On ( G O,O )-fuzzy rough sets derived from overlap and grouping functions

Abstract

Rough sets, as a powerful tool to deal with uncertainties and inaccuracies in data analysis, have been continuously concerned and studied by many scholars since it was put forward, especially the research on various rough set models. On the other hand, overlap and grouping functions, as two newly aggregation operators and mathematical model to handle the problems involving in information fusion, have been successfully applied in many real-life problems. In this paper, based on overlap and grouping functions, we propose a new fuzzy rough set model named (G_O, O)-fuzzy rough sets and consider its characterizations along with topological properties. Properly speaking, firstly, we utilize QL-operators (and also QL-implications) constructed from overlap and grouping functions and fuzzy negations to define the lower approximation operator in (G_O, O)-fuzzy rough set model named G_O-lower fuzzy rough approximation operator and the upper approximation operator in (G_O, O)-fuzzy rough set model is considered as the O-upper fuzzy rough approximation operator in (I_O, O)-fuzzy rough set model proposed by Qiao recently. Secondly, we discuss lots of basic properties of (G_O, O)-fuzzy rough sets, especially for the properties of G_O-lower fuzzy rough approximation operator. Thirdly, we focus on the relationship between (G_O, O)-fuzzy rough sets and concrete fuzzy relations. Finally, we give the topological properties of the upper and lower approximation operators in (G_O, O)-fuzzy rough set model.

Keywords

overlap function grouping function fuzzy relation fuzzy topology

1 Introduction and motivation

1.1 A short introduction of overlap and grouping functions

Overlap and grouping functions, as two sorts of not necessary associative binary aggregation operators, have been continuously investigated by many scholars since they were, respectively, proposed by Bustince, Fernández, Mesiar et al. [1] in 2010 and Bustince, Pagola, Mesiar et al. [2] in 2012. As a result, after nearly ten years of development, these two types of newly aggregation operators have achieved a series of research results in theory and practical applications.

In practical applications, it is well known that overlap and grouping functions have been successfully used to address many practical issues, for instance, the applications on decision making [3], image processing [4], classification [5 –7], fuzzy community detection problems [8], wavelet-fuzzy power quality diagnosis systems [9] and so on.

In theory, there exists a vast literature which involve in various respects of overlap and grouping functions, such as distribution equations [10], corresponding implications [11], concepts generalization [12, 13], generator pairs [14] and so on.

1.2 A brief review on the development of rough sets

Rough sets was originally introduced by Pawlak [15] in 1982. Nowadays, it has been successfully treated as a mathematical method of handling imprecision and vague information in data analysis. Because of this, the research on the new rough set model and its related properties along with applications has never been interrupted in the past four decades. In particular, one of the main line of research on rough set models is the extended research on existing models, which include the extension of relations [16 –28], objects [29 –32], the domain of discourse [33 –35] and etc.

In addition, it is well known that the theory of rough sets has been successfully utilized for various aspects of practical issues, for instance, in data analysis [36], decision making [37, 38], conflict analysis [39], information process [40] and etc.

On the other hand, it is worth mentioning that theoretical research on new rough set models has been mainly focused on the characterization of approximate operators in the model and the research on their topological properties [41 –44].

1.3 The motivation of our research

On the one hand, fuzzy implications are taken to define the lower approximate operators in rough set models when they are considered in fuzzy status and various certain fuzzy implications can induce different fuzzy rough set models. Among them, the most widely used are the three types of basic fuzzy implications induced by t-norms and t-conorms, that is, residual implications, S-implications and QL-implications. On the other hand, based on overlap and grouping functions, we also can derive three types of basic fuzzy implications and call them R_O-implications, (G, N)-implications and QL-implications generated by tuples (O, G, N). Meanwhile, using R_O-implications, Qiao [45] proposed a new fuzzy rough set model named (I_O, O)-fuzzy rough sets recently.

In addition, it should be pointed out that when Dimuro, Bedregal, Bustince et al. [46] discussed the QL-operators and QL-implications generated by tuples (O, G, N) of overlap and grouping functions and fuzzy negations, they stated that “although the standard QL-operators constructed from t-norms and t-conorms have not been applied in any field, and, thus, have no importance in the study of fuzzy operators derived from the implication given in p → q ≡ ¬ p ∨ (p ∧ q), on the contrary, QL-operators constructed from overlap and grouping functions (and also the derived QL-implication functions) play an important role in the generation of fuzzy subsethood and entropy measures, which can be applied in several areas, such as clustering, fuzzy relational databases, intelligent systems, fuzzy decision making, image processing.”. At the same time, Qiao and Hu [47] used QL-operators constructed from left continuous t-norms and right continuous t-conorms to define the lower approximation operator and obtained a new rough set model, namely (⊥ _⊤, ⊤)-fuzzy rough sets based on left continuous t-norms and right continuous t-conorms.

Therefore, whether it is from the perspective of theoretical research or practical application, it is essential and meaningful to define the lower approximation operator in one class of rough set model by using the QL-operators derived from overlap and grouping functions and fuzzy negations (and also the derived QL-implications). However, so far, as far as we are aware, there is no related research using QL-operators derived from overlap and grouping functions and fuzzy negations (and also the derived QL-implications) to give the lower approximation operator to propose a fuzzy rough set model. So, as a supplement of this field, in this paper, we propose a new fuzzy rough set model on the basis of QL-operators derived from overlap and grouping functions and fuzzy negations (and also the derived QL-implications) and give a systemic study both on the characterization and topological properties of this new model. At the same time, in order to show that the relationship between the proposed new fuzzy rough set model and the existing relation based fuzzy rough set models more specifically, we provide the following Fig. 1 on the basis of the corresponding figure given in [47].

Fig. 1

The extension process of rough sets on the basis of relations (the first item in these parentheses is relation and the second represents object, and notations ⊥, ⊤ , I, & , ra, ⊙ , O, G represent t-conorms, t-norms, fuzzy implications, binary operators and residual implications on complete residuated lattices, binary operators on complete co-residuated lattices, overlap functions and grouping functions, respectively).

The rest of this paper are organized as follows. Section 2 briefly reviews several basic definitions and used results in this paper. Section 3 proposes the notion of (G_O, O)-fuzzy rough sets based on overlap and grouping functions and study some basic properties of (G_O, O)-fuzzy rough sets. Section 4 systematically discusses the connections between (G_O, O)-fuzzy rough sets and different types of fuzzy relations. Section 5 shows the topological properties of (G_O, O)-fuzzy rough sets. Section 6 summarizes this work.

