Hemimetric-based λ-valued fuzzy rough sets

Abstract

A λ-subset, or a [0,λ]-valued fuzzy subset, is a mapping from a nonempty set to the interval [0,λ]. In this paper, we use the notion of hemimetrics, a kind of distance functions, as the basic structure to define and study fuzzy rough set model of λ-subsets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper/lower approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the related Lukasiewicz logical systems. We show that upper definable sets, lower definable sets and definable sets are equivalent, and they form an Alexandrov fuzzy topology. A processing of a λ-subset via fuzzy upper/lower approximation operators can actually considered as a processing of the related image, and thus has potential applications in image processing.

Keywords

Hemimetric λ-subset fuzzy rough set fuzzy upper/lower approximation operator definable set

1 Introduction

Rough set theory, originally proposed by Pawlak [19–21], provides a systematic approach for the classification of objects through an indiscernibility relation. Nowadays, the theory and methods of rough sets have been widely applied in process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis and other fields [6, 14, 22, 23, 36].

Rough sets were originated in classification problem. An equivalence relation, the equivalent concept of a partition, is the simplest form of an indiscernibility. However, the classical rough set theory can not deal with some granularity problems of information tables in the real world, and thus extensions including tolerance relations [8, 26] and similarity relations [27] are needed. We can call them (binary) relation-based rough sets. Besides from the relation-based rough sets, coverings and neighborhood systems/operators are alternative basic structures to obtain rough set models [7, 35]. In the environment of set-valued analysis, neighborhood systems/operators are always associated with coverings. A covering is a generalization of a partition, every member can be considered as an abstract of equivalence classes in the related equivalence relation. Both of covering-based and neighborhood system/operator-based rough sets are indeed the granule-based approaches, where discrete points in the universe are firstly combined into granules and then are used to describe roughness. A third kind of generalized models is obtained when in the subsystem-based definition, we replace the σ-algebra of subsets by a pair of systems: a closure system and its dual system [9, 34]. However, the above mentioned generalizations are no longer equivalent and thus yield different generalized rough set models and image process.

The real world is full of uncertainty, including fuzziness, roughness and probability. Since it is hard to define an equivalence relation for a given data set, fuzzification becomes another important method to get extensions of the classical rough set models. In the framework of relation-based fuzzy rough set theory, various fuzzy generalizations of approximation operators have been proposed and investigated. Considering the unit interval [0,1] as the truth value table, many researchers proposed various types of fuzzy rough sets and provided axiomatic characterizations of fuzzy rough approximation operators [16–18, 28–31]. In fact, the unit interval [0,1] can not be supplied as the truth value table any more in the partial ordering setting. Conforming to this trend, Radzikowska et al. [24] and She et al. [25] chose a complete residuated lattice L to investigate L-fuzzy rough approximation operators. In the framework of covering-based fuzzy rough set, a family of fuzzy subsets of the universe is used to define the concept of fuzzy coverings, from which different pairs of upper and lower approximation rough operators can be constructed [12, 15]. Like the classical setting, fuzzy neighborhood systems are also strongly associated to fuzzy covering. For example in the recent paper [10], Dąŕeer, Cornelis and Godo introduced the notion of fuzzy neighborhood systems based on a given fuzzy coverings, as well as the notion of the fuzzy minimal and maximal descriptions. Sixteen different fuzzy neighborhood operators are defined and studied therein. Besides the above methods, the hemimetric-base fuzzy rough set [32,33] is a new model of fuzzy rough set which has potential applications in fuzzy clustering. Some approaches to fuzzy rough set theory in recent years include containment neighborhoods [1], somewhere dense sets [2], subset neighborhoods [3], E-neighborhoods [4], fuzzy soft coverings [5].

We know that every two dimensional grayscale image can be regarded as a $[0, 255] \cap ℤ$ -valued fuzzy subset of the Euclidean plane, and every two dimensional color-scale image can be regarded as a composition of three grayscale images [33]. For a positive real number λ, the λ-subset (specifically, [0, λ]-fuzzy subset) in this paper can be regarded as an abstraction of the image, and a processing of λ-subset is actually equivalent to a processing of the related image. In [32,33], Yao et al. used this idea to define and lattice-valued fuzzy rough sets and [0,1]-valued fuzzy rough sets based on hemimetrics and study their applications in clustering analysis and image processing. This paper will follow the step of [32,33] and continue to propose a fuzzy rough set model of λ-subsets based on hemimetrics and explore its basic properties to further applications.

