A kind of new rough set: Rough soft sets and rough soft rings

Abstract

The aim of this paper is to lay a foundation for providing a rough soft tool in considering many problems that contain uncertainties. We put forward the concepts of rough soft rings and rough idealistic soft rings. Some basic operations on rough soft rings are discussed. Some good examples are explored. Special attention is paid to rough prime idealistic soft rings. Finally, we investigate some properties of products and sums of rough soft rings.

Keywords

Rough soft set rough soft ring congruence Pawlak approximation space rough prime idealistic soft ring

1 Introduction

It is well-known that rough set theory introduced by Pawlak [45, 46], a new mathematical approach to deal with uncertain knowledge, has recently received wide attention on the research clues in both of theory and applications. There are at least two methods for the development of this theory, the constructive and axiomatic approaches. In constructive methods, rough set theory renders a set-theoretic definition of knowledge based on equivalence relations.

Nowadays, rough set theory has been applied to many areas, such as knowledge discovery, machine learning, data analysis, approximate classification, conflict analysis, and so on, see [14 , 55]. We know the basic requirement of Pawlak’s approximations is an equivalent relation among the objects under consideration. Some time due to incomplete information such as equivalence relation is not easy to establish. Therefore, some more general models have been proposed, for example, see [6 , 60–66]. Some researchers applied this theory to algebraic structures. Kuroki [33] introduced the concept of rough ideals in a semigroup. Davvaz [15, 16] applied it to rings. Further, Kazanci et al. [30, 53] discussed this topic. The other algebraic structures, see [9 , 56].

In modern time, the mathematical modelling and manipulating of various types of uncertainties has become an increasingly important issue in solving complicated problems arisings in a wide rang of areas such as engineering, economics, environmental science, medicine and social sciences. Although probability theory, fuzzy set theory [57] and rough set theory [45] are well-known and effective tools for dealing with vagueness and uncertainty, each of them has certain inherent limitations, which are pointed out by Molodtsov [44]. Based on the reason, he proposed soft set theory, as a new mathematical approach to vagueness and uncertainty, which is free from the difficulty affecting the above-mentioned methods. Since then there has been a rapid growth of interest in soft sets and their various applications, such as [1–3 , 67].

Hybrid models combing fuzzy sets with rough sets have arisen in various guises in different setting. For instance, Dubois and Prade [20] investigated rough fuzzy sets and fuzzy rough set. Feng [23] combined fuzzy sets, rough sets and soft sets all together, which gave rise to some new concepts, such as rough soft sets, soft rough sets, soft rough sets, soft rough fuzzy sets. Maji et al. [40] proposed fuzzy soft set theory. Now, many researchers continued to study these topics, for examples, see [4 , 59]. In particular, Zhan investigated the ideal theory on hemirings, most of relevant conclusions have been already demonstrated in Zhan’s book, which is referredto [58].

In this paper, we investigate rough soft rings. This paper is organized as follows: In Section 2, we recall some concepts and results on rough sets and soft sets. In Section 3, we put forward the concept of rough soft rings. In Section 4, we investigate some characterizations of rough idealistic soft rings. Finally, we investigate some properties of products and sums of rough soft rings in Section 5.

2 Preliminaries

In this section, we recall some basis notions relevant to soft sets and rough sets. Mododtsov [44] defined the notion of a soft set as follows: Let U be an initial universe and E a set of parameters. The power set of U is denoted by P (U) and A ⊆ U.

Definition 2.1. [44] A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U).

Definition 2.2. [7] Let (F, A) and (G, B) be two soft sets over U.

(1) The extended intersection of (F, A) and (G, B), denoted by (F, A) ⊓ _ɛ (F, B), is defined as the soft set (H, C), where C = A ∪ B, and ∀e ∈ C, $\begin{matrix} H (e) = {\begin{matrix} F (e) if e \in A ∖ B, \\ G (e) if e \in B ∖ A, \\ F (e) \cap G (e) if e \in A \cap B . \end{matrix} \end{matrix}$

(2) The restricted intersection of (F, A) and (G, B), denoted by (F, A) Cap (F, B), is defined as the soft set (H, C), where C = A ∩ B and H (c) = F (c) ∩ G (c) for all c ∈ C.

(3) The extended union of (F, A) and (G, B), denoted by $(F, A) \tilde{\cup} (F, B)$ , is defined as the soft set (H, C), where C = A ∪ B, and ∀e ∈ C, $\begin{matrix} H (e) = {\begin{matrix} F (e) if e \in A ∖ B, \\ G (e) if e \in B ∖ A, \\ F (e) \cup G (e) if e \in A \cap B . \end{matrix} \end{matrix}$

(4) The restricted union of (F, A) and (G, B), denoted by $(F, A) \cup_{R} (F, B)$ , is defined as the soft set (H, C), where C = A ∩ B and H (c) = F (c) ∪ G (c) for allc ∈ C.

Note that restricted intersection was also known as bi-intersection in Feng et al. [22], and extended union was at first introduced and called union by Maji et al. [41].

Definition 2.3. [41] For a soft set (F, A), the set Supp (F, A) = {x ∈ A|F (x) ≠ ∅} is called the support of the soft set (F, A). Thus, the null soft set is indeed a soft set with an empty support and we say that a soft set (F, A) is non-null if Supp (F, A)≠ ∅.

Throughout this paper, let R be a ring and A a non-empty set. η will refer to an arbitrary binary relation between an element of A and an element of R, that is, η is a subset of A × R. A set-valued function F : A → P (R) can be defined as F (x) = {y ∈ R| (x, y) ∈ η} for all x ∈ A. The pair (F, A) is then a soft set over R, which is derived from the relation η.

