A note on rough multisets

Abstract

Multisets are the nature extension of classical sets. Rough multisets are multisets in rough set context. This note studies rough multisets. We illustrate that Definition 9.1 in [4] is imperfect or irrational via an example. In order to improve this definition, we redefine rough multisets along with a illustrative example. Moreover, we obtain some properties of rough multisets based on new definition. Finally, we give applications of rough multisets.

Keywords

Multiset Multiset relation rough multiset roughness multiset topology

1 Introduction

Zermelo-Fraenkel set theory (ZF theory) is the classical theory based on first-order logic and it is the foundation of mathematics [6, 7]. In first-order logic, a formal theory MST (multi-set theory), which contains ZF as a special case is conceived.

In some situations we may want a structure which is a family of objects in the same sense as a set but in which redundancy counts. A multiset or bag provides a framework in which to study these sets in which redundant elements count.

A multiset or bag is a family of objects in which repetition of elements is significant. From a practical point of view, a multiset is very useful as it arises in many areas of mathematics and computer science. For example, the zeros of meromorphic functions and invariants of matrices in a canonical form are both multisets.

Multisets or bags as the nature extension of classical sets were proposed by Cerf et al. [2] in 1971, Peterson [11] in 1976 and Yager [16] in 1986. A naive concept of multiset was formalized by Blizard [1].

Some scholars studied the extension of multisets or bags. For example, Yager [16] introduced the concept of fuzzy bags; Miyamoto [10] discussed relationship between multisets and fuzzy multisets; Wang et al. [15] proposed event multisets and gave discovering process models based on them; Girish and John [5] presented partially ordered multisets and their chains and antichains; Yang et al. [17] defined a multi-fuzzy soft set and considered its application in decision making; Lizasoain et al. [9] introduced lattice fuzzy multisets and investigated generalized Atanassov¡¯s operators defined on them.

Rough set theory, proposed by Pawlak [12, 13], is a mathematical tool for dealing with inaccuracy and uncertainty in data analysis. The lower and upper approximation operators are constructed by using an equivalence relation on the universe. Using these approximations, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules.

The classical rough approximations are based on equivalence relations, but this requirement is not satisfied in some situations. Thus, the classical rough approximations may be extended to equivalence multiset relations.

Multiset topology is a family of multisets which satisfies the topological axioms. Binary relation based on a multiset is said to be a multiset relation. Girish and John [4] investigated multiset topologies induced by multiset relations and introduced the concept of rough multisets.

The main purpose of this note is to propose new definition of rough multisets along and their applications for roughness and multiset topology.

The remaining part of this note is organized as follows. In Section 2, we recall some basic notions about multisets or bags, and Pawlak rough sets. In Section 3, we give an example to illustrate that Definition 9.1 in [4] is imperfect or irrational. In Section 4, we introduce a new definition of rough multisets along with a illustrative example and some properties. In Section 5, we give applications of rough multisets. In Section 6, we summarize this note.

2 Preliminaries

In this section, we recall some basic notions about multisets and Pawlak rough sets.

Throughout this paper, X denotes a nonempty finite set called the universe, N represents the nature number set, N ∪ {0} means the set of nonnegative integers, 2^X denotes the set of all subsets of X and I^X denotes the set of all fuzzy sets on X.

2.1 Multisets or bags

Definition 2.1. ([8]) Given the universe X. A multiset or bag M drawn from X is represented by a function count C_M defined as C_M : X → N ∪ {0}.

In order to facilitate, $C_{M} (x) is denoted by M (x) (x \in X) .$

If M (x) = m, then x appears m times in M, we denoted it by m/x ∈ M or x ∈ ^mM.

Remark 2.2. In [8], the concept of bags is defined as follows:

Given the universe X. A Bag M drawn from X is represented by a function count C_M defined as C_M : X → N ∪ {0};

Given the universe X. A bag M drawn from X is represented by a function count C_M defined as C_M : X → {0, 1}.

In fact, a bag can be seen as a special kind of Bag.