2 A λ-valued fuzzy rough set model

In this section, based on a hemimetric on a set X, we shall define a pair of fuzzy rough approximation operators on the family of all λ-subsets of X, and then investigate their properties.

Let X be a nonempty set. By a λ-subset of X, we mean a mapping A : X ⟶ [0, λ], which is an extension of fuzzy subsets. The family of all λ-subsets of X is denoted by $F_{λ} (X)$ . The constant λ-subset with a value a ∈ [0, λ] is denoted by a_X. The operations and the order relation on $F_{λ} (X)$ are pointwisely defined as usual. The λ-subsets can be considered as a model of gray and color images (λ = 255) and land surfaces (λ = 8848).

Definition 2.1. [13] Let X be a nonempty set. A mapping d : X × X ⟶ [0, + ∞) is called a hemimetric on X if

(M1) d (x, x) =0 (∀ x ∈ X);

(M2) d (x, z) ⩽ d (x, y) + d (y, z) (∀ x, y, z ∈ X).

The pair (X, d) is called a hemimetric space.

Definition 2.2. ([33] for λ = 1, [32] for a complete residuated lattice as the valued lattice) Let (X, d) be a hemimetric space. Define two operators ${\bar{Apr}}_{d}, {\underline{Apr}}_{d} : F_{λ} (X) ⟶ F_{λ} (X)$ respectively by: for every x ∈ X, ${\bar{Apr}}_{d} (A) (x) = ⋁_{y \in X} A (y) - d (y, x);$ ${\underline{Apr}}_{d} (A) (x) = ⋀_{y \in X} A (y) + d (x, y) .$ The operators ${\bar{Apr}}_{d}, {\underline{Apr}}_{d}$ are respectively called the λ-valued fuzzy upper rough approximation operator (upper operator, for short) and the λ-valued fuzzy lower rough approximation operator (lower operator, for short) on X induced by the hemimetric d.

The upper/lower operators in fact can be considered as a special kind of closure/interior operators of an Alexandrov fuzzy topological space respectively.

Proposition 2.3. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ , a ∈ [0, λ] and ${A_{i} ∣ i \in I} \subseteq F_{λ} (X)$ , it holds that

(U1) $A ⩽ {\bar{Apr}}_{d} (A) ⩽ λ_{X}$ ;

(U2) ${\bar{Apr}}_{d} (a_{X}) = a_{X}$ ;

(U3) ${\bar{Apr}}_{d} (⋁_{i} A_{i}) = ⋁_{i} {\bar{Apr}}_{d} (A_{i})$ ;

(U4) ${\bar{Apr}}_{d} ({\bar{Apr}}_{d} (A)) = {\bar{Apr}}_{d} (A)$ .

Proof. (U1) Clearly, ${\bar{Apr}}_{d} (A) ⩽ λ_{X}$ . For every x ∈ X, $\begin{matrix} {\bar{Apr}}_{d} (A) (x) & = & ⋁_{y \in X} [A (x) - d (x, y)] \\ ⩾ & A (x) - d (x, x) \\ = & A (x) - 0 = A (x) . \end{matrix}$

Hence, ${\bar{Apr}}_{d} (A) ⩾ A$ .

(U2) By (U1), ${\bar{Apr}}_{d} (a_{X}) ⩾ a_{X}$ . For every x ∈ X, ${\bar{Apr}}_{d} (a_{X}) = ⋁_{y \in X} a - d (y, x) ⩽ a$ . Hence ${\bar{Apr}}_{d} (a_{X}) = a_{X}$ .

(U3) For every x ∈ X,

$\begin{matrix} {\bar{Apr}}_{d} (⋁_{i} A_{i}) (x) & = & ⋁_{y \in X} [⋁_{i} A (y) - d (y, x)] \\ = & ⋁_{i} ⋁_{y \in X} A_{i} (y) - d (y, x) \\ = & ⋁_{i} {\bar{Apr}}_{d} (A_{i}) (x) . \end{matrix}$

Hence, ${\bar{Apr}}_{d} (⋁_{i} A_{i}) = ⋁_{i} {\bar{Apr}}_{d} (A_{i})$ .