2 Preliminaries

In this section, let us first recall some definitions and their related properties which are indispensable for the reader to follow the work.

Definition 2.1. (see [1]) A bivariate function O : [0, 1] ² ⟶ [0, 1] is said to be an overlap function if it satisfies the following conditions:

O is commuative;

O (a, b) =0 iff ab = 0;

O (a, b) =1 iff ab = 1;

O is increasing;

O is continuous.

Moreover, an overlap function O is said to be satisfied 1-section deflation [48] if

O (1, b) ≤ b for every b ∈ [0, 1], and the 1-section inflation [48] if

b ≤ O (1, b) for every b ∈ [0, 1], and the exchange principle [48] if

O (a, O (b, c)) = O (b, O (a, c)) for every a, b, c ∈ [0, 1].

In the following example, we list some usual overlap functions from [2, 14].

Example 2.2.

Any positive and continuous t-norm¹\footnote []¹A bivariate function T : [0, 1] ² ⟶ [0, 1] is said to be a triangular norm (t-norm, for short) if it is commutative, associative, increasing and 1 is the neutral element. Meanwhile, a t-norm T is said to be continuous, if it is continuous in both arguments at the same time and positive if T (a, b) =0, then either a = 0 or b = 0 [49]. is an overlap function. In particular, the minimum t-norm T_M (a, b) = min(a, b) is an overlap function. And, here, we call it minimum overlap function and denoted by O_M.

For any p > 0, the function O_p : [0, 1] ² → [0, 1] given, for all a, b ∈ [0, 1], by $O_{p} (a, b) = a^{p} b^{p}$ (1) is an overlap function.

The function O_mM : [0, 1] ² → [0, 1] given, for all a, b ∈ [0, 1], by $O_{mM} (a, b) = min {a, b} max {a^{2}, b^{2}}$ (2) is an overlap function.

Lemma 2.3. (see [45]) A bivariate function O : [0, 1] ² ⟶ [0, 1] is an overlap function satisfying (O8). Then 1 is the neutral element of O.

Definition 2.4. (see [2]) A bivariate function G : [0, 1] ² ⟶ [0, 1] is said to be a grouping function if it satisfies the following conditions:

G is commuative;

G (a, b) =0 iff a = b = 0;

G (a, b) =1 iff a = 1 or b = 1;

G is increasing;

G is continuous.

Moreover, a grouping function G is said to be satisfied 0-section inflation [50] if

b ≤ G (0, b) for every b ∈ [0, 1], and the 0-section deflation [50] if

G (0, b) ≤ b for every b ∈ [0, 1].

In the following example, we list some usual grouping functions from [14].

Example 2.5.

Any positive and continuous t-conorm²\footnote []²A bivariate function S : [0, 1] ² ⟶ [0, 1] is said to be a triangular conorm (t-conorm, for short) if it is commutative, associative, increasing and 0 is the neutral element. Meanwhile, a t-conorm S is said to be continuous, if it is continuous in both arguments at the same time and positive if S (a, b) =1, then either a = 1 or b = 1 [49]. is a grouping function.

For any p > 0, the function G_p : [0, 1] ² → [0, 1] given, for all a, b ∈ [0, 1], by $G_{p} (a, b) = 1 - (1 - a)^{p} (1 - b)^{p}$ (3) is a grouping function.

The function G_{M
_p} : [0, 1] ² → [0, 1] given, for all a, b ∈ [0, 1], by $G_{M_{p}} (a, b) = max {a^{p}, b^{p}}$ (4) is a grouping function.

Definition 2.6. (see [51]) A function N : [0, 1] ⟶ [0, 1] is said to be a fuzzy negation, if the following conditions hold:

Boundary conditions: N (0) =1 and N (1) =0.

N is decreasing: if x ≤ y then N (y) ≤ N (x).

A fuzzy negation is said to be strong if it is involutive, that is,

N (N (x)) = x.

Definition 2.7. (see [46]) A function I : [0, 1] ² ⟶ [0, 1] is said to be a QL-operator constructed from tuples (O, G, N) if there exist an overlap function O : [0, 1] ² ⟶ [0, 1], a grouping function G : [0, 1] ² ⟶ [0, 1] and a fuzzy negation N : [0, 1] ⟶ [0, 1], such that $I (x, y) = G (N (x), O (x, y))$ (5) for all x, y ∈ [0, 1]. We denote such QL-operator by I_O,G,N, and say that I_O,G,N is a QL-operator constructed from the tuple (O, G, N).

Lemma 2.8. (see [46]) A QL-operator constructed from the tuple (O, G, N) is a fuzzy implication function if and only if G (N (x) , x) =1 for any x ∈ [0, 1].

Let X be a nonempty universe. Then a mapping A : X ⟶ [0, 1] is said to be a fuzzy set on X [52]. The family of all fuzzy sets on X is represented as [0, 1] ^X. In addition, some needed symbols related to fuzzy set are listed here, they mainly come from [53 –55]. The fuzzy subset of X with constant value a is denoted by a_X. For all a ∈ [0, 1] and X′⊂X, let a_X′ denote the special fuzzy set written as a_X′ (x) = a for every x ∈ X′ and a_X′ (x) =0 for every x ∉ X′.

Definition 2.9. (see [45]) Let O : [0, 1] ² ⟶ [0, 1] be an overlap function and X be a nonempty universe. Then the operation O_X : [0, 1] ^X × [0, 1] ^X ⟶ [0, 1] ^X based on the overlap function O is given, for all fuzzy subsets A, B ∈ [0, 1] ^X and x ∈ X, by $O_{X} (A, B) (x) = O (A (x), B (x)) .$ (6) Similar to Definition 2.9, we give the definition of grouping function based operations as follows.

Definition 2.10. Let G : [0, 1] ² ⟶ [0, 1] be a grouping function and X be a nonempty universe. Then the operation G_X : [0, 1] ^X × [0, 1] ^X ⟶ [0, 1] ^X based on the grouping function G is given, for all fuzzy subsets A, B ∈ [0, 1] ^X and x ∈ X, by $G_{X} (A, B) (x) = G (A (x), B (x)) .$ (7)

The operations on fuzzy sets are given as follows [47]:

A^N (x) = N (A (x)) for any A ∈ [0, 1] ^X and x ∈ X,

(⋂ _i∈ΛA_i) (x) = ⋀ _i∈ΛA_i (x) for any index set Λ, A_i ∈ [0, 1] ^X and x ∈ X,

(⋃ _i∈ΛA_i) (x) = ⋁ _i∈ΛA_i (x) for any index set Λ, A_i ∈ [0, 1] ^X and x ∈ X.