Since the universe X is a finite set, according to Zermelo axiom of choice or Zorn lemma, the universe X can be ordered (see [14, 18]). Thus, for convenience, we may view the universe X as an order set in this paper. Specifically, given X = {x₁, x₂, …, x_n}, we default that X is an order set (X, ≤), i.e., $x_{1} \leq x_{2} \leq \dots \leq x_{n} .$

Given X = {x₁, x₂, …, x_n}. If M (x_i) = m_i (i = 1, 2, ⋯ , n), then M is denoted by ${m_{1} / x_{1}, m_{2} / x_{2}, \dots, m_{n} / x_{n}},$ i.e., $M = {m_{1} / x_{1}, m_{2} / x_{2}, \dots, m_{n} / x_{n}} .$

$| M | = m_{1} + m_{2} + \dots + m_{n}$ is defined as the cardinality of M.

According to the above instructions, M can be represented by $v (M) = (m_{1}, m_{2}, \dots, m_{n}) .$

In this paper, denote $X_{m} = {m / x_{1}, m / x_{2}, \dots, m / x_{n}} (m \in N),$

$\tilde{\emptyset} = {0 / x_{1}, 0 / x_{2}, \dots, 0 / x_{n}} .$

Obviously, v (X_m) = (m, m, …, m) , $v (\tilde{\emptyset}) = (0, 0, \dots, 0) .$

Definition 2.3. ([8]) Given the universe X. Let A and B be two multisets drawn from X. Then the following are defined:

(1) A = B⇔ ∀ x ∈ X, A (x) = B (x) ;

(2) A⊆ B ⇔ ∀ x ∈ X, A (x) ≤ B (x) ;

(3) P = A∪ B ⇔ ∀ x ∈ X, P (x) = A (x) ∨ B (x) ;

(4) P = A ∩ B ⇔ ∀ x ∈ X, P (x) = A (x) ∧ B (x) .

(5) P = A⊕ B ⇔ ∀ x ∈ X, P (x) = A (x) + B (x) ;

(6) P = A ⊖ B ⇔ ∀ x ∈ X, P (x) = (A (x) - B (x)) ∨0 .

Recall that (a₁, a₂, ⋯⋯ , a_n) is called a vector if for any i ≤ n, a_i is a real number. We write $\vec{a} = (a_{1}, a_{2}, \dots \dots, a_{n})$ .

Let $\vec{a} = (a_{1}, a_{2}, \dots \dots, a_{n}), \vec{b} = (b_{1}, b_{2},$ ⋯⋯ , b_n). Define $\vec{a} = \vec{b} \Leftrightarrow \forall i, a_{i} = b_{i}; \vec{a} ⪯ \vec{b} \Leftrightarrow \forall i, a_{i} \leq b_{i} .$

Obviously, $\vec{a} = \vec{b} \Leftrightarrow \vec{a} ⪯ \vec{b} and \vec{b} ⪯ \vec{a} .$

Let A and B be two multisets drawn from X. Then

$A = B \Leftrightarrow v (A) = v (B); A \subseteq B \Leftrightarrow v (A) ⪯ v (B) .$

Definition 2.4. ([4, 8]) Given the universe X. Suppose m ∈ N. Denote [X] ^m = {M : M is a multiset drawn from X and ∀ x ∈ X, M (x) ≤ m} . Then [X] ^m is called the multiset space or bag space over X.

If X = {x₁, x₂, ⋯ , x_n}, then [X] ^m = {{m₁/x₁, m₂/x₂, ⋯ , m_n/x_n} : m_i ∈ {0, 1, 2, ⋯ , m} , i ∈ {1, 2, ⋯ , n}} . Obviously, $A \in [X]^{m} \Leftrightarrow A \subseteq X_{m} .$

The relations and operations on multiset spaces or bag spaces can be defined as follows:

(1) [X] ^m = [Y] ⁿ⇔ ∀ X = Y, m = n ;