(U4) By (U1), ${\bar{Apr}}_{d} ({\bar{Apr}}_{d} (A)) ⩾ {\bar{Apr}}_{d} (A)$ . For every z ∈ X, $\begin{matrix} {\bar{Apr}}_{d} ({\bar{Apr}}_{d} (A)) (z) \\ = & ⋁_{y \in X} {\bar{Apr}}_{d} (A) (y) - d (y, z) \\ = & ⋁_{y \in X} [⋁_{x \in X} A (x) - d (x, y)] - d (y, z) \\ = & ⋁_{y \in X} ⋁_{x \in X} A (x) - [d (x, y) + d (y, z)] \\ ⩽ & ⋁_{x \in X} A (x) - d (x, z) \\ = & {\bar{Apr}}_{d} A (z) . \end{matrix}$

Hence, ${\bar{Apr}}_{d} ({\bar{Apr}}_{d} (A)) = {\bar{Apr}}_{d} (A)$ . □

Proposition 2.4. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ , a ∈ [0, λ] and ${A_{i} ∣ i \in I} \subseteq F_{λ} (X)$ , it holds that

(L1) $0_{X} ⩽ {\underline{Apr}}_{d} (A) ⩽ A$ ;

(L2) ${\underline{Apr}}_{d} (a_{X}) = a_{X}$ ;

(L3) ${\underline{Apr}}_{d} (⋀_{i} A_{i}) = ⋀_{i} {\underline{Apr}}_{d} (A_{i})$ ;

(L4) ${\underline{Apr}}_{d} ({\underline{Apr}}_{d} (A)) = {\underline{Apr}}_{d} (A)$ .

Proof. (L1) Clearly, ${\underline{Apr}}_{d} (A) ⩾ 0_{X}$ . For every x ∈ X, $\begin{matrix} {\underline{Apr}}_{d} (A) (x) & = & ⋀_{y \in X} A (y) + d (x, y) \\ ⩽ & A (x) + d (x, x) = A (x) . \end{matrix}$

Hence, ${\underline{Apr}}_{d} (A) ⩽ A$ .

(L2) By (L1), ${\underline{Apr}}_{d} (a_{X}) ⩽ a_{X}$ . For every x ∈ X, ${\underline{Apr}}_{d} (a_{X}) (x) = ⋀_{y \in X} a + d (x, y) ⩾ a$ . Hence, ${\underline{Apr}}_{d} (a_{X}) = a_{X}$ .

(L3) For every x ∈ X, $\begin{matrix} {Apr}_{d} (⋀_{i} A_{i}) (x) \\ = & ⋀_{y \in X} (⋀_{i} A_{i} (y) + d (x, y)) \\ = & ⋀_{i} ⋀_{y \in X} A_{i} (y) + d (x, y) \\ = & ⋀_{i} {\underline{Apr}}_{d} (A_{i}) (x) . \end{matrix}$

Hence, ${Apr}_{d} (⋀_{i} A_{i}) = ⋀_{i} {\underline{Apr}}_{d} (A_{i})$ .

(L4) By (L1), we have ${\underline{Apr}}_{d} ({\underline{Apr}}_{d} (A)) ⩽ {\underline{Apr}}_{d} (A)$ . For every z ∈ X, $\begin{matrix} {\underline{Apr}}_{d} ({\underline{Apr}}_{d} (A)) (z) \\ = & ⋀_{y \in X} {\underline{Apr}}_{d} (A) (y) + d (y, z) \\ = & ⋀_{y \in X} [⋀_{x \in X} A (x) + d (x, y)] + d (y, z) \\ = & ⋀_{y \in X} ⋀_{x \in X} A (x) + [d (x, y) + d (y, z)] \\ ⩾ & ⋀_{x \in X} A (x) + d (x, z) \\ = & {\underline{Apr}}_{d} A (z) . \end{matrix}$

Hence, ${\underline{Apr}}_{d} ({\underline{Apr}}_{d} (A)) = {\underline{Apr}}_{d} (A)$ . □

Proposition 2.5. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ , it holds that