For every A, B ∈ [0, 1] ^X, the notation A ⪯ B means that A (x) ≤ B (x) for every x ∈ X.

Let X and Y are two nonempty universes. Then a fuzzy set R : X × Y ⟶ [0, 1] is said to be a fuzzy relation from X to Y [52]. If X = Y, then R is called a fuzzy relation on X.

Definition 2.11. (see [45, 56]) Let R be a fuzzy relation on X. Then R is called

serial if ⋁_y∈XR (x, y) =1 for every x ∈ X,

reflexive if R (x, x) =1 for every x ∈ X,

symmetric if R (x, y) = R (y, x) for every x, y ∈ X,

O-transitive if O (R (x, y) , R (y, z)) ≤ R (x, z) for every x, y, z ∈ X,

O-Euclidean if O (R (x, y) , R (x, z)) ≤ R (y, z) for every x, y, z ∈ X.

Moreover, R is called an O-fuzzy preorder if it is reflexive and O-transitive. Also, if R is an O-fuzzy preorder on X, then the pair (X, R) is called an O-fuzzy preorder set [45]. Beyond that, R is called an O-similarity fuzzy relation if it is reflexive, symmetric and O-transitive.

For each fuzzy relation R on X and z ∈ X, there are two unique fuzzy sets [z] _R, [z] ^R : X ⟶ [0, 1] given, respectively, by [z] _R (x) = R (x, z) and [z] ^R (x) = R (z, x) for any x ∈ X [57]. Moreover, for all fuzzy relation R on X, the fuzzy relation R^-1 on X is defined, for every x, y ∈ X, by R^-1 (x, y) = R (y, x).

Definition 2.12. (see [58]) A mapping i : [0, 1] ^X ⟶ [0, 1] ^X is said to be a fuzzy Čech interior operator on X, if the following conditions hold.

i (1_X) =1_X.

i (A) ⪯ A for every A ∈ [0, 1] ^X.

i (A ∩ B) = i (A) ∩ i (B) for each A, B ∈ [0, 1] ^X.

A mapping i^♯ : [0, 1] ^X ⟶ [0, 1] ^X is said to be a fuzzy semi-Čech interior operator on X, if it only meets items (I1) and (I3) in Definition 2.12.

Definition 2.13. (see [59]) Let X be a nonempty set. Then a subset η ⊆ [0, 1] ^X is said to be a fuzzy topology on X, if the following conditions hold.

0_X, 1_X ∈ η.

A ∩ B ∈ η for all A, B ∈ η.

⋃_i∈ΛA_i ∈ η for every {A_i} _i∈Λ ⊆ η. A fuzzy topological space (X, η) is said to be an Alexandroff space [59] if

⋂_i∈ΛA_i ∈ η for every {A_i} _i∈Λ ⊆ η.

A subset ω ⊆ [0, 1] ^X is said to be a fuzzy semi-topology on a set X, if it meets items T1) and T2) in Definition 2.13 [47].

3 (G_O, O)-fuzzy rough sets derived from overlap and grouping functions

Let X, Y be two nonempty universes and R be a fuzzy relation from X to Y. Then the triple (X, Y, R) is said to be a fuzzy approximation space. If X = Y and R is a fuzzy relation on X, then we call (X, R) a fuzzy approximation space [47].

Definition 3.1. Let (X, Y, R) be a fuzzy approximation space. Then the two mappings ${\underline{R}}_{G_{O}}, {\bar{R}}_{O} : [0, 1]^{Y} ⟶ [0, 1]^{X}$ given, for any A ∈ [0, 1] ^Y and x ∈ X, by: ${\underline{R}}_{G_{O}} (A) (x) = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), A (y)))$ (8) and ${\bar{R}}_{O} (A) (x) = ⋁_{y \in Y} O (R (x, y), A (y)),$ (9) are called G_O-lower and O-upper fuzzy rough approximation operators of (X, Y, R), respectively. Moreover, the pair $({\underline{R}}_{G_{O}} (A), {\bar{R}}_{O} (A))$ is called the (G_O, O)-fuzzy rough set of A with regard to (X, Y, R).

Remark 3.2.

It should be noted that the O-upper fuzzy rough approximation operator in (G_O, O)-fuzzy rough set model given in Definition 3.1 is same with the upper fuzzy rough approximation operator in (I_O, O)-fuzzy rough set model given in Definition 3.1 in [45]. Therefore, in the sequel, we just discuss the related properties of lower fuzzy rough approximation operator in (G_O, O)-fuzzy rough set model.

In Definition 3.1, if the overlap function O degrades into the minimum overlap function O_M and R becomes an O_M-similarity fuzzy relation on X, then the O-upper fuzzy rough approximation operator in (G_O, O)-fuzzy rough set model given in Definition 3.1 is same with the upper approximation operator in fuzzy rough set model shown in [60]. In other words, the O-upper fuzzy rough approximation operator in (G_O, O)-fuzzy rough set model given in Definition 3.1 can be regarded as a generalization of the upper approximation operator in fuzzy rough set model given in [60].

Proposition 3.3. Let (X, Y, R) be a fuzzy approximation space. Then, for any A, B ∈ [0, 1] ^Y and {A_i : i ∈ Λ} ⊆ [0, 1] ^Y (Λ is any index set), below items come into existence.

${\bar{R}}_{O} (0_{Y}) = 0_{X}$ .

A ⪯ B implies ${\underline{R}}_{G_{O}} (A) ⪯ {\underline{R}}_{G_{O}} (B)$ .

$⋃_{i \in Λ} {\bar{R}}_{O} (A_{i}) = {\bar{R}}_{O} (⋃_{i \in Λ} A_{i})$ and $⋃_{i \in Λ} {\underline{R}}_{G_{O}} (A_{i}) ⪯ {\underline{R}}_{G_{O}} (⋃_{i \in Λ} A_{i})$ .

${\bar{R}}_{O} (⋂_{i \in Λ} A_{i}) ⪯ ⋂_{i \in Λ} {\bar{R}}_{O} (A_{i})$ and ${\underline{R}}_{G_{O}} (⋂_{i \in Λ} A_{i}) = ⋂_{i \in Λ} {\underline{R}}_{G_{O}} (A_{i})$ .