(2) [X] ^m ⊔ [Y] ⁿ = {A ⊔ B : A ∈ [X] ^m, Y ∈ [Y] ⁿ} , where $(A ⊔ B) (x) = {\begin{matrix} A (x), x \in X - Y, \\ A (x) \lor B (x), x \in X \cap Y, \\ B (x), x \in Y - X; \end{matrix}$

(3) [X] ^m ⊓ [Y] ⁿ = {A ⊓ B : A ∈ [X] ^m, Y ∈ [Y] ⁿ} , where $(A ⊓ B) (x) = A (x) \land B (x), x \in X \cap Y;$

(4) [X] ^m ⊕ [Y] ⁿ = {A ⊕ B : A ∈ [X] ^m, Y ∈ [Y] ⁿ} , where $(A \oplus B) (x) = {\begin{matrix} A (x), x \in X - Y, \\ A (x) + B (x), x \in X \cap Y, \\ B (x), x \in Y - X; \end{matrix}$

(5) [X] ^m ⊖ [Y] ⁿ = {A ⊖ B : A ∈ [X] ^m, Y ∈ [Y] ⁿ} , where $(A ⊖ B) (x) = {\begin{matrix} A (x), x \in X - Y, \\ (A (x) - B (x)) \lor 0, x \in X \cap Y, \\ B (x), x \in Y - X . \end{matrix}$

Obviously, $[X]^{m} ⊔ [Y]^{n} \subseteq [X \cup Y]^{m \lor n},$ $[X]^{m} ⊓ [Y]^{n} \subseteq [X \cap Y]^{m \lor n},$ $[X]^{m} \oplus [Y]^{n} \subseteq [X \cup Y]^{m + n},$ $[X]^{m} ⊖ [Y]^{n} \subseteq [X \cup Y]^{m \lor n} .$

Definition 2.5. ([4, 8]) Given the universe X. Suppose m ∈ N. Let [X] ^m be the multiset space over X. Then for any multiset A ∈ [X] ^m, the complement of A in [X] ^m, denoted by A^c, is defined as $\forall x \in X, A^{c} (x) = m - A (x) .$

Definition 2.6. ([3]) Suppose that M₁ and M₂ are two multisets drawn from X. Then Cartesian product of M₁ and M₂ is defined as

$M_{1} \times M_{2} = {(m / x, n / y) / mn : x \in^{m} M_{1}, y \in^{n} M_{2}} .$

Obviously, for every x, y ∈ X, C_M₁×M₂ (x, y) = C_{M
₁} (x) C_{M
₂} (y) or (M₁ × M₂) (x, y) = M₁ (x) M₂ (y) .

Definition 2.7. ([3]) Suppose that M is the multiset drawn from X. Then R is called a multiset relation on M, if R ⊆ M × M (i.e., for any x, y ∈ X, C_R (x, y) ≤ C_M×M (x, y) or R (x, y) ≤ (M × M) (x, y)).

Given x, y ∈ X. If m ≤ M (x), n ≤ M (y) and (m/x, n/y)/mn ∈ R, then we say that m/x is R-related to n/y, which is denoted by m/xRn/y.

Definition 2.8. ([3]) Let M be a multiset drawn from X. Suppose that R is a multiset relation on M. Then R is called

(1) reflexive, if m/xRm/x for any m/x ∈ M;

(2) symmetric, if m/xRn/y implies n/yRm/x;

(3) transitive, if m/xRn/y and n/yRk/z imply m/xRk/z.

Let R be a multiset relation on M. Then R is called an equivalence multiset relation on M, if R is reflexive, symmetric and transitive.

Example 2.9 Let X = {x, y, z}. Suppose that M = {3/x, 4/y, 5/z} is a multiset drawn from X. Put R = {(3/x, 3/x)/9, (3/x, 4/y)/12, (3/x, 5/z)/15, (4/y, 3/x)/12, (4/y, 4/y)/16, (4/y, 5/z)/20, (5/z, 3/x)/15, (5/z, 4/y)/15, (5/z, 5/z)/25} .