Let ρ be an equivalence relation over R. Then, the pair (R, ρ) is usually called a Pawlak approximation space [45].

By an equivalence relation, we can define twooperations: $\begin{matrix} ρ_{*} A = {x \in R | [x]_{ρ} \subseteq A}, \\ ρ^{*} A = {x \in R | [x]_{ρ} \cap A \neq \emptyset} \end{matrix}$ assigning to every subset A ⊆ U, two sets ρ_*A and ρ^*A are called the lower and upper approximation of A with respect to (R, ρ).

Definition 2.4. [45] Let (R, ρ) be a Pawlak approximation space. A subset A ⊆ R is called definable if ρ_*A = ρ^*A. Otherwise, A is called a rough set.

Feng et al. [23] introduced the following concept via rough sets and soft sets.

Definition 2.5. [23] Let (R, ρ) be a Pawlak approximation space and $G = (F, A)$ be a soft set over R. The lower and upper rough approximations of $G = (F, A)$ with respect to (R, ρ) are denoted by $ρ_{*} (G) = (F_{*}, A)$ and $ρ^{*} (G) = (F^{*}, A)$ , which are soft sets over R with the set-valued mappings given by $F_{*} (x) = ρ_{*} (F (x)) = {y \in R | [y]_{ρ} \subseteq F (x)}$ and $F^{*} (x) = ρ^{*} (F (x)) = {y \in R | [y]_{ρ} \cap F (x) \neq \emptyset},$ where x ∈ A. The operators ρ^* and ρ_* are called the upper and lower rough approximation operators on soft sets. If $ρ_{*} (G) = ρ^{*} (G)$ , $G$ is called definable, otherwise $G$ is called a rough soft set.

Theorem 2.6. [23] Let (R, ρ) be a Pawlak approximation space and $G = (F, A)$ be a soft set over R. Then,

$ρ_{*} (G) \subseteq G \subseteq ρ^{*} (G)$ ,

$ρ_{*} (ρ_{*} (G)) = ρ_{*} (G)$ ,

$ρ^{*} (ρ^{*} (G)) = ρ^{*} (G)$ ,

$ρ^{*} (ρ_{*} (G)) = ρ_{*} (G)$ ,

$ρ_{*} (ρ^{*} (G)) = ρ^{*} (G)$ .

Theorem 2.7. [23] Let (R, ρ) be a Pawlak approximation space, and $G = (F, A)$ and $I = (H, B)$ be two soft sets over R. Then,

$ρ_{*} (G I) = ρ_{*} (G) ρ_{*} (I)$ ,

$ρ_{*} (G ⊓_{ɛ} I) = ρ_{*} (G) ⊓_{ɛ} ρ_{*} (I)$ ,

$ρ^{*} (G I) \subseteq ρ^{*} (G) ρ^{*} (I)$ ,

$ρ^{*} (G ⊓_{ɛ} I) \subseteq ρ^{*} (G) ⊓_{ɛ} ρ^{*} (I)$ ,

$ρ_{*} (G \cup_{R} I) \supseteq ρ_{*} (G) \cup_{R} ρ_{*} (I)$ ,

$ρ_{*} (G \tilde{\cup} I) \supseteq ρ_{*} (G) \tilde{\cup} ρ_{*} (I)$ ,

$ρ^{*} (G \cup_{R} I) = ρ^{*} (G) \cup_{R} ρ^{*} (I)$ ,

$ρ^{*} (G \tilde{\cup} I) = ρ^{*} (G) \tilde{\cup} ρ^{*} (I)$ ,

$G \subseteq I \Rightarrow ρ_{*} (G) \subseteq ρ_{*} (I)$ and $ρ^{*} (G) \subseteq ρ^{*} (I)$ .

Further terminology, see [2 , 27].

An equivalence relation ρ on R is called a congruence on R if ∀v ∈ R, (a, b) ∈ ρ ⇒ (a + x, b + x) ∈ ρ, (ax, bx) ∈ ρ and (xa, xb) ∈ ρ.

A congruence ρ is called complete if [a] _ρ · [b] _ρ = [ab] _ρ for all a, b ∈ R.

3 Rough soft rings

In this section, we investigate some properties of rough soft rings.

Definition 3.1. A non-null soft set $G = (F, A)$ over R is called a lower (upper) rough soft ring over R if $ρ_{*} (G) (ρ^{*} (G))$ is a soft ring over R, this is, F_* (x) (F^* (x)) is a subring of R for all x ∈ Supp (F_*, A) (Supp (F^*, A)). Moreover, $G$ is called a rough soft ring if $ρ_{*} (G)$ and $ρ^{*} (G)$ are soft rings over $G$ .

Example 3.2. Let R = {0, a, b, c} be a ring with the following Cayley tables:

+	0	a	b	c
0	0	a	b	c
a	a	0	c	b
b	b	c	0	a
c	c	b	a	0

·	a	b	c
0	0	0	0
a	a	0	a
b	0	b	b
c	a	b	c

Let ρ be a congruence on R with ρ-congruence classes [0] _ρ = {0, b} and [a] _ρ = {a, c}.

Define a soft set $G = (F, A)$ over R, where A = R and F : A → P (R) is a set-valued function defined by

F (x) = {y ∈ R| (x, y) ∈ η ⇔ xy ∈ {0, a}} for all x ∈ A.

Then, F (0) = F (a) = R and F (b) = F (c) = {0, a}. Thus, F_* (0) = F^* (a) = F^* (b) = F^* (c) = R.

F_* (0) = F_* (a) = R and F_* (b) = F_* (c) =∅.

Hence, $G$ is a rough soft ring over R.