Proof. These conclusions can be derived from Definition 3.1 by a routine way and thus the details are omitted. □ Proposition 3.4. Let (X, Y, R) be a fuzzy approximation space and O be an overlap function having 1 as neutral element. If G (N (x) , x) =1 for any x ∈ [0, 1], then below statement hold. ${\underline{R}}_{G_{O}} (1_{Y}) = 1_{X} .$

Proof. For every x ∈ X, by Lemma 2.3, we infer that

$\begin{matrix} {\underline{R}}_{G_{O}} (1_{Y}) (x) & = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), 1_{Y} (y))) \\ = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), 1)) \\ = ⋀_{y \in Y} G (N (R (x, y)), R (x, y)) \\ = 1 \\ = 1_{X} (x) . \end{matrix}$

Thus, ${\underline{R}}_{G_{O}} (1_{Y}) = 1_{X} .$ □ Proposition 3.5. Let (X, Y, R) be a fuzzy approximation space, O be an overlap function satisfying (O6) and G be a grouping function satisfying (G6). Then, for each x ∈ X and a ∈ [0, 1], below statement hold. $\begin{matrix} O (⋀_{y \in Y} R (x, y), a) \leq {\underline{R}}_{G_{O}} (a_{Y}) (x) \\ \leq G (⋀_{y \in Y} R^{N} (x, y), a) . \end{matrix}$

Proof. For every x ∈ X and a ∈ [0, 1], by item (O4) of Definition 2.1, item (G5) of Definition 2.4 and item (O6), we infer that $\begin{matrix} {\underline{R}}_{G_{O}} (a_{Y}) (x) & = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), a_{Y} (y))) \\ = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), a)) \\ \leq ⋀_{y \in Y} G (N (R (x, y)), O (1, a)) \\ \leq ⋀_{y \in Y} G (N (R (x, y)), a) \\ = ⋀_{y \in Y} G (R^{N} (x, y), a) \\ = G (⋀_{y \in Y} R^{N} (x, y), a) . \end{matrix}$ On the other hand, for every x ∈ X and a ∈ [0, 1], by item (O5) of Definition 2.1, items (G4) and (G5) of Definition 2.4 and item (G6), we deduce that $\begin{matrix} {\underline{R}}_{G_{O}} (a_{Y}) (x) & = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), a_{Y} (y))) \\ = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), a)) \\ \geq ⋀_{y \in Y} G (0, O (R (x, y), a)) \\ = G (0, ⋀_{y \in Y} O (R (x, y), a)) \\ \geq ⋀_{y \in Y} O (R (x, y), a) \\ = O (⋀_{y \in Y} R (x, y), a) . \end{matrix}$ Therefore, $O (⋀_{y \in Y} R (x, y), a) \leq {\underline{R}}_{G_{O}} (a_{Y}) (x) \leq G (⋀_{y \in Y} R^{N} (x, y), a)$ for each x ∈ X and a ∈ [0, 1].

□ Proposition 3.6. Let (X, Y, R) be a fuzzy approximation space. Then below statement hold. ${\underline{R}}_{G_{O}} (a_{Y}) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, a_{X}))$ for all a ∈ [0, 1] iff ${\underline{R}}_{G_{O}} (0_{Y}) = 0_{X} .$

Proof. On the one hand, if ${\underline{R}}_{G_{O}} (a_{Y}) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, a_{X}))$ for all a ∈ [0, 1], then, from item (O2) of Definition 2.1 and item (G2) of Definition 2.4, one infers that ${\underline{R}}_{G_{O}} (0_{Y}) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, 0_{X})) = G_{X} (0_{X}, 0_{X}) = 0_{X}$ . Thus, it holds that ${\underline{R}}_{G_{O}} (0_{Y}) = 0_{X} .$ On the other hand, if ${\underline{R}}_{G_{O}} (0_{Y}) = 0_{X}$ , then, for every x ∈ X, from item (O2) of Definition 2.1 and item (G5) of Definition 2.4, it holds that $\begin{matrix} 0 & = 0_{X} (x) \\ = {\underline{R}}_{G_{O}} (0_{Y}) (x) \\ = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), 0_{Y} (y))) \\ = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), 0)) \\ = ⋀_{y \in Y} G (N (R (x, y)), 0) \\ = G (⋀_{y \in Y} N (R (x, y)), 0) . \end{matrix}$ Furthermore, by item (G2) of Definition 2.4, it holds that ⋀_y∈YN (R (x, y)) =0. And, for all a ∈ [0, 1], from Definitions 2.9 and 2.10, item (G5) of Definition 2.4 and item (O4) of Definition 2.1, it follows that $\begin{matrix} {\underline{R}}_{G_{O}} (a_{Y}) (x) & = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), a_{Y} (y))) \\ = ⋀_{y \in Y} G (N (R (x, y)), O (R (x, y), a)) \\ \leq ⋀_{y \in Y} G (N (R (x, y)), O (1, a)) \\ = G (⋀_{y \in Y} N (R (x, y)), O (1, a)) \\ = G (0, O (1, a)) \\ = G (0_{X} (x), O (1_{X} (x), a_{X} (x))) \\ = G (0_{X} (x), O_{X} (1_{X}, a_{X}) (x)) \\ = G_{X} (0_{X}, O_{X} (1_{X}, a_{X})) (x) . \end{matrix}$

Thus, ${\underline{R}}_{G_{O}} (a_{Y}) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, a_{X}))$ for all a ∈ [0, 1] iff ${\underline{R}}_{G_{O}} (0_{Y}) = 0_{X} .$ □ Proposition 3.7. Let (X, Y, R) be a fuzzy approximation space. Then below statements hold.

${\underline{R}}_{G_{O}} (a_{Y}) ⪯ a_{X}$ for all a ∈ [0, 1].

${\underline{R}}_{G_{O}} (0_{Y}) = 0_{X} .$

Then (1) ⇒ (2), and if O is an overlap function having 1 as the neutral element and G is a grouping function having 0 as the neutral element, then (1) ⇔ (2).

Proof. It can be checked in a similar way as that of Proposition 3.6 and the details are omitted. □

4 Connections between (G_O, O)-fuzzy rough sets and fuzzy relations

In this section, the connections between G_O-lower and O-upper fuzzy rough approximation operators and some of specific fuzzy relations are investigated, for instance, serial fuzzy relations, reflexive fuzzy relations, symmetric fuzzy relations, O-transitive fuzzy relations and O-Euclidean fuzzy relations.