(1) ${\bar{Apr}}_{d} ({\underline{Apr}}_{d} (A)) = {\underline{Apr}}_{d} (A);$

(2) ${\underline{Apr}}_{d} ({\bar{Apr}}_{d} (A)) = {\bar{Apr}}_{d} (A) .$

Proof. (1) $\begin{matrix} {\bar{Apr}}_{d} ({\underline{Apr}}_{d} (A)) (x) \\ = & ⋁_{y \in X} {\underline{Apr}}_{d} (A) (y) - d (y, x) \\ = & ⋁_{y \in X} [⋀_{z \in X} A (z) + d (y, z)] - d (y, x) \\ = & ⋁_{y \in X} ⋀_{z \in X} [A (z) + d (y, z) - d (y, x)] \\ = & ⋁_{y \in X} ⋀_{z \in X} A (z) + [d (y, z) - d (y, x)] \\ ⩽ & ⋀_{z \in X} A (z) + d (x, z) \\ = & {\underline{Apr}}_{d} A (x) . \end{matrix}$

Together with (U1), ${\bar{Apr}}_{d} ({\underline{Apr}}_{d} (A)) = {\underline{Apr}}_{d} (A)$ .

(2) $\begin{matrix} {\underline{Apr}}_{d} ({\bar{Apr}}_{d} (A)) (x) \\ = & ⋀_{y \in X} {\bar{Apr}}_{d} (A) (y) + d (x, y) \\ = & ⋀_{y \in X} [⋁_{z \in X} A (z) - d (z, y)] + d (x, y) \\ = & ⋀_{y \in X} ⋁_{z \in X} [A (z) - d (z, y) + d (x, y)] \\ = & ⋀_{y \in X} ⋁_{z \in X} [A (z) - (d (z, y) - d (x, y))] \\ ⩾ & ⋁_{z \in X} A (z) - d (z, x) \\ = & {\bar{Apr}}_{d} A (x) . \end{matrix}$

Together with (L1), ${\underline{Apr}}_{d} ({\bar{Apr}}_{d} (A)) = {\bar{Apr}}_{d} (A) .$ □

3 Definable sets and the related topological structures

In this section, we will study the definability of λ-subsets and then investigate properties of the related topological structures.

Definition 3.1. Let (X, d) be a hemimetric space. A λ-subset A of X is called

(1) upper definable if ${\bar{Apr}}_{d} (A) = A$ , or equivalently, ${\bar{Apr}}_{d} (A) ⩽ A$ .

(2) lower definable if ${\underline{Apr}}_{d} (A) = A$ , or equivalently, ${\underline{Apr}}_{d} (A) ⩾ A$ .

(3) definable if it is both upper definable and lower definable.

Proposition 3.2. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ , the following statements are equivalent:

(1) A is upper definable;

(2) A (x) - A (y) ⩽ d (x, y) for all x, y ∈ X;

(3) A is lower definable. □

Proof. $\begin{matrix} (1) A is upper definable \\ \Leftrightarrow & {\bar{Apr}}_{d} (A) ⩽ A \\ \Leftrightarrow & \forall y \in X, {\bar{Apr}}_{d} (A) (y) ⩽ A (y) \\ \Leftrightarrow & \forall y \in X, ⋁_{x \in X} A (x) - d (x, y) ⩽ A (y) \\ \Leftrightarrow & \forall x, y \in X, A (x) - d (x, y) ⩽ A (y) \\ \Leftrightarrow & (2) \forall x, y \in X, A (x) - A (y) ⩽ d (x, y) \\ \Leftrightarrow & \forall x, y \in X, A (x) ⩽ A (y) + d (x, y) \\ \Leftrightarrow & \forall x \in X, A (x) ⩽ ⋀_{y \in X} A (y) + d (x, y) \\ \Leftrightarrow & \forall x \in X, A (x) ⩽ {\underline{Apr}}_{d} A (x) \\ \Leftrightarrow & A ⩽ {\underline{Apr}}_{d} A \\ \Leftrightarrow & (3) A is lower definable . □ \end{matrix}$

By Proposition 3.2, we know that upper definable sets, lower definable sets and definable sets are equivalent to each other. We denote the family of all definable sets by $D F_{λ} (X)$ . As has been stated above, the upper/lower operator is a closure/interior operator of an Alexandrov fuzzy topological space respectively. The following result will show the related Alexandrov fuzzy topology is exactly $D F_{λ} (X)$ .