Then R is an equivalence multiset relation on M. For convenience, R can be represented by $M (R) = (\begin{matrix} (3, 3) & (3, 4) & (3, 5) \\ (4, 3) & (4, 4) & (4, 5) \\ (5, 3) & (5, 4) & (5, 5) \end{matrix}) .$

Definition 2.10. (Definition 3.7 in [3]) Let M be a multiset drawn from X. Given that R is a multiset relation on M. Then the R-relative multiset of m/x, is denoted by R (m/x), is defined as follows: $R (m / x) = {n / y : \exists k, k / xRn / y} .$ If R is equivalence, then we denote [m/x] _R = R (m/x).

2.2 Pawlak rough sets

Rough set theory, proposed by Pawlak [12], is a mathematical tool for disposing uncertainty of knowledge.

Let R be an equivalence relation on X. Then the pair (X, R) is called a Pawlak approximation space. Based on (X, R), a pair of operations $\underline{R}$ , $\bar{R}$ : 2^X → 2^X is defined as follows: $\underline{R} (A) = {x \in X : [x]_{R} \subseteq A},$ $\bar{R} (A) = {x \in X : [x]_{R} \cap A \neq \emptyset} .$ Then $\underline{R} (A)$ and $\bar{R} (A)$ are called the lower and upper approximates of A ∈ 2^X with respect to (X, R), respectively.

The following definition is used to deal with the lower and upper approximates of a given fuzzy set.

Definition 2.11. ([19]) Let (X, R) be a Pawlak approximation space. Based on (X, R), define ${\underline{A}}_{R} (x) = ⋀_{y \in [x]_{R}} A (y),$ ${\bar{A}}_{R} (x) = ⋁_{y \in [x]_{R}} A (y) (x \in X) .$ Then ${\underline{A}}_{R}$ and ${\bar{A}}_{R}$ are called the lower and upper approximates of A ∈ I^X with respect to (X, R), respectively.

3 An example

In this section, we provide an example to show that there is a flaw for the definition of rough multiset proposed in [4].

Definition 3.1. (Definition 9.1 in [4]) Let M be the multiset drawn from X. Suppose that R is an equivalence multiset relation on M. For any A ⊆ M, lower and upper multiset approximations R_L (A) and R_U (A), are defined by $R_{L} (A) = {m / x : [m / x]_{R} \subseteq A},$ $R_{U} (A) = {m / x : [m / x]_{R} \cap A \neq \tilde{\emptyset}} .$

If R_L (A) ≠ R_U (A), then A is called a rough multiset with respect to R; if R_L (A) = R_U (A), then A is called a definable multiset with respect to R.

This definition means that if [m/x] _R ⊆ A or v ([m/x] _R) ⪯ v (A), then R_L (A) (x) = m. It also means that if $[m / x]_{R} \cap A \neq \tilde{\emptyset}$ , then R_U (A) (x) = m.

Meanwhile, this definition means that if [m/x] _R⊊A or v ([m/x] _R) ⋠v (A), then R_L (A) (x) =0. It also means that if $[m / x]_{R} \cap A = \tilde{\emptyset}$ , then R_U (A) (x) =0.

The following example shows the fact that Definition 9.1 in [4] is imperfect or irrational.

Example 3.2 (Continued from Example 5.8 in [3]) Given X = {x, y, z}. Let M = {4/x, 5/y, 7/z} be the multiset drawn from X. Then v (M) = (4, 5, 7).

Put $M (R) = (\begin{matrix} (4, 4) & (4, 5) & (0, 0) \\ (5, 4) & (5, 5) & (0, 0) \\ (0, 0) & (0, 0) & (7, 7) \end{matrix}) .$ Then R is an equivalence multiset relation on M.

Obviously [1/x] _R = [2/x] _R = [3/x] _R = [4/x] _R = {4/x, 5/y}.

Then v ([1/x] _R) = v ([2/x] _R) = v ([3/x] _R) = v ([4/x] _R) = (4, 5, 0).

Pick A = {4/x, 5/y, 1/z}. Then v (A) = (4, 5, 1). Since v (A) ⪯ v (M), we have A ⊆ M.