Example 3.3. Let $ℤ_{6} = {0, 1, 2, 3, 4, 5}$ be the ring of integers modulo 6 and ρ be a congruence on $ℤ_{6}$ with ρ-congruence classes

[0] _ρ = {0, 2, 4} and [1] _ρ = {1, 3, 5}.

Define a soft set $G = (F, A)$ over $ℤ_{6}$ , where $A = ℤ_{6}$ and F : A → P (R) is a set-valued function defined by

$F (x) = {y \in ℤ_{6} | x η y \Leftrightarrow x + y \in {0, 1, 2}}$ for all x ∈ A.

Then, F (0) = {0, 1, 2}, F (1) = {0, 1, 5}, F (2) = {0, 4, 5}, F (3) = {3, 4, 5}, F (4) = {2, 3, 4} and F (5) = {1, 2, 3}. Thus, $F_{*} (x) = ρ^{*} (F (x)) = {y \in ℤ_{6} | [y]_{ρ} \cap F (x) \neq \emptyset} = ℤ_{6}$ , for all x ∈ A, is a subring of $ℤ_{6}$ . Hence, $G$ is an upper rough soft ring over $ℤ_{6}$ .

Lemma 3.4. Let (F, A) and (H, B) be two soft rings. Then, we have

(F, A) Cap (H, B) is a soft ring over R if it is non-null.

(F, A) ⊓ _ɛ (H, B) is a soft ring over R if it is non-null.

Proof. (1) See Theorem 3.6 (2) in [7].

(2) By Definition 2.2, (F, A) ⊓ _ɛ (H, B) = (K, C), where C = A ∪ B, ∀e ∈ C. Then, $K (e) = {\begin{matrix} F (e) if e \in A ∖ B, \\ H (e) if e \in B ∖ A, \\ F (e) \cap H (e) if e \in A \cap B . \end{matrix}$

By hypothesis, Supp (K, C)≠ ∅, Then, there exists e ∈ C such that e ∈ Supp (K, C). For all e ∈ C, F (e) , H (e) and F (e) ∩ H (e) are non-empty. For any cases, K (e) is a subring of R.

Hence (F, A) ⊓ _ɛ (H, B) is a soft ring over R. □

Theorem 3.5. Assume that (R, ρ) is a Pawlak approximation space. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R such that $ρ_{*} (G)$ and $ρ_{*} (I)$ are soft rings over R. Then,

$ρ_{*} (G I)$ is a soft ring over R if $ρ_{*} (G) ρ_{*} (I)$ is non-null.

$ρ_{*} (G ⊓_{ɛ} I)$ is a soft ring over R if $ρ_{*} (G) ⊓_{ɛ} ρ_{*} (I)$ is non-null.

Proof. We only show that (1) holds. By Lemma 3.4, $ρ_{*} (G) ρ_{*} (I)$ is a soft ring over R. By Theorem 2.7, $ρ_{*} (G I) = ρ_{*} (G) ρ_{*} (I)$ , and so $ρ_{*} (G I)$ is a soft ring over R. □

Remark 3.6. The above theorem may not be true for upper rough soft rings.

Example 3.7. Let $ℤ_{4} = {0, 1, 2, 3}$ be the ring of integers modulo 4 and ρ be a congruence on $ℤ_{4}$ with ρ-congruence classes [0] _ρ = {0, 2} and [1] _ρ = {1, 3}.

Define two soft sets $G = (F, A)$ and $I = (H, B)$ over $ℤ_{4}$ , where $A = B = ℤ_{4}$ and

F (0) = F (1) = {0, 1}, F (2) = F (3) = {2, 3} and H (0) = H (3) = {0, 3}, H (1) = H (2) = {1, 2}.

Denote $ρ^{*} (G) = (F^{*}, A)$ and $ρ^{*} (I) = (H^{*}, B)$ . By calculation, $F^{*} (x) = ℤ_{4}$ , ∀x ∈ A and $H^{*} (x) = ℤ_{4}$ , ∀x ∈ B. Hence, $ρ^{*} (G)$ and $ρ^{*} (I)$ are soft rings over $ℤ_{4}$ .

(1) It is clear that $ρ^{*} (G) ρ^{*} (I)$ is non-null. By Lemma 3.4, $ρ^{*} (G) ρ^{*} (I)$ is a soft ring over $ℤ_{4}$ .

Denote $G I = (K, C)$ , where $C = A \cap B = ℤ_{4}$ . By calculation, K (0) = {0}, K (1) = {2} and K (3) = {3}.

Denote $ρ^{*} (G I) = (K^{*}, C)$ and $K^{(} x) = {y \in ℤ_{4} | [y]_{ρ} \cap K (x) \neq \emptyset}$ .

Then, K^* (0) = K^* (2) = {0, 2} and K^* (1) = K^* (3) = {1, 3}.

This means that K^* (1) and K^* (3) are not subrings of $ℤ_{4}$ . Hence $ρ^{*} (G I)$ is not a soft ring over $ℤ_{4}$ .

(2) is similar to (1), we can cheek that $ρ^{*} (G ⊓_{ɛ} I)$ is not a soft ring over $ℤ_{4}$ .

Lemma 3.8. Let (F, A) and (H, B) be two soft rings. Then,

$(F, A) \cup_{R} (H, B)$ is a soft ring over R if (F, A) ⊆ (H, B) or (F, A) ⊇ (H, B).

$(F, A) \tilde{\cup} (H, B)$ is a soft ring over R if A∩ B ≠ ∅.

$(F, A) \tilde{\cup} (H, B)$ is a soft ring over R if (F, A) ⊆ (H, B) or (F, A) ⊇ (H, B).