Proposition 3.3. Let (X, d) be a hemimetric space. Then $D F_{λ} (X)$ is a stratified Alexandrov fuzzy topology, that is,

(DF1) $a_{X} \in D F_{λ} (X)$ (∀ a ∈ [0, λ]);

(DF2) $⋀_{i} A_{i}, ⋁_{i} A_{i} \in D F_{λ} (X)$ $(\forall {A_{i} ∣ i \in I} \in D F_{λ} (X))$ ;

Proof. It is straightforward by Proposition 3.2(2). □

Proposition 3.4. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ , it holds that

(1) ${\bar{Apr}}_{d} (A) = ⋀ {B \in D F_{λ} (X) ∣ A ⩽ B}$ ;

(2) ${\underline{Apr}}_{d} (A) = ⋁ {C \in D F_{λ} (X) ∣ C ⩽ A}$ .

Proof. (1) By (U1), we get that $A ⩽ {\bar{Apr}}_{d} (A)$ , and due to (U4), we have ${\bar{Apr}}_{d} (A) ⩽ {\bar{Apr}}_{d} ({\bar{Apr}}_{d} (A))$ . Hence, ${\bar{Apr}}_{d} (A) \in D F_{λ} (X)$ . If $B \in D F_{λ} (X)$ and A ⩽ B, then ${\bar{Apr}}_{d} (A) ⩽ {\bar{Apr}}_{d} (B)$ . According to (U2), $B = {\bar{Apr}}_{d} (B)$ , so ${\bar{Apr}}_{d} (A) ⩽ B$ . Hence, ${\bar{Apr}}_{d} (A) = ⋀ {B \in D F_{λ} (X) ∣ A ⩽ B}$ ;

(2) By (L1), we get that ${\underline{Apr}}_{d} (A) ⩽ A$ , and due to (L4), we have ${\underline{Apr}}_{d} ({\underline{Apr}}_{d} (A)) ⩽ {\underline{Apr}}_{d} (A)$ . Hence, ${\underline{Apr}}_{d} (A) \in D F_{λ} (X)$ . If $C \in D F_{λ} (X)$ and C ⩽ A, then ${\underline{Apr}}_{d} (C) ⩽ {\underline{Apr}}_{d} (A)$ . According to (L2), $C = {\underline{Apr}}_{d} (C)$ , so $C ⩽ {\underline{Apr}}_{d} (A)$ . Hence, ${\underline{Apr}}_{d} (A) = ⋁ {C \in D F_{λ} (X) ∣ C ⩽ A}$ . □

4 Logical description of upper/lower operators

We will study hemimetric-based λ-valued fuzzy rough sets. In order to give some logical descriptions of definitions and formulas, we need logical operations on [0, λ]. We know that for the unit interval, the most useful logical system is the Lukasiewicz system, which is a concrete MV-algebraic system. Since the truth value table is [0, λ], we firstly will extend those operations on [0,1] to [0, λ] as follows:

Difference: a ⊖ b = max {a - b, 0};

Conjuction: a ⊗ b = max {a + b - λ, 0};

Implication: a → b = min {λ - a + b, λ};

Negation: ¬a = λ - a.

Then the pair (⊗,→) forms a Galois adjoint, that is, a ⊗ b ⩽ c iff a ⩽ b → c (∀ a, b, c ∈ [0, λ]) . The quadruple ([0, λ] , ⊗ , ¬ , 0) becomes an MV-algebra.

Proposition 4.1. For all a, b, c ∈ [0, λ], it holds that

(1) a ⊕ b = ¬ (¬ a ⊗ ¬ b) = (¬ a) → b;

(2) a ⊗ b = ¬ (¬ a ⊕ ¬ b) = a ⊖ (¬ b);

(3) a → b = ¬ a ⊗ b = ¬ (¬ a ⊖ b);

(4) a ⊖ b = a ⊗ ¬ b = ¬ (¬ a ⊕ b);

(5) (a ⊗ b) ⊖ c = a ⊗ (b ⊖ c);

(6) (b ⊖ a) ⊖ c = b ⊖ (a ⊕ c);

(7) (a → b) ⊕ c = a → (b ⊕ c).

For the purpose of logical description, we need the λ-truncation of hemimetrics.

Proposition 4.2. Let (X, d) be a hemimetric space. Define d^* : X × X → [0, λ] by

d^{*} (x, y) = min {d (x, y), λ} (\forall x, y \in X),

called the λ-truncation of d. Then

(1) d^* (x, x) =0 (∀ x ∈ X);

(2) d^* (x, z) ⩽ d^* (x, y) ⊕ d^* (y, z) (∀ x, y, z ∈ X).