Note that v ([1/x] _R) ⪯ v (A), v ([2/x] _R) ⪯ v (A), v ([3/x] _R) ⪯ v (A), v ([4/x] _R) ⪯ v (A). Then $R_{L} (A) (x) = 1 or 2 or 3 or 4 .$

Thus, Definition 9.1 in [4] is imperfect or irrational.

Note that Definitions 9.2 and 9.5 in [4] are both equivalent to Definition 9.1 in [4]. Then, Definitions 9.2 and 9.5 in [4] are both imperfect or irrational. Since Theorems 9.3 and 9.6 in [4] are both based on Definition 9.5 in [4], we can conclude that these two theorems are not necessarily right.

4 Suggested modifications

In this section, we propose a new rough multiset model and give its properties.

Note that a multiset is essentially a mapping. Then similar to Definition 2.11, we give the following definition.

Definition 4.1. Let X = {x₁, x₂, ⋯ , x_n}. Given that $M = {m_{1} / x_{1}, m_{2} / x_{2}, \dots, m_{n} / x_{n}}$ is a multiset drawn from X. Suppose that R is an equivalence multiset relation on M.

Denote $Γ_{k}^{i} = {x \in X : \exists n_{k}^{i} \leq M (x), n_{k}^{i} / x \in [k / x_{i}]_{R}}$ $(1 \leq k \leq m_{i}, 1 \leq i \leq n),$ $Γ^{i} = ⋃_{k = 1}^{m_{i}} Γ_{k}^{i} (1 \leq i \leq n) .$

For any A ⊆ M and i ∈ {1, 2, ⋯ , n}, define $R_{L} (A) (x_{i}) = ⋀_{x \in Γ^{i}} A (x), R_{U} (A) (x_{i}) = ⋁_{x \in Γ^{i}} A (x),$ ${Bnd}_{R} (A) = R_{U} (A) ⊖ R_{L} (A) .$ Then R_L (A) and R_U (A) are called the lower and upper multiset approximation of A with respect to R, respectively, and Bnd_R (A) is called the boundary of rough multiset.

If R_L (A) ≠ R_U (A), then A is called a rough multiset with respect to R; if R_L (A) = R_U (A), then A is called a definable multiset with respect to R.

Example 4.2 (Continued from Example 3.2) We have $v (M) = (4, 5, 7), v (A) = (4, 5, 1),$ $M (R) = (\begin{matrix} (4, 4) & (4, 5) & (0, 0) \\ (5, 4) & (5, 5) & (0, 0) \\ (0, 0) & (0, 0) & (7, 7) \end{matrix}) .$

Denote $x_{1} = x, x_{2} = y, x_{3} = z .$

We have v ([1/x₁] _R) = v ([2/x₁] _R) = v ([3/x₁] _R) = (0, 0, 0) , v ([4/x₁] _R) = (4, 5, 0);v ([1/x₂] _R) = v ([2/x₂] _R) = v ([3/x₂] _R) = v ([4/x₂] _R) = v ([5/x₂] _R) = (4, 5, 0);v ([1/x₃] _R) = v ([2/x₃] _R) = v ([3/x₃] _R) = v ([4/x₃] _R) = v ([5/x₃] _R) = v ([6/x₃] _R) = v ([7/x₃] _R) = (0, 0, 7).

Then $Γ_{1}^{1} = Γ_{2}^{1} = Γ_{3}^{1} = Γ_{4}^{1} = {x_{1}, x_{2}};$ $Γ_{1}^{2} = Γ_{2}^{2} = Γ_{3}^{2} = Γ_{4}^{2} = Γ_{5}^{2} = {x_{1}, x_{2}};$ $Γ_{1}^{3} = Γ_{2}^{3} = Γ_{3}^{3} = Γ_{4}^{3} = Γ_{5}^{3} = Γ_{6}^{3} = Γ_{7}^{3} = {x_{3}} .$