Proof. We denote $(F, A) \cup_{R} (H, B) = (K, C)$ , where C = A ∩ B and K (c) = F (c) ∪ H (c) , ∀ c ∈ C. If (F, A) ⊆ (H, B), then A ⊆ B and F (c) ⊆ H (c) , ∀ c ∈ A, and so C = A and F (c) = H (c) , ∀ c ∈ A. Since H (c) is a subring for all c ∈ C. This implies that (F, A) ∪ _R (H, B) is a soft ring over R. The case for (F, A) ⊇ (H, B) is also true.

(2) See Theorem 3.8 in [3].

(3) We denote $(F, A) \tilde{\cup} (H, B) = (K, C)$ , where C = A ∪ B, ∀ c ∈ C. Then,

$K (c) = {\begin{matrix} F (c) if c \in A ∖ B, \\ H (c) if c \in B ∖ A, \\ F (c) \cap H (c) if c \in A \cap B . \end{matrix}$

If (F, A) ⊆ (H, B), then A ⊆ B and F (c) ⊆ H (c), for all c ∈ C.

Since H (c) is a subring over R for all c ∈ C, then K (c) is a subring over R for all c ∈ C. This implies that $(F, A) \tilde{\cup} (H, B)$ is a soft ring over R. □

Theorem 3.9. Assume that (R, ρ) is a Pawlak approximation space. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R such that $ρ^{*} (G)$ and $ρ^{*} (I)$ are soft rings over R. Then,

$ρ^{*} (G \cup_{R} I)$ is a soft ring over R if $ρ^{*} (G) \subseteq ρ^{*} (I)$ or $ρ^{*} (G) \supseteq ρ^{*} (I)$ .

$ρ^{*} (G \tilde{\cup} I)$ is a soft ring over R if A∩ B = ∅.

$ρ^{*} (G \tilde{\cup} I)$ is a soft ring over R if $ρ^{*} (G) \subseteq ρ^{*} (I)$ or $ρ^{*} (G) \supseteq ρ^{*} (I)$ .

Proof. (1) If $ρ^{*} (G) \subseteq ρ^{*} (I)$ or $ρ^{*} (G) \supseteq ρ^{*} (I)$ , then it follows from Lemma 3.8 that $ρ^{*} (G) \cup_{R} ρ^{*} (I)$ is a soft ring over R. By Theorem 2.7(7), $ρ^{*} (G \cup_{R} I) = ρ^{*} (G) \cup_{R} ρ^{*} (I)$ . This implies that $ρ^{*} (G \cup_{R} I)$ is a soft ring over R.

Proves of (2) and (3) are similar to that of (1). □

Remark 3.10. The above theorem may not be true for lower rough soft rings.

Example 3.11. Let $ℤ_{9} = {0, 1, 2, 3, 4, 5, 6, 7, 8}$ be the ring of integers modulo 9 and ρ be a congruence on $ℤ_{9}$ with ρ-congruence classes

[0] _ρ = {0, 3, 6}, [1] _ρ = {1, 4, 7} and [2] _ρ = {2, 5, 8}.

Define two soft sets $G = (F, A)$ and $I = (H, B)$ over $ℤ_{9}$ , where

A = B = {0, 3, 6} and F (0) = {0, 1, 3, 4, 6}, F (3) = {0, 2, 6}, F (6) = {0, 3, 6, 7} and H (0) = {0, 1, 3, 6, 7}, H (3) = {0, 5, 6}, H (6) = {0, 1, 3, 4, 6} .

We denote $ρ_{*} (G) = (F_{*}, A)$ and $ρ_{*} (I) = (H_{*}, B)$ . By calculation, we have

F_* (0) = {0, 3, 6}, F_* (3) =∅, F_* (6) = {0, 3, 6} and H_* (0) = {0, 3, 6}, H_* (3) =∅, H_* (6) = {0, 3, 6}. Then, $ρ_{*} (G)$ and $ρ_{*} (I)$ are soft rings over $ℤ_{9}$ . By Lemma 3.8, $ρ_{*} (G) \tilde{\cup} ρ_{*} (I)$ and $ρ_{*} (G) \cup_{R} ρ_{*} (I)$ are soft rings over $ℤ_{9}$ .

Now, we denote $G \cup_{R} I = (K, C)$ , where C = A ∩ B = {0, 3, 6}.

By calculation, K (0) = {0, 1, 3, 4, 6, 7}, K (3) = {0, 2, 5, 6} and K (6) = {0, 1, 3, 4, 6, 7}.

Denote $ρ^{*} (G \cup_{R} I) = (K_{*}, C)$ , C = A ∩ B = {0, 3, 6}.

By calculation, we have K_* (0) = K_* (6) = {0, 1, 3, 4, 6, 7} and K_* (3) =∅. This implies that K_* (0) and K_* (6) are not subrings of $ℤ_{9}$ . Therefore, $ρ_{*} (G \cup_{R} I)$ is not a soft ring over $ℤ_{9}$ .

Lemma 3.12. [16] Let ρ be a congruence on R. If a, b ∈ R, then

$\begin{matrix} (1) [a]_{ρ} + [b]_{ρ} = [a + b]_{ρ}, \\ (2) [- a]_{ρ} = - [a]_{ρ}, \\ (3) [a]_{ρ} \cdot [b]_{ρ} \subseteq [ab]_{ρ} . \end{matrix}$

Theorem 3.13. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. If $G = (F, A)$ is a soft ring over R, then $G$ is an upper rough soft ringover R.