We call d^* an ⊕-hemimetric on X.

Proof. We here only prove (2). For all x, y, z ∈ X, $\begin{matrix} d^{*} (x, y) \oplus d^{*} (y, z) \\ = & min {d^{*} (x, y) + d^{*} (y, z), λ} \\ = & min {min {d (x, y), λ} + min {d (y, z), λ}, λ} \\ = & min {d (x, y) + d (y, z), d (x, y) + λ, d (y, z) + λ, λ, λ} \\ = & min {d (x, y) + d (y, z), λ} \\ ⩾ & min {d (x, z), λ} \\ = & d^{*} (x, z) . □ \end{matrix}$

Proposition 4.3. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ , For every x ∈ X, it holds that

(1) ${\bar{Apr}}_{d} (A) (x) = ⋁_{y \in X} [A (y) ⊖ d^{*} (y, x)]$ ;

(2) ${\underline{Apr}}_{d} A (x) = ⋀_{y \in X} [A (y) \oplus d^{*} (x, y)]$ .

Proof. (1) $\begin{matrix} ⋁_{y \in X} [A (y) ⊖ d^{*} (y, x)] \\ = & ⋁_{y \in X} max {A (y) - d^{*} (y, x), 0} \\ = & ⋁_{y \in X} max {A (y) - min {d (y, x), λ}, 0} \\ = & ⋁_{y \in X} max {A (y) - d (y, x), A (y) - λ, 0} \\ = & ⋁_{y \in X} max {A (y) - d (y, x), 0} \\ = & max {⋁_{y \in X} A (y) - d (y, x), 0} \\ = & max {{\bar{Apr}}_{d} (A) (x), 0} \\ = & {\bar{Apr}}_{d} (A) (x) . \end{matrix}$

(2) $\begin{matrix} ⋀_{y \in X} [A (y) \oplus d^{*} (y, x)] \\ = & ⋀_{y \in X} min {A (y) + d^{*} (y, x), λ} \\ = & ⋀_{y \in X} min {A (y) + min {d (y, x), λ}, λ} \\ = & ⋀_{y \in X} min {A (y) + d (y, x), d (y, x) + λ, λ} \\ = & ⋀_{y \in X} min {A (y) + d (y, x), λ} \\ = & min {⋀_{y \in X} A (y) + d (y, x), λ} \\ = & min {{\underline{Apr}}_{d} A (x), λ} \\ = & {\underline{Apr}}_{d} A (x) . □ \end{matrix}$

If a, b are two complex numbers on the complex plane, we know that a - b refers to a vector (an arrow) from b to a. In this regard, we write a ⊖ b = : b ⇝ a.

Proposition 4.4. Let (X, d) be a hemimetric space. Then for all $A \in F_{λ} (X)$ and a ∈ L, it holds that

(U5) ${\bar{Apr}}_{d} (a \otimes A) = a \otimes {\bar{Apr}}_{d} (A)$ ;

(U6) ${\bar{Apr}}_{d} (a ⇝ A) = a ⇝ {\bar{Apr}}_{d} (A)$ ;

(L5) ${\underline{Apr}}_{d} (a \oplus A) = a \oplus {\underline{Apr}}_{d} (A)$ ;

(L6) ${\underline{Apr}}_{d} (a \to A) = a \to {\underline{Apr}}_{d} (A)$ .

Proof. (U5) $\begin{matrix} {\bar{Apr}}_{d} (a \otimes A) (x) \\ = & ⋁_{y \in X} (a \otimes A) (y) - d (y, x) \\ = & ⋁_{y \in X} max {a + A (y) - λ, 0} - d (y, x) \\ = & ⋁_{y \in X} max {a + A (y) - λ - d (y, x), - d (y, x)} \\ = & ⋁_{y \in X} max {a + A (y) - λ - d (y, x), \\ = & max {a + ⋁_{y \in X} [A (y) - d (y, x)] - λ, 0}, \\ = & a \otimes {\bar{Apr}}_{d} (A) (x) . \end{matrix}$

Hence, ${\bar{Apr}}_{d} (a \otimes A) = a \otimes {\bar{Apr}}_{d} (A)$ .