Proposition 4.1. Let (X, R) be a fuzzy approximation space and N be a strong fuzzy negation. Then below items are equivalent.

R is serial.

${\underline{R}}_{G_{O}} (0_{X}) = 0_{X}$ .

${\underline{R}}_{G_{O}} (a_{X}) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, a_{X}))$ for all a ∈ [0, 1].

Proof. (1) ⇒ (2) For each x ∈ X, from item (O2) of Definition 2.1, items (G2) and (G5) of Definition 2.4, items (N1) and (N2) of Definition 2.6 and item (N3), it holds that $\begin{matrix} {\underline{R}}_{G_{O}} (0_{X}) (x) & = ⋀_{y \in X} G (N (R (x, y)), O (R (x, y), 0_{X} (y))) \\ = ⋀_{y \in X} G (N (R (x, y)), O (R (x, y), 0)) \\ = ⋀_{y \in X} G (N (R (x, y)), 0) \\ = G (⋀_{y \in X} N (R (x, y)), 0) \\ = G (N (⋁_{y \in X} R (x, y)), 0) \\ = G (N (1), 0) \\ = G (0, 0) \\ = 0 \\ = 0_{X} (x) . \end{matrix}$ Hence, we get that ${\underline{R}}_{G_{O}} (0_{X}) = 0_{X}$ . (2) ⇒ (1) For each x ∈ X, by item (O2) of Definition 2.1 and item (G5) of Definition 2.4, it follows immediately from the proof of (1) ⇒ (2) that $G (N (⋁_{y \in X} R (x, y)), 0) = 0 .$ Furthermore, from item (G2) of Definition 2.4, it holds that N (⋁ _y∈XR (x, y)) = 0. And thus, by item (N1) of Definition 2.6 and item (N3), one deduces that ⋁_y∈XR (x, y) =1. Thus, R is serial. (2) ⇔ (3) It follows immediately from Proposition 3.6. Therefore, we obtain that (1) ⇔ (2) ⇔ (3).

Proposition 4.2. Let (X, R) be a fuzzy approximation space and N be a strong fuzzy negation. Consider below items.

R is reflexive.

${\underline{R}}_{G_{O}} (A) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, A))$ for all A ∈ [0, 1] ^X.

Then (1) ⇒ (2), and if G (N (x) , x) =1 for any x ∈ [0, 1] and O satisfies (O7), then (1) ⇔ (2).

Proof. (1) ⇒ (2) For all x ∈ X, by item (N1) of Definition 2.6 and Definitions 2.9 and 2.10, one has that $\begin{matrix} {\underline{R}}_{G_{O}} (A) (x) & = ⋀_{y \in X} G (N (R (x, y)), O (R (x, y), A (y))) \\ \leq G (N (R (x, x)), O (R (x, x), A (x))) \\ = G (N (1), O (1, A (x))) \\ = G (0, O (1_{X} (x), A (x))) \\ = G (0_{X} (x), O_{X} (1_{X}, A) (x)) \\ = G_{X} (0_{X}, O_{X} (1_{X}, A)) (x) . \end{matrix}$ Therefore, we get that ${\underline{R}}_{G_{O}} (A) ⪯ G_{X} (0_{X}, O_{X} (1_{X}, A))$ for all A ∈ [0, 1] ^X. And now, we check that (2) ⇒ (1) under the condition of that G (N (x) , x) =1 for any x ∈ [0, 1] and O satisfies (O7).

For any x ∈ X, let A = 1_X-{x}. Then, by Definitions 2.9 and 2.10, items (G2) and (G3) of Definition 2.4 and item (O2) of Definition 2.1, we infer that $\begin{matrix} 0 & = G (0, 0) \\ = G (0, O (1, 0)) \\ = G (0_{X} (x), O (1_{X} (x), 1_{X - {x}} (x))) \\ = G (0_{X} (x), O_{X} (1_{X}, 1_{X - {x}}) (x)) \\ = G_{X} (0_{X}, O_{X} (1_{X}, 1_{X - {x}})) (x) \\ \geq {\underline{R}}_{G_{O}} (1_{X - {x}}) (x) \\ = ⋀_{y \in X} G (N (R (x, y)), O (R (x, y), 1_{X - {x}} (y))) \\ = ⋀_{y \neq x} G (N (R (x, y)), O (R (x, y), 1_{X - {x}} (y))) \\ ⋀ G (N (R (x, x)), O (R (x, x), 0)) \\ = ⋀_{y \neq x} G (N (R (x, y)), O (R (x, y), 1)) \\ ⋀ G (N (R (x, x)), O (R (x, x), 0)) \\ \geq ⋀_{y \neq x} G (N (R (x, y)), R (x, y)) \\ ⋀ G (N (R (x, x)), O (R (x, x), 0)) \\ = G (N (R (x, x)), O (R (x, x), 0)) \\ = G (N (R (x, x)), 0), \end{matrix}$ that is, G (N (R (x, x)) , 0) =0. And, from item (G2) of Definition 2.4, it holds that N (R (x, x)) =0. Further, by item (N1) of Definition 2.6 and item (N3), one gets that R (x, x) =1. Hence, R is reflexive.

From Proposition 4.2, we can obtain the following corollary immediately.

Corollary 4.3. Let (X, R) be a fuzzy approximation space, N be a strong fuzzy negation, O be an overlap function having 1 as the neutral element and G be a grouping function satisfying (G7). Consider below items.

R is reflexive.

${\underline{R}}_{G_{O}} (A) ⪯ A$ for all A ∈ [0, 1] ^X.

Then (1) ⇒ (2), and if G (N (x) , x) =1 for any x ∈ [0, 1], then (1) ⇔ (2).

Proposition 4.4. Let (X, R) be a fuzzy approximation space, N be a strong fuzzy negation, O be an overlap function having 1 as the neutral element and G (N (x) , x) =1 for any x ∈ [0, 1]. Consider below items.

R is symmetric.

${\underline{R}}_{G_{O}} (1_{X - {y}}) (x) = {\underline{R}}_{G_{O}} (1_{X - {x}}) (y)$ for any x, y ∈ X.

Then (1) ⇒ (2), and if G (0, -) is strict increasing, then (1) ⇔ (2).