Proof. Since $G = (F, A)$ is a soft ring over R, then F (x) is a subring of R for all x ∈ Supp (F, A). We denote $ρ^{*} (G) = (F^{*}, A)$ and ∀x ∈ A, F^* (x) = ρ^* (F (x)) = {y ∈ R| [y] _ρ ∩ F (x) ≠ ∅}. Let a, b ∈ F^* (x), then [a] _ρ∩ F (x) ≠ ∅ and [b] _ρ∩ F (x) ≠ ∅, then there exist c, d ∈ R such that c ∈ [a] _ρ ∩ F (x) and d ∈ [b] _ρ ∩ F (x), that is, c ∈ [a] _ρ, d ∈ [b] _ρ and c, d ∈ F (x).

Since F (x) is a subring of R, then cd, c - d ∈ F (x). By Lemma 3.12, c - d ∈ [a] _ρ - [b] _ρ = [a - b] _ρ and cd ∈ [a] _ρ · [b] _ρ = [ab] _ρ. Thus, c - d ∈ [a - b] _ρ ∩ F (x) and cd ∈ [ab] _ρ ∩ F (x), which implies, [a - b] _ρ∩ F (x) ≠ ∅ and [ab] _ρ∩ F (x) ≠ ∅. Then, a - b, ab ∈ F^* (x). This shows that F^* (x) is a subring of R for all x ∈ Supp (F^*, A). Hence, $ρ^{*} (G)$ is a soft ringover R, and so, $G$ is an upper rough soft ringover R.

Theorem 3.14. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. Assume that $G = (F, A)$ is a soft ring over R. If $ρ^{*} (G)$ is non-null, then $ρ^{*} (G) = G .$

Proof. By Theorem 2.6(1), we have $ρ^{*} (G) \subseteq G$ . Since $ρ^{*} (G)$ is non-null, then Supp (F_*, A)≠ ∅, and so F_* (x)≠ ∅, which implies, for some c ∈ F_* (x). Thus,

[0] _ρ = [c + (- c)] _ρ = [c] _ρ + [- c] _ρ = [c] _ρ + (- [c] _ρ) ⊆ F (x) + F (x) = F (x).

For all y ∈ F (x). Since F (x) is a subring of R, then y + [0] _ρ ⊆ y + F (x) = F (x). Moreover,

z ∈ y + [0] _ρ ⇔ z - y ∈ [0] _ρ ⇔ (z - y, 0) ∈ ρ ⇔ (z, y) ∈ ρ ⇔ z ∈ [y] _ρ.

This implies, [y] _ρ ⊆ F (x), and so, y ∈ F_* (x). This shows that F (x) ⊆ F_* (x) for allx ∈ Supp (F_*, A). Thus, $G \subseteq ρ_{*} (G)$ . □

Corollary 3.15. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. Assume that $G = (F, A)$ is a soft ring over R. If $ρ_{*} (G)$ is non-null, then $G$ is a rough soft ring over R.

4 Rough idealistic soft rings

In this section, we investigate some characterizations of rough idealistic soft rings.

Definition 4.1. A non-soft set $G = (F, A)$ over R is called a lower (upper) rough idealistic soft ring over R if $ρ_{*} (G)$ ( $ρ^{*} (G)$ ) is an idealistic soft ring over R, that is, F_* (x) (F^* (x)) is an ideal of R for all x ∈ Supp (F_*, A) (Supp (F^*, A)). Moreover, $G$ is called a rough idealistic soft ring if $ρ_{*} (G)$ and $ρ^{*} (G)$ are idealistic soft ringsover R.

Example 4.2. Let $R = ℤ_{15}$ be the ring of integers modulo 15 and ρ be a congruence on $ℤ_{15}$ with ρ-congruence classes [0] _ρ = {0, 5, 10}, [1] _ρ = {1, 6, 11}, [2] _ρ = {2, 7, 12}, [3] _ρ = {3, 8, 13} and [4] _ρ = {4, 9, 14}.

Define a soft set $G = (F, A)$ over $ℤ_{15}$ , where A = {0, 5, 10} and $F : A \to P (ℤ_{15})$ is a set-valued function defined by F (0) = {0, 5, 10}, F (5) = {0, 5} and F (0) = {0, 10}. By calculation, we have

F_* (0) = {0, 5, 10}, F_* (5) = F_* (10) =∅ and F^* (0) = F^* (5) = F^* (10) = {0, 5, 10}. This implies that F_* (x) (F^* (x)) is an ideal of R for all x ∈ Supp (F_*, A) (Supp (F^*, A)). Hence, $G$ is a rough idealistic soft ring over R.

Example 4.3. Consider the ring R as in Example 3.2. Let ρ be a congruence on R with ρ-congruence classes [0] _ρ = {0, b} and [a] _ρ = {a, c}.

Define a soft set $G = (F, A)$ over R, where A = R and F : A → P (R) is a set-valued function defined by

F (x) = {y ∈ R| (x, y) ∈ η ⇔ y = xⁿ for some $n \in ℕ$ } for all x ∈ A, where xⁿ = xx ⋯ x means the n-fold product of x and x⁰ = 0.

Then, F (0) = {0}, F (a) = {0, a}, F (b) = {0, b} and F (c) = {0, c}. By calculation, F^* (0) = {0, b}, F^* (a) = F^* (b) = F^* (c) = R and F_* (0) = F_* (a) = F_* (c) =∅, F_* (b) = {0, b}. This implies that F^* (x) (F_* (x)) is an idealistic soft ring over R for all x ∈ Supp (F^*, A) (Supp (F_*, A)). Hence $G$ is a rough idealistic soft ring over R.

Remark 4.4. Since every idealistic soft ring over any ring R may not be a soft ring over R (see [22]), we know that every rough idealistic soft ring over R may not be a rough idealistic soft ring over R.

Theorem 4.5. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. If $G = (F, A)$ is an idealistic soft ring over R, then $G$ is an upper rough idealistic soft ring over R.