(U6) $\begin{matrix} {\bar{Apr}}_{d} (a ⇝ A) (x) \\ = & ⋁_{y \in X} (a ⇝ A) (y) - d (y, x) \\ = & ⋁_{y \in X} max {A (y) - a, 0} - d (y, x) \\ = & ⋁_{y \in X} max {A (y) - a - d (y, x), - d (y, x)} \\ = & ⋁_{y \in X} max {A (y) - a - d (y, x), 0} \\ = & max {⋁_{y \in X} [A (y) - d (y, x)] - a, 0} \\ = & {\bar{Apr}}_{d} (A) (x) ⊖ a \\ = & a ⇝ {\bar{Apr}}_{d} (A) (x) . \end{matrix}$

Hence, ${\bar{Apr}}_{d} (a ⇝ A) = a ⇝ {\bar{Apr}}_{d} (A)$ .

(L5) $\begin{matrix} {\underline{Apr}}_{d} (a \oplus A) (x) \\ = & ⋀_{y \in X} (a \oplus A) (y) + d (x, y) \\ = & ⋀_{y \in X} min {a + A (y), λ} + d (x, y) \\ = & ⋀_{y \in X} min {a + A (y) + d (x, y), d (x, y) + λ} \\ = & ⋀_{y \in X} min {a + A (y) + d (x, y), λ} \\ = & min {⋀_{y \in X} A (y) + d (x, y) + a, λ} \\ = & a \oplus {\underline{Apr}}_{d} (A) (x) . \end{matrix}$

Hence, ${\underline{Apr}}_{d} (a \oplus A) = a \oplus {\underline{Apr}}_{d} (A)$ .

(L6) $\begin{matrix} {\underline{Apr}}_{d} (a \to A) (x) \\ = & ⋀_{y \in X} (a \to A) (y) + d (x, y) \\ = & ⋀_{y \in X} min {λ - a + A (y), λ} + d (x, y) \\ = & ⋀_{y \in X} min {λ - a + A (y) + d (x, y), λ + d (x, y)} \\ = & ⋀_{y \in X} min {λ - a + A (y) + d (x, y), λ} \\ = & min {λ - a + ⋀_{y \in X} [A (y) + d (x, y)], λ} \\ = & a \to {\underline{Apr}}_{d} (A) (x) . \end{matrix}$

Hence, ${\underline{Apr}}_{d} (a \to A) = a \to {\underline{Apr}}_{d} (A)$ . □

5 Conclusions

In this paper, we use the hemimetrics as the basic structure to define and study fuzzy rough set model of λ-subsets by using the usual addition and subtraction. We define a pair of fuzzy approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the Lukasiewicz logical systems. We show that upper definable sets, lower definable sets and definable sets are equivalent, and they form an Alexandrov fuzzy topology. In the future, we will follow [33] to study further applications in digital image processing.

Footnotes

Acknowledgment

This paper is supported by NNSF of China (12111540250, 12231007), NSF of Hebei Province (A2020208008), Jiangsu Provincial Innovative and Entrepreneurial Talent Support Plan (JSSCRC2021521).

References

Al-shami

T.M.

, An improvement of rough sets’ accuracy measure using containment neighborhoods with a medical application, Inf Sci 569 (2021), 110–124.

Al-shami

T.M.

, Improvement of the approximations and accuracy measure of a rough set using somewhere dense sets, Soft Comput 25 (2021), 14449–14460.

Al-shami

T.M.

and Ciucci

, Subset neighborhood rough sets, Knowl-Based Syst 237 (2022), 107868.

Al-shami

T.M.

, Fu

W.-Q.

and Abo-Tabl

E.A.

, New rough approximations based on E-neighborhoods, Complexity (2021), Article ID 6666853, 6 pages.

Atef

, Ali

M.I.

and Al-shami

T.M.

, Fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications, Comput & Appl Math 40(4) (2021), 115.

Belohlavek

, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers, New York USA, 2002.

Bonikowski

, Bryniarski

and Wybraniec-Skardowska

, Extensions and intentions in the rough set theory, Inf Sci 107(1–4) (1998), 149–167.

Cattaneo

and Ciucci

, Algebraic structures for rough sets, Lect Notes Comput Sc 3135 (2004), 208–252.

Comer

, An algebraic approach to the approximation of information, Fundam Inform 14(4) (1991), 492–502.

10.

D’eer

, Cornelis

and Godo

, Fuzzy neighborhood operators based on fuzzy coverings, Fuzzy Sets Syst 312 (2017), 17–35.

