preliminaries

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

4. Approximation of Soft Substructures in Quantales

5. Relationship between Soft Quantale Homomorphism and Their Approximation

6. Comparison

7. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

Figures

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Soft Substructures in Quantales and Their Approximations Based on Soft Relations

Abstract

1. Introduction

2. Preliminaries

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

4. Approximation of Soft Substructures in Quantales

5. Relationship between Soft Quantale Homomorphism and Their Approximation

6. Comparison

7. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

Figures

References

Related

Information