So $Γ^{1} = Γ_{1}^{1} \cup Γ_{2}^{1} \cup Γ_{3}^{1} \cup Γ_{4}^{1} = {x_{1}, x_{2}};$ $Γ^{2} = Γ_{1}^{2} \cup Γ_{2}^{2} \cup Γ_{3}^{2} \cup Γ_{4}^{2} \cup Γ_{5}^{2} = {x_{1}, x_{2}};$ $Γ^{3} = Γ_{1}^{3} \cup Γ_{2}^{3} \cup Γ_{3}^{3} \cup Γ_{4}^{3} \cup Γ_{5}^{3} \cup Γ_{6}^{3} \cup Γ_{7}^{3} = {x_{3}} .$

Thus $\begin{matrix} R_{L} (A) (x_{1}) = A (x_{1}) \land A (x_{2}) = 4 \land 5 = 4, \\ R_{L} (A) (x_{2}) = A (x_{1}) \land A (x_{2}) = 4 \land 5 = 4, \\ R_{L} (A) (x_{3}) = A (x_{3}) = 1; \\ R_{U} (A) (x_{1}) = A (x_{1}) \lor A (x_{2}) = 4 \lor 5 = 5, \\ R_{U} (A) (x_{2}) = A (x_{1}) \lor A (x_{2}) = 4 \lor 5 = 5, \\ R_{U} (A) (x_{3}) = A (x_{3}) = 1 . \end{matrix}$

Hence $R_{L} (A) = {4 / x_{1}, 4 / x_{2}, 1 / x_{3}} = {4 / x, 4 / y, 1 / z},$ $R_{U} (A) = {5 / x_{1}, 5 / x_{2}, 1 / x_{3}} = {5 / x, 5 / y, 1 / z} .$

Theorem 4.3 Given the universe X. Let M is a multiset drawn from X. Suppose that R is an equivalence multiset relation on M. Then (1) ∀ A ⊆ M, R_L (A) ⊆ A ⊆ R_U (A). (2) ∀ A₁, A₂ ⊆ M, R_L (A₁ ∩ A₂) = R_L (A₁) ∩ R_L (A₂) , R_U (A₁ ∪ A₂) = R_U (A₁) ∪ R_U (A₂). (3) ∀ A₁, A₂ ⊆ M, R_L (A₁) ∪ R_L (A₂) ⊆ R_L (A₁ ∪ A₂) , R_U (A₁ ∩ A₂) ⊆ R_U (A₁) ∩ R_U (A₂). (4) ∀ A ⊆ M, R_U (R_U (A)) = R_L (R_U (A)) = R_U (A) , R_L (R_L (A)) = R_U (R_L (A)) = R_L (A). (5) $R_{L} (M) = M, R_{U} (\tilde{\emptyset}) = \tilde{\emptyset} .$

Proof. (1) This is obvious. $\begin{matrix} (2) R_{L} (A_{1} \cap A_{2}) (x) & = ⋀_{x \in Γ^{i}} (A_{1} \cap A_{2}) (x) \\ = ⋀_{x \in Γ^{i}} [A_{1} (x) \land A_{2} (x)] \\ = [⋀_{x \in Γ^{i}} A_{1} (x)] \land [⋀_{x \in Γ^{i}} A_{2} (x)] \\ = R_{L} (A_{1}) (x) \land R_{L} (A_{2}) (x), \end{matrix}$ $\begin{matrix} R_{U} (A_{1} \cup A_{2}) (x) & = ⋁_{x \in Γ^{i}} (A_{1} \cup A_{2}) (x) \\ = ⋁_{x \in Γ^{i}} [A_{1} (x) \lor A_{2} (x)] \\ = [⋁_{x \in Γ^{i}} A_{1} (x)] \lor [⋁_{x \in Γ^{i}} A_{2} (x)] \\ = R_{U} (A_{1}) (x) \lor R_{U} (A_{2}) (x) . \end{matrix}$ (3) (R_L (A₁) ∪ R_L (A₂)) (x)

= R_L (A₁) (x) ∨ R_LA₂ (x)

= [⋀ _x∈ΓⁱA₁ (x)] ∨ [⋀ _x∈Γⁱ (A₂) (x)]