11.

D’eer

, Cornelis

and Yao

Y.Y.

, A semantically sound approach to Pawlak rough sets and covering-based rough sets, Int J Approx Reason 78(11) (2016), 62–72.

12.

Deng

T.Q.

, Chen

Y.M.

, Xu

W.L.

and Dai

Q.H.

, A novel approach to fuzzy rough sets based on a fuzzy covering, Inf Sci 177(11) (2007), 2308–2326.

13.

Goubault-Larrecq

, Non-Hausdorff Topology and Domain Theory, Cambridge University Press, Cambridge, 2013.

14.

Leung

, Wu

W.Z.

and Zhang

W.X.

, Knowledge acquisition in incomplete information systems: a rough set approach, Eur J Oper Res 168(1) (2006), 164–180.

15.

T.J.

, Leung

and Zhang

W.X.

, Generalized fuzzy rough approximation operators based on fuzzy coverings, Int J Approx Reason 48(3) (2008), 836–856.

16.

Liu

G.L.

, The axiomatization of the rough set upper approximation operations, Fundam Inform 69(3) (2006), 331–342.

17.

J.S.

, Leung

, Zhao

H.Y.

and Feng

, Generalized fuzzy roughsets determined by a triangular norm, Inf Sci 178(16) (2008), 3203–3213.

18.

Morsi

N.N.

and Yakout

M.M.

, Axiomatics for fuzzy rough sets, Fuzzy Sets Syst 100(1– 3) (1998), 327–342.

19.

Pawlak

, Rough sets, Int J Comput Inform Sci 11(5) (1982), 341–356.

20.

Pawlak

, Rough set and fuzzy set, Fuzzy Sets Syst, 17(1) (1985), 99–102.

21.

Pawlak

, Rough Set: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, Netherlands, 1991.

22.

Polkowski

and Skowron

, Rough Sets in Knowledge Discovery, Physica-Verlag, Heidelberg, Germany, 1998.

23.

Polkowski

and Skowron

, Rough-neuro computing, Lect Notes Comput Sci 2005 (2000), 57–64.

24.

Radzikowska

A.M.

and Kerre

E.E.

, Fuzzy rough sets based on residuated lattices, Lect Notes Comput Sc 3135 (2005), 278–296.

25.

She

and Wang

, An axiomatic approach of fuzzy rough sets based on residuated lattices, Comput Math Appl 58(1) (2009), 189–201.

26.

Skowron

and Stepaniuk

, Tolerance approximation spaces, Fundam Inform 27(2– 3) (1996), 245–253.

27.

Slowinski

and Vanderpooten

, A generalized definition of rough approximations based on similarity, IEEE Trans Knowl Data Eng 12(2) (2000), 331–336.

28.

Thiele

, On axiomatic characterization of fuzzy approximation operators I, In: Proceedings of the 2nd International Conference of Rough Sets and Current Trends in Computing, RSCTC 2000 Banff, Canada, (2000), 277–285.

29.

Thiele

, On axiomatic characterization of fuzzy approximation operators II, In: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic (2001), 330–335.

30.

W.Z.

, Leung

and Shao

M.W.

, Generalized fuzzy rougha pproximation operators determined by fuzzy implicators, Int J Approx Reason 54(9) (2013), 1388–1409.

31.

W.Z.

, Xu

Y.H.

, Shao

M.W.

and Wang

G.Y.

, Axiomatic characterizations of (S, T)-fuzzy rough approximation operators, Inf Sci 334–335 (2016), 17–43.

32.

Yao

, She

Y.H.

and Lu

L.X

, Metric-based L-fuzzy rough sets: Approximation operators and definable sets, Knowl-Based Syst 163 (2019), 91–102.

33.

Yao

, Zhang

G.X.

and Zhou

C.-J.

, Real-valued hemimetric-based fuzzy rough sets and an application in image process, Fuzzy Sets and Systems 459 (2023), 201–219.

34.

Yao

Y.Y.

and Yao

, Covering based rough sets approximations, Inf Sci 200 (2012), 91–107.

35.

Zakowski

, Approximations in the space (u, p), Demonstr Math 16(3) (1983), 761–769.

36.

Zhong

, Yao

Y.Y.

and Ohshima

, Peculiarity oriented multi database mining, IEEE Trans Knowl Data Eng 15(4) (2003), 952–960.