Proof. Since $G = (F, A)$ is an idealistic soft ring over R, then F (x) is an ideal of R for all x ∈ Supp (F, A). We denote ρ^* (G) = (F^*, A) and ∀x ∈ A, F^* (x) = ρ^* (F (x)) = {y ∈ R| [y] _ρ ∩ F (x) ≠ ∅}.

Let a, b ∈ F^* (x) and r ∈ R, then [a] _ρ∩ F (x) ≠ ∅ and [b] _ρ∩ F (x) ≠ ∅, then there exist c, d ∈ R such that c ∈ [a] _ρ ∩ F (x) and d ∈ [b] _ρ ∩ F (x), that is, c ∈ [a] _ρ, d ∈ [b] _ρ and c, d ∈ F (x). Since F (x) is an ideal of R, then rc, cr, c - d ∈ F (x). By Lemma 3.12, c - d ∈ [a] _ρ - [b] _ρ = [a - b] _ρ, rc ∈ r · [a] _ρ = [ra] _ρ and cr ∈ [a] _ρ · r = [ar] _ρ. Thus, we have c - d ∈ [a - b] _ρ ∩ F (x), rc ∈ [ra] _ρ ∩ F (x) and cr ∈ [ar] _ρ ∩ F (x), which implies, [a - b] _ρ∩ F (x) = ∅ and [a] _ρ∩ F (x) = ∅. Then, a - b, ra, ar ∈ F^* (x). This shows that F^* (x) is an ideal of R for all x ∈ Supp (F^*, A). Hence, $ρ^{*} (G)$ is an idealistic soft ring over R, and so, $G$ is an upper rough idealistic soft ring over R. □

Theorem 4.6. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. Assume that $G = (F, A)$ is an idealistic soft ring over R. If $ρ_{*} (G)$ is non-null, then $ρ_{*} (G) = G$ .

Proof. It is a consequence of Theorems 3.14 and 4.5. □

Corollary 4.7. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. If $G = (F, A)$ is an idealistic soft ring over R and $ρ_{*} (G)$ is non-null, then $G$ is a rough idealistic soft ring over R.

Recall that an ideal I of a commutative ring R is prime if xy ∈ I implies x ∈ I or y ∈ I. In Theorems 4.8 and 4.9, we suppose that R is a commutativering.

Theorem 4.8. Let (R, ρ) be a Pawlak approximation space and ρ a complete congruence on R. If $G = (F, A)$ is a prime idealistic soft ring over R, then $G$ is an upper rough prime idealistic soft ring over R.

Proof. By Theorem 4.5, $G$ is an upper rough idealistic soft ring over R.

Since $G = (F, A)$ is a prime idealistic soft ring over R, then F (x) is a prime ideal of R for all x ∈ Supp (F, A). We denote $ρ^{*} (G) = (F^{*}, A)$ and ∀x ∈ A, F^* (x) = ρ^* (F (x)) = {y ∈ R| [y] _ρ ∩ F (x) ≠ ∅}.

Let ab ∈ F^* (x), then [ab] _ρ∩ F (x) = [a] _ρ · [b] _ρ ∩ F (x) ≠ ∅. Then, there exist a′ ∈ [a] _ρ and b′ ∈ [b] _ρ such that a′b′ ∈ F (x). Since F (x) is prime of R, then a′ ∈ F (x) or b′ ∈ F (x). Hence [a] _ρ∩ F (x) ≠ ∅ or [b] _ρ∩ F (x) ≠ ∅, and so a ∈ F^* (x) or b ∈ F^* (x). This implies that $ρ^{*} (G)$ is a prime idealistic ring over R. That is, $G$ is an upper rough prime idealistic soft over R. □

The following shows that we need not completecongruences.

Theorem 4.9. Let (R, ρ) be a Pawlak approximation space and ρ a congruence on R. If $G = (F, A)$ is a prime idealistic soft ring over R and $ρ_{*} (G)$ is non-null, then $G$ is a lower rough prime idealistic soft ringover R.

Proof. By Theorem 4.6, $G$ is a lower rough idealistic soft ring over R.

Since $G = (F, A)$ is a prime idealistic soft ring over R, then F (x) is a prime ideal of R for all x ∈ Supp (F, A). We denote ρ_* (G) = (F_*, A) and ∀x ∈ A, F^* (x) = ρ_* (F (x)) = {y ∈ R| [y] _ρ ⊆ F (x)}.

Let ab ∈ F_* (x), then [ab] _ρ ⊆ F (x). By Lemma 3.12, [a] _ρ · [b] _ρ ⊆ [ab] _ρ, and so [a] _ρ · [b] _ρ ⊆ F (x). Since F (x) is a prime ideal of R for all x ∈ Supp (F, A), then [a] _ρ ⊆ F (x) or [b] _ρ ⊆ F (x). This means that a ∈ F_* (x) or b ∈ F_* (x), and so F_* (x) is a prime ideal of R for all x ∈ Supp (F_*, A). Thus, $G$ is a lower rough prime idealistic soft ring over R.

5 Products and sums of rough soft rings

In this section, we investigate the product and the sum of two rough soft rings.

Definition 5.1. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over a ring (R, + , ·) with C = A∩ B ≠ ∅. The product is defined as $G ⨀ I = (F, A) ⨀ (H, B) = (K, C)$ , where K (c) = F (c) · H (c) for all c ∈ C. The sum is defined as $G oplus I = (F, A) oplus (H, B) = (K, C)$ , where K (c) = F (c) + H (c) for all c ∈ C.

Theorem 5.2. Let (R, ρ) be a Pawlak approximation space, where (R, + , ·) is a ring and ρ is a congruence on R. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R with C = A∩ B ≠ ∅. Then, $ρ^{*} (G) \cdot ρ^{*} (I) \subseteq ρ^{*} (G ⨀ I)$ .