≤ ⋀ _x∈Γⁱ [A₁ (x) ∨ A₂ (x)]

= R_L (A₁ ∪ A₂) (x) , $\begin{matrix} R_{U} (A_{1} \cap A_{2}) (x) & = ⋁_{x \in Γ^{i}} (A_{1} \cap A_{2}) (x) \\ \leq [⋁_{x \in Γ^{i}} A_{1} (x)] \land [⋁_{x \in Γ^{i}} A_{2} (x)] \\ = R_{U} (A_{1}) (x) \land R_{U} (A_{2}) (x) \\ = (R_{U} (A_{1}) \cap R_{U} (A_{2})) (x) . \end{matrix}$ (4) $\begin{matrix} R_{U} (R_{U} (A)) (x) & = ⋁_{x \in Γ^{i}} (R_{U} (A)) (x) \\ = ⋁_{x \in Γ^{i}} (⋁_{x \in Γ^{i}} A) (x) \\ = ⋀_{x \in Γ^{i}} (⋁_{x \in Γ^{i}} A) (x) \\ = R_{L} (R_{U} (A)) (x) \\ = ⋁_{x \in Γ^{i}} A (x) \\ = R_{U} (A) (x), \end{matrix}$ $\begin{matrix} R_{L} (R_{L} (A)) (x) & = ⋀_{x \in Γ^{i}} (R_{L} (A)) (x) \\ = ⋀_{x \in Γ^{i}} (⋀_{x \in Γ^{i}} A) (x) \\ = ⋁_{x \in Γ^{i}} (⋀_{x \in Γ^{i}} A) (x) \\ = R_{U} (R_{L} (A)) (x) \\ = ⋀_{x \in Γ^{i}} A (x) \\ = R_{L} (A) (x) . \end{matrix}$

(5) $R_{L} (M) (x) = ⋀_{x \in Γ^{i}} M (x) = M (x),$ $R_{U} (\tilde{\emptyset}) (x) = ⋁_{x \in Γ^{i}} \tilde{\emptyset} (x) = 0 .$ □

Theorem 4.4 Given the universe X and a natural number m. Suppose that R_m is an equivalence multiset relation on X_m. If A ⊆ X_m, then $R_{L} (A^{c}) = (R_{U} (A))^{c}, R_{U} (A^{c}) = (R_{L} (A))^{c} .$

Proof. $\begin{matrix} R_{L} (A^{c}) (x) & = ⋀_{x \in Γ^{i}} (m - A (x)) = m - ⋁_{x \in Γ^{i}} A (x) \\ = m - R_{U} (A) (x) = (R_{U} (A))^{c} (x), \end{matrix}$ $\begin{matrix} R_{U} (A^{c}) (x) & = ⋁_{x \in Γ^{i}} (m - A (x)) = m - ⋀_{x \in Γ^{i}} A (x) \\ = m - R_{L} (A) (x) = (R_{L} (A))^{c} (x) . \end{matrix}$ □

5 Applications of rough multisets

As applications of rough multisets, accuracy, roughness of a rough multiset, and multiset topology are given in this section.

Definition 5.1. Let M be a multiset drawn from X. Given that R is an equivalence multiset relation on M. Suppose that A ⊆ M is a rough multiset with respect to R. Then accuracy and roughness of A are expressed as α_R (A) and ρ_R (A), respectively. They are defined as follows: $α_{R} (A) = \frac{| R_{L} (A) |}{| R_{U} (A) |}, ρ_{R} (A) = 1 - α_{B} (A) .$

Example 5.2 (Continued from Example 4.2) We have $R_{L} (A) = {4 / x_{1}, 4 / x_{2}, 1 / x_{3}} = {4 / x, 4 / y, 1 / z},$ $R_{U} (A) = {5 / x_{1}, 5 / x_{2}, 1 / x_{3}} = {5 / x, 5 / y, 1 / z} .$ Then $α_{R} (A) = \frac{| R_{L} (A) |}{| R_{U} (A) |} = \frac{4 + 4 + 1}{5 + 5 + 1} = \frac{9}{11},$ $ρ_{R} (A) = 1 - \frac{9}{11} = \frac{2}{11} .$

Definition 5.3. ([4]) Given the universe X and a natural number m. τ ⊆ [X] ^m is called a multiset topology on X_m, if

(i) $X_{m}, \tilde{\emptyset} \in τ$ ;

(ii) A, B ∈ τ implie A ∩ B ∈ τ;

(iii) {A_j : j ∈ J} ⊆ τ implies ⋃ _j∈JA_j ∈ τ.