Proof. (1) ⇒ (2) For each x, y ∈ X, by item (O2) of Definition 2.1, items (G2) and (G3) of Definition 2.4, one has that $\begin{matrix} {\underline{R}}_{G_{O}} (1_{X - {y}}) (x) \\ = ⋀_{z \in X} G (N (R (x, z)), O (R (x, z), 1_{X - {y}} (z))) \\ = ⋀_{z \neq y} G (N (R (x, z)), O (R (x, z), 1_{X - {y}} (z))) \\ ⋀ G (N (R (x, y)), O (R (x, y), 0)) \\ = ⋀_{z \neq y} G (N (R (x, z)), O (R (x, z), 1)) \\ ⋀ G (N (R (x, y)), O (R (x, y), 0)) \\ = ⋀_{z \neq y} G (N (R (x, z)), R (x, z)) \\ ⋀ G (N (R (x, y)), O (R (x, y), 0)) \\ = G (N (R (x, y)), O (R (x, y), 0)) \\ = G (N (R (x, y)), 0) \\ = G (N (R (y, x)), 0) \\ = G (N (R (y, x)), O (R (y, x), 0)) \\ = ⋀_{z \neq x} G (N (R (y, z)), R (y, z)) \\ ⋀ G (N (R (y, x)), O (R (y, x), 0)) \\ = ⋀_{z \neq x} G (N (R (y, z)), O (R (y, z), 1)) \\ ⋀ G (N (R (y, x)), O (R (y, x), 0)) \\ = ⋀_{z \neq x} G (N (R (y, z)), O (R (y, z), 1_{X - {x}} (z))) \\ ⋀ G (N (R (y, x)), O (R (y, x), 0)) \\ = ⋀_{z \in X} G (N (R (y, z)), O (R (y, z), 1_{X - {x}} (z))) \\ = {\underline{R}}_{G_{O}} (1_{X - {x}}) (y) . \end{matrix}$ Therefore, we get that ${\underline{R}}_{G_{O}} (1_{X - {y}}) (x) = {\underline{R}}_{G_{O}} (1_{X - {x}}) (y)$ for any x, y ∈ X. Next, we prove (2) ⇒ (1) under the condition of that G (0, -) is strict increasing. For every x, y ∈ X, by item (O2) of Definition 2.1, it follows immediately from the proof of (1) ⇒ (2) that $\begin{matrix} G (N (R (x, y)), 0) & = {\underline{R}}_{G_{O}} (1_{X - {y}}) (x) \\ = {\underline{R}}_{G_{O}} (1_{X - {x}}) (y) \\ = G (N (R (y, x)), 0) . \end{matrix}$ Thus, one gets that N (R (x, y)) = N (R (y, x)). Furthermore, from item (N3), one concludes that R (x, y) = R (y, x) for any x, y ∈ X, that is, R is symmetric.

Proposition 4.5. Let (X, R) be a fuzzy approximation space, R be serial, N be a strong fuzzy negation, O be an overlap function satisfying (O8) and G be a grouping function satisfying (G7). Consider below items.

R is O-transitive.

${\underline{R}}_{G_{O}} ({\bar{R}}_{O} (A)) ⪯ {\bar{R}}_{O} (A)$ for all A ∈ [0, 1] ^X.

Then (1) ⇒ (2).

Proof. Notice that the proof of this conclusion can be obtained from Proposition 6.8 in [47] under the condition of that overlap function satisfying (O8) and, for the sake of completeness, here, we also give the verification from the overlap function viewpoint.

For all A ∈ [0, 1] ^X and x ∈ X, by item (G5) of Definition 2.4, items (N1) and (N2) of Definition 2.6 and item (N3), items (O1) and (O5) of Definition 2.1 and items (O8) and (G7), one infers that

$\begin{matrix} {\underline{R}}_{G_{O}} ({\bar{R}}_{O} (A)) (x) \\ = ⋀_{y \in X} G (N (R (x, y)), O (R (x, y), {\bar{R}}_{O} (A) (y))) \\ = ⋀_{y \in X} G (N (R (x, y)), O (R (x, y), ⋁_{z \in X} O (R (y, z), A (z)))) \\ = ⋀_{y \in X} G (N (R (x, y)), ⋁_{z \in X} O (R (x, y), O (R (y, z), A (z)))) \\ = ⋀_{y \in X} G (N (R (x, y)), ⋁_{z \in X} O (R (x, y), O (A (z), R (y, z)))) \\ = ⋀_{y \in X} G (N (R (x, y)), ⋁_{z \in X} O (A (z), O (R (x, y), R (y, z)))) \\ = ⋀_{y \in X} G (N (R (x, y)), ⋁_{z \in X} O (O (R (x, y), R (y, z)), A (z))) \\ \leq ⋀_{y \in X} G (N (R (x, y)), ⋁_{z \in X} O (R (x, z), A (z))) \\ = ⋀_{y \in X} G (N (R (x, y)), {\bar{R}}_{O} (A) (x)) \\ = G (⋀_{y \in X} N (R (x, y)), {\bar{R}}_{O} (A) (x)) \\ = G (N (⋁_{y \in X} R (x, y)), {\bar{R}}_{O} (A) (x)) \\ = G (N (1), {\bar{R}}_{O} (A) (x)) \\ = G (0, {\bar{R}}_{O} (A) (x)) \\ \leq {\bar{R}}_{O} (A) (x) . \end{matrix}$

Hence, we get that ${\underline{R}}_{G_{O}} ({\bar{R}}_{O} (A)) ⪯ {\bar{R}}_{O} (A)$ for all A ∈ [0, 1] ^X. □

From Proposition 4.5, we can obtain the following corollary immediately.

Corollary 4.6 Let (X, R) be a fuzzy approximation space, R be reflexive, N be a strong fuzzy negation, O be an overlap function satisfying (O8) and G be a grouping function satisfying (G7). Consider below items.

R is O-transitive.

${\underline{R}}_{G_{O}} ({\bar{R}}_{O} (A)) ⪯ {\bar{R}}_{O} (A)$ for all A ∈ [0, 1] ^X.

Then (1) ⇒ (2).

Proposition 4.7. Let (X, R) be a fuzzy approximation space, R be reflexive, G be a grouping function satisfying (G7). Consider below items.

R is O-transitive.

${\underline{R}}_{G_{O}} ({\underline{R}}_{G_{O}} ([x]_{R})) ⪯ {\underline{R}}_{G_{O}} ([x]_{R})$ for any x ∈ X.

Then (1) ⇒ (2).