Proof. By Definition 5.1, we have $G ⨀ I = (F, A) ⨀ (H, B) = (K, C)$ , where K (x) = F (x) · H (x) for all x ∈ C and $ρ^{*} (G ⨀ I) = (K^{*}, C)$ , K^* (x) = ρ^* (K (x)) = ρ^* (F (x) · H (x)).

In order to show that $ρ^{*} (G) \cdot ρ^{*} (I) \subseteq ρ^{*} (G ⨀ I)$ , we need to prove that ρ^* (F (x)) · ρ^* (G (x)) ⊆ ρ^* (F (x) · G (x)) holds.

Let y ∈ ρ^* (F (x)) · ρ^* (H (x)), then $y = \sum_{i = 1}^{n} a_{i} b_{i}$ , a_i ∈ ρ^* (F (x)) and b_i ∈ ρ^* (H (x)). Thus, [a_i] _ρ∩ F (x) ≠ ∅ and [b_i] _ρ∩ H (x) ≠ ∅. Then, there exist c_i, d_i ∈ R such that c_i ∈ [a_i] _ρ ∩ F (x) and d_i ∈ [b_i] _ρ ∩ H (x), and so c_iρa_i and d_iρb_i. Since ρ is congruence, c_id_iρa_ib_i, $(\sum_{i = 1}^{n} c_{i} d_{i}) ρ (\sum_{i = 1}^{n} a_{i} b_{i})$ and $\sum_{i = 1}^{n} c_{i} d_{i} \in F (x) \cdot H (x)$ . Hence $(\sum_{i = 1}^{n} a_{i} b_{i})_{ρ} \cap F (x) \cdot H (x) \neq \emptyset$ . This implies y ∈ ρ^* (F (x) · H (x)) = K^* (x), and so ρ^* (F (x)) · ρ^* (H (x)) ⊆ ρ^* (F (x) · H (x)) .

Therefore, $ρ^{*} (G) \cdot ρ^{*} (I) \subseteq ρ^{*} (G ⨀ I)$ . □

Theorem 5.3. Let (R, ρ) be a Pawlak approximation space, where (R, + , ·) is a ring and ρ is a congruence on R. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R with C = A∩ B ≠ ∅. Then, $ρ^{*} (G) + ρ^{*} (I) \subseteq ρ^{*} (G oplus I)$ .

Proof. By Definition 5.1, we have

$G oplus I = (F, A) oplus (H, B) = (K, C)$ , where K (x) = F (x) + H (x) for all x ∈ C and $ρ^{*} (G oplus I) = (K^{*}, C)$ , K^* (x) = ρ^* (K (x)) = ρ^* (F (x) + H (x)).

To show that $ρ^{*} (G) + ρ^{*} (I) \subseteq ρ^{*} (G oplus I)$ , we prove ρ^* (F (x)) + ρ^* (H (x)) ⊆ ρ^* (F (x) + H (x)) holds.

For any y ∈ ρ (F (x)) + ρ (H (x)), then there exist a ∈ ρ^* (F (x)) and b ∈ ρ^* (H (x)) such that y = a + b. Hence [a] _ρ∩ F (x) ≠ ∅ and [b] _ρ∩ F (x) ≠ ∅. This implies that there exist c ∈ [a] _ρ ∩ F (x) and d ∈ [b] _ρ ∩ H (x), and so c ∈ [a] _ρ, d ∈ [b] _ρ and c ∈ F (x), d ∈ H (x). Since ρ is congruence (c + d) ρ (a + b). Since c ∈ F (x) and d ∈ H (x), c + d ∈ F (x) + H (x). Hence [a+ b] _ρ ∩ (F (x) + H (x)) ≠ ∅, and so [y] _ρ∩ (F (x) + H (x)) ≠ ∅. This implies, y ∈ ρ^* (F (x) + H (x)), and so ρ^* (F (x)) + ρ^* (H (x)) ⊆ ρ^* (F (x) + H (x)). Therefore $ρ^{*} (G) + ρ^{*} (I) \subseteq ρ^{*} (G oplus I)$ .

Remark 5.4. The converses of Theorems 5.2 and 5.3 may not be true as shown in the following examples:

Example 5.5. Let R = {0, a, b, c} be a ring with the following Cayley tables:

+	0	a	b	c
0	0	a	b	c
a	a	0	c	b
b	b	c	0	a
c	c	b	a	0

·	a	b	c
0	0	0	0
a	0	0	0
b	0	a	a
c	0	a	a

Let ρ be a congruence on R with ρ-congruence classes: [0] _ρ = {0, a} and [b] _ρ = {b, c}. Let $G = (F, A)$ and $I = (H, B)$ , be two soft sets over R, where A = B = R and for all x ∈ C = A ∩ B, F (x) = {b} and H (x) = {c}. $G ⨀ I = (K, C)$ and K (x) = F (x) · H (x) = {a} for all x ∈ C.

Denote that $ρ^{*} (G) = (F^{*}, A)$ , $ρ^{*} (I) = (H^{*}, B)$ and $ρ^{*} (G ⨀ I) = (K^{*}, C) .$

By calculation, we have F^* (x) = ρ^* (F (x)) = {y ∈ R| [y] _ρ ∩ F (x) ≠ ∅} = {b, c} for all x ∈ A. H^* (x) = ρ^* (H (x)) = {y ∈ R| [y] _ρ ∩ H (x) ≠ ∅} = {b, c} for allx ∈ B. K^* (x) = ρ^* (F (x) · H (x)) = {y ∈ R| [y] _ρ ∩ (F (x) · H (x)) ≠ ∅} = {0, a}.