The interior and closure of A ∈ [X] ^m, denoted by int_τ (A) and cl_τ (A), respectively, are defined as follows: ${int}_{τ} (A) = ⋃ {B \in τ : B \subseteq A},$ ${cl}_{τ} (A) = ⋂ {B^{c} \in τ : X_{m} \supseteq B \supseteq A} .$

Proposition 5.4. Given the universe X and a natural number m. Suppose that R_m is an equivalence multiset relation on X_m. Denote $τ_{R_{m}} = {A \subseteq X_{m} : R_{L} (A) = A} .$ Then τ_{R
_m} is a multiset topology.

Proof. This holds by Theorems 4.3 and 4.4.□

Theorem 5.5 Given the universe X and a natural number m. Suppose that R_m is an equivalence multiset relation on X_m. Then ${int}_{τ_{R_{m}}} \leq (R_{m})_{L} \leq (R_{m})_{U} \leq {cl}_{τ_{R_{m}}},$ where “≤" means the size of an operator.

Proof. By Theorem 4.4, (R_m) _L ≤ (R_m) _U.

Given X = {x₁, x₂, ⋯ , x_n}. Then v (X_m) = (m, m, ⋯ , m). Denote $Γ_{k}^{i} = {x \in X : n_{k}^{i} / x \in [k / x_{i}]_{R}} (1 \leq k \leq m, 1 \leq i \leq n),$ $Γ^{i} = ⋃_{k = 1}^{m} Γ_{k}^{i} (1 \leq i \leq n) .$ For any A ⊆ X_m and i ∈ {1, 2, ⋯ , n}, we have $\begin{matrix} {int}_{τ_{R_{m}}} (A) (x_{i}) & = ⋁ {B (x_{i}) : B \in τ_{R_{m}}, B \subseteq A} \\ = ⋁_{B \subseteq A} (R_{m})_{L} (B) (x_{i}) \\ = ⋁_{B \subseteq A} ⋀_{x \in Γ^{i}} B (x) \\ \leq ⋀_{x \in Γ^{i}} ⋁_{B \subseteq A} B (x) \\ = ⋀_{x \in Γ^{i}} A (x) \\ = (R_{m})_{L} (A) (x_{i}) \end{matrix}$ Thus int_{τ
_{R
_m}} ≤ (R_m) _L.

Since cl_{τ
_{R
_m}} is the dual of int_{τ
_{R
_m}}, by Theorem 4.5 and int_{τ
_{R
_m}} ≤ (R_m) _L, we can be concluded that (R_m) _U ≤ cl_{τ
_{R
_m}} .□

6 Conclusions

Based on Example 3.2, it can be concluded that Definition 9.1 in [4] is imperfect or irrational. In order to improve this definition, we have proposed a new rough multiset model along with a illustrative example and some properties. Moreover, we have given applications of new rough multisets.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions, which have helped immensely in improving the quality of the paper. This work is supported by National Natural Science Foundation of China (11971420), Special Scientific Research Project of Young Innovative Talents in Guangxi (2019AC20052), Natural Science Foundation of Guangxi (2019JJA110036, AD19245102, 2018GXNSFDA294003, 2018GXNSFDA294134), Guangxi Higher Education Institutions of China (Document No.[2019] 52), Guangxi Higher Education Reform Project (2020 XJJGZD17), Research Project of Institute of Big Data in Yulin (YJKY03) and Engineering Project of Undergraduate Teaching Reform of Higher Education in Guangxi (2017JGA179).

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