Proof. For every fixed x ∈ X, from item (N1) of Definition 2.6 and item (G7), it holds that $\begin{matrix} {\underline{R}}_{G_{O}} ({\underline{R}}_{G_{O}} ([x]_{R})) (z) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), {\underline{R}}_{G_{O}} ([x]_{R}) (y))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), \\ ⋀_{z^{'} \in X} G (N (R (y, z^{'})), O (R (y, z^{'}), [x]_{R} (z^{'}))))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), \\ ⋀_{z^{'} \in X} G (N (R (y, z^{'})), O (R (y, z^{'}), R (z^{'}, x))))) \\ \leq ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), \\ ⋀_{z^{'} \in X} G (N (R (y, z^{'})), R (y, x)))) \\ \leq ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), \\ G (N (R (y, y)), R (y, x)))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), G (N (1), R (y, x)))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), G (0, R (y, x)))) \\ \leq ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), R (y, x))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), [x]_{R} (y))) \\ = {\underline{R}}_{G_{O}} ([x]_{R}) (z) \end{matrix}$

for any z ∈ X.

Thus, we achieve that ${\underline{R}}_{G_{O}} ({\underline{R}}_{G_{O}} ([x]_{R})) ⪯ {\underline{R}}_{G_{O}} ([x]_{R})$ for any x ∈ X. □ Proposition 4.8. Let (X, R) be a fuzzy approximation space, N be a strong fuzzy negation, O be an overlap function satisfying (O8), G be a grouping function satisfying (G7) and R be serial and symmetric. Consider below items.

R is O-Euclidean.

${\underline{R}}_{G_{O}} ({\bar{R}}_{O} (A)) ⪯ {\bar{R}}_{O} (A)$ for all A ∈ [0, 1] ^X.

Then (1) ⇒ (2).

Proof. It can be checked in a similar way as that of Proposition 4.5 and the details are omitted. □ Proposition 4.9. Let (X, R) be a fuzzy approximation space, R be reflexive and G be a grouping function satisfying (G7). Consider below items.

R is O-Euclidean.

${\underline{R}}_{G_{O}} ({\underline{R^{- 1}}}_{G_{O}} ([x]_{R})) ⪯ {\underline{R}}_{G_{O}} ([x]_{R})$ for any x ∈ X.

Then (1) ⇒ (2).

Proof. For every fixed x ∈ X, from item (N1) of Definition 2.6 and item (G7), one has that $\begin{matrix} {\underline{R}}_{G_{O}} ({\underline{R^{- 1}}}_{G_{O}} ([x]_{R})) (z) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), {\underline{R^{- 1}}}_{G_{O}} ([x]_{R}) (y))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), ⋀_{z^{'} \in X} G (N (R^{- 1} (y, z^{'})), \\ O (R^{- 1} (y, z^{'}), [x]_{R} (z^{'}))))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), ⋀_{z^{'} \in X} G (N (R^{- 1} (y, z^{'})), \\ O (R^{- 1} (y, z^{'}), R (z^{'}, x))))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), ⋀_{z^{'} \in X} G (N (R (z^{'}, y)), \\ O (R (z^{'}, y), R (z^{'}, x))))) \\ \leq ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), ⋀_{z^{'} \in X} G (N (R (z^{'}, y)), \\ R (y, x)))) \\ \leq ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), G (N (R (y, y)), R (y, x)))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), G (N (1), R (y, x)))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), G (0, R (y, x)))) \\ \leq ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), R (y, x))) \\ = ⋀_{y \in X} G (N (R (z, y)), O (R (z, y), [x]_{R} (y))) \\ = {\underline{R}}_{G_{O}} ([x]_{R}) (z) \end{matrix}$ for any z ∈ X.

Thus, we achieve that ${\underline{R}}_{G_{O}} ({\underline{R^{- 1}}}_{G_{O}} ([x]_{R})) ⪯ {\underline{R}}_{G_{O}} ([x]_{R})$ for any x ∈ X. □

5 Topological properties of (G_O, O)-fuzzy rough sets

In this section, we discuss the topological properties of G_O-lower fuzzy rough approximation operator in (G_O, O)-fuzzy rough sets.

Proposition 5.1. Let (X, R) be a fuzzy approximation space, R be reflexive, G be a grouping function satisfying (G7) and O be an overlap function having 1 as neutral element. If G (N (x) , x) =1 for any x ∈ [0, 1], then ${\underline{R}}_{G_{O}}$ is a fuzzy Čech interior operator.

Proof. It follows immediately from Proposition 3.4, item (4) of Proposition 3.3 and Corollary 4.3. □

Proposition 5.2. Let (X, R) be a fuzzy approximation space and O be an overlap function having 1 as neutral element. If G (N (x) , x) =1 for any x ∈ [0, 1], then ${\underline{R}}_{G_{O}}$ is a fuzzy semi-Čech interior operator.

Proof. It follows immediately from Proposition 3.4, item (4) of Proposition 3.3. □

Proposition 5.3. Let (X, R) be a fuzzy approximation space, R be serial, N be a strong fuzzy negation and O be an overlap function having 1 as neutral element. If G (N (x) , x) =1 for any x ∈ [0, 1], then $R_{G_{O}} = {{\underline{R}}_{G_{O}} (A) | A \in [0, 1]^{X}}$ is a fuzzy semi-topology.

Proof. The items T1) and T2) in Definition 2.13 are checked as follows. T1) From Propositions 4.1 and 3.4, we obtain that $0_{X}, 1_{X} \in R_{G_{O}} .$ T2) For ${\underline{R}}_{G_{O}} (A_{1}), {\underline{R}}_{G_{O}} (A_{2}) \in R_{G_{O}}$ , by item (4) Proposition 3.3, it follows that ${\underline{R}}_{G_{O}} (A_{1}) ⋂ {\underline{R}}_{G_{O}} (A_{2}) = {\underline{R}}_{G_{O}} (A_{1} ⋂ A_{2}) .$

Thus, we get that ${\underline{R}}_{G_{O}} (A_{1}) ⋂ {\underline{R}}_{G_{O}} (A_{2}) \in R_{G_{O}} .$

Therefore, $R_{G_{O}}$ is a fuzzy semi-topology. □

Proposition 5.4. Let (X, R) be a fuzzy approximation space, R be reflexive, N be a strong fuzzy negation, G be a grouping function satisfying (G7), O be an overlap function having 1 as neutral element. If G (N (x) , x) =1 for any x ∈ [0, 1], then $D_{G_{O}} (R) = {A \in [0, 1]^{X} | {\underline{R}}_{G_{O}} (A) = A}$ is an Alexandroff fuzzy topology.

Proof. The items T1) and T3) in Definition 2.13 and T4) are checked as follows.

T1) From Propositions 4.1 and 3.4, we obtain that $0_{X}, 1_{X} \in D_{G_{O}} (R) .$ T3) For all ${A_{i}}_{i \in Λ} \subseteq D_{G_{O}} (R)$ , from item (3) of Proposition 3.3 and Corollary 4.3, it holds that $\begin{matrix} ⋃_{i \in Λ} A_{i} & = ⋃_{i \in Λ} {\underline{R}}_{G_{O}} (A_{i}) \\ ⪯ {\underline{R}}_{G_{O}} (⋃_{i \in Λ} A_{i}) \\ ⪯ ⋃_{i \in Λ} A_{i} . \end{matrix}$ Thus, we get that $⋃_{i \in Λ} A_{i} \in D_{G_{O}} (R) .$ T4) For every ${A_{i}}_{i \in Λ} \subseteq D_{G_{O}} (R)$ , by item (4) Proposition 3.3, it follows that $\begin{matrix} ⋂_{i \in Λ} A_{i} & = ⋂_{i \in Λ} {\underline{R}}_{G_{O}} (A_{i}) \\ = {\underline{R}}_{G_{O}} (⋂_{i \in Λ} A_{i}) . \end{matrix}$ Thus, we get that $⋂_{i \in Λ} A_{i} \in D_{G_{O}} (R) .$

Hence, $D_{G_{O}} (R)$ is an Alexandroff fuzzy topology.

□ Lemma 5.5. (see [45]) Let (X, R) be a fuzzy approximation space. Then below items are equivalent.

R is serial.

${\bar{R}}_{O} (1_{X}) = 1_{X}$ .

${\bar{R}}_{O} (a_{X}) = O_{X} (1_{X}, a_{X})$ for all a ∈ [0, 1].

Lemma 5.6. (see [45]) Let (X, R) be a fuzzy approximation space and O be an overlap function satisfying (O7). Then below items are equivalent.

R is reflexive.

$A ⪯ {\bar{R}}_{O} (A)$ for all A ∈ [0, 1] ^X.

Proposition 5.7. Let (X, R) be a fuzzy approximation space, R be reflexive, N be a strong fuzzy negation, O be an overlap function having 1 as neutral element and G be a grouping function satisfying (G7). If G (N (x) , x) =1 for any x ∈ [0, 1], then $D (R) = {A \in [0, 1]^{X} | {\underline{R}}_{G_{O}} (A) = {\bar{R}}_{O} (A)}$ is an Alexandroff fuzzy topology.

Proof. The items T1) and T3) in Definition 2.13 and T4) are checked as follows. T1) From item (1) of Proposition 3.3, Propositions 3.4 and 4.1 and Lemma 5.5, we obtain that $0_{X}, 1_{X} \in D (R) .$ T3) For each ${A_{i}}_{i \in Λ} \subseteq D (R)$ , by item (3) of Proposition 3.3, Corollary 4.3 and Lemma 5.6, it holds that $\begin{matrix} {\underline{R}}_{G_{O}} (⋃_{i \in Λ} A_{i}) & ⪯ ⋃_{i \in Λ} A_{i} \\ ⪯ {\bar{R}}_{O} (⋃_{i \in Λ} A_{i}) \\ = ⋃_{i \in Λ} {\bar{R}}_{O} (A_{i}) \\ = ⋃_{i \in Λ} {\underline{R}}_{G_{O}} (A_{i}) \\ ⪯ {\underline{R}}_{G_{O}} (⋃_{i \in Λ} A_{i}) . \end{matrix}$ Thus, we get that $⋃_{i \in Λ} A_{i} \in D (R) .$ T4) For all ${A_{i}}_{i \in Λ} \subseteq D (R)$ , by item (4) of Proposition 3.3, Corollary 4.3 and Lemma 5.6, it follows that $\begin{matrix} {\bar{R}}_{O} (⋂_{i \in Λ} A_{i}) & ⪯ ⋂_{i \in Λ} {\bar{R}}_{O} (A_{i}) \\ = ⋂_{i \in Λ} {\underline{R}}_{G_{O}} (A_{i}) \\ = {\underline{R}}_{G_{O}} (⋂_{i \in Λ} A_{i}) \\ ⪯ ⋂_{i \in Λ} A_{i} \\ ⪯ {\bar{R}}_{O} (⋂_{i \in Λ} A_{i}) . \end{matrix}$ Thus, we get that ${\bar{R}}_{O} (⋂_{i \in Λ} A_{i}) = {\underline{R}}_{G_{O}} (⋂_{i \in Λ} A_{i})$ ,that is, $⋂_{i \in Λ} A_{i} \in D (R) .$

Therefore, $D (R)$ is an Alexandroff fuzzy topology.

□

Lemma 5.8. (see [45]) Let (X, R) be a fuzzy approximation space and R be serial. Then $D_{O} (R) = {A \in [0, 1]^{X} | {\bar{R}}_{O} (A) = A}$ is an Alexandroff fuzzy topology.

Proposition 5.9. Let (X, R) be a fuzzy approximation space, R be reflexive, N be a strong fuzzy negation, G be a grouping function satisfying (G7) and O be an overlap function having 1 as neutral element. If G (N (x) , x) =1 for any x ∈ [0, 1], then $D (R) = D_{G_{O}} (R) \cap D_{O} (R)$ .

Proof. On the one hand, for any $A \in D_{G_{O}} (R) \cap D_{O} (R)$ , it is visible that $A \in D (R)$ . On the other hand, for any $A \in D (R)$ , by Corollary 4.3 and Lemma 5.6, one has that ${\underline{R}}_{G_{O}} (A) ⪯ A ⪯ {\bar{R}}_{O} (A) .$ Hence, we obtain that ${\underline{R}}_{G_{O}} (A) = A = {\bar{R}}_{O} (A)$ , that is, $A \in D_{G_{O}} (R) \cap D_{O} (R)$ .

Therefore, $D (R) = D_{G_{O}} (R) \cap D_{O} (R)$ . □

6 Conclusions

This paper mainly focuses on the research topic of fuzzy rough set model based on fuzzy logic connectives and their induced fuzzy implications. We consider a new fuzzy