Hence $ρ^{*} (G) \cdot ρ^{*} (I) = {a}$ and $ρ^{*} (G ⨀ I) = {0, a}$ . Thus, $ρ^{*} (G) \cdot ρ^{*} (I) \neq ρ^{*} (G ⨀ I)$ .

Example 5.6. Consider the ring as in the above example. Let ρ be a congruence on R with ρ-congruence classes: [0] _ρ = {0, a, b} and [c] _ρ = {c}.

Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R, where A = B = R and for all x ∈ C = A ∩ B, F (x) = H (x) = {c}.

$G oplus I = (K, C)$ and K (x) = F (x) + H (x) = {0} for all x ∈ C.

Denote that $ρ^{*} (G) = (F^{*}, A)$ , $ρ^{*} (I) = (H^{*}, B)$ and $ρ^{*} (G oplus I) = (K^{*}, C)$ .

By calculation, we have F^* (x) = ρ^* (F (x)) = {y ∈ R| [y] _ρ ∩ F (x) ≠ ∅} = {c} for all x ∈ A.

H^* (x) = ρ^* (H (x)) = {y ∈ R| [y] _ρ ∩ H (x) ≠ ∅} = {c} for all x ∈ B.

K^* (x) = ρ^* (F (x) + H (x)) = {y ∈ R| [y] _ρ ∩ (F (x) + H (x)) ≠ ∅} = {0, a, b}.

Hence $ρ^{*} (G) + ρ^{*} (I) = {0}$ and $ρ^{*} (G oplus I) = {0, a, b}$ . Thus, $ρ^{*} (G) + ρ^{*} (I) \neq ρ^{*} (G oplus I)$ .

Now, we give lower rough soft sets over rings.

Theorem 5.7. Let (R, ρ) be a Pawlak approximation space, where (R, + , ·) is a ring and ρ is a congruence on R. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R with A∩ B ≠ ∅. Then, $ρ_{*} (G) \cdot ρ_{*} (I) \subseteq ρ_{*} (G oplus I)$ .

Proof. By Definition 5.1, we have

$G ⨀ I = (F, A) ⨀ (H, B) = (K, C)$ , where K (x) = F (x) · H (x) for all x ∈ C and $ρ_{*} (G ⨀ I) = (K_{*}, C)$ , K_* (x) = ρ_* (K (x)) = ρ_* (F (x) · H (x)).

To show that $ρ_{*} (G) \cdot ρ_{*} (I) \subseteq ρ_{*} (G ⨀ I)$ , we need prove ρ_* (F (x)) · ρ_* (H (x)) ⊆ ρ_* (F (x) · H (x)) holds.

Let y ∈ ρ_* (F (x)) · ρ_* (H (x)), then $y = \sum_{i = 1}^{n} a_{i} b_{i}$ , a_i ∈ ρ_* (F (x)) and b_i ∈ ρ_* (H (x)). Thus, [a_i] _ρ ⊆ F (x) and [b_i] _ρ ⊆ H (x). By Lemma 3.12, $[y]_{ρ} = [\sum_{i = 1}^{n} a_{i} b_{i}]_{ρ} \subseteq \sum_{i = 1}^{n} [a_{i}]_{ρ} [b_{i}]_{ρ} \subseteq F (x) \cdot H (x)$ . Hence y ∈ ρ_* (F (x) · H (x)). This implies that ρ_* (F (x)) · ρ_* (H (x)) ⊆ ρ_* (F (x) · H (x)). Therefore, $ρ_{*} (G) \cdot ρ_{*} (I) \subseteq ρ_{*} (G ⨀ I)$ .

Theorem 5.8. Let (R, ρ) be a Pawlak approximation space, where (R, + , ·) is a ring and ρ is a congruence on R. Let $G = (F, A)$ and $I = (H, B)$ be two soft sets over R with A∩ B ≠ ∅. Then, $ρ_{*} (G) + ρ_{*} (I) \subseteq ρ_{*} (G oplus I)$ .

Proof. By Definition 5.1, we have

$G oplus I = (F, A) oplus (H, B) = (K, C)$ , where K (x) = F (x) + H (x) for all x ∈ C and $ρ_{*} (G oplus I) = (K_{*}, C)$ , K_* (x) = ρ_* (K (x)) = ρ_* (F (x) + H (x)).

To show that $ρ_{*} (G) + ρ_{*} (I) \subseteq ρ_{*} (G oplus I)$ , we only need prove ρ_* (F (x)) + ρ_* (H (x)) ⊆ ρ_* (F (x) + H (x)) holds.

For any y ∈ ρ_* (F (x)) + ρ_* (H (x)), then y = a + b for some a ∈ ρ_* (F (x)) and b ∈ ρ_* (H (x)). Hence[a] _ρ ⊆ F (x) and [b] _ρ ⊆ H (x), and so [a] _ρ + [b] _ρ ⊆ F (x) + H (x). By Lemma 3.12, [a + b] _ρ = [a] _ρ + [b] _ρ ⊆ F (x) + H (x). That is, [y] _ρ ⊆ F (x) + H (x). Thus, y ∈ ρ_* (F (x) + H (x)). This implies that ρ_* (F (x)) + ρ_* (H (x)) ⊆ ρ_* (F (x) + H (x)). Therefore, $ρ_{*} (G) + ρ_{*} (I) \subseteq ρ_{*} (G oplus I)$ .

We believe that the research along this direction can be continued, and in fact, some results in this paper have already constituted a foundation for further investigation concerning the further development of rings. As future work, connections between rough soft sets and various types of generalized rough set models could be explored.

Acknowledgments

This research is partially supported by a grant of National Natural Science Foundation of China (11461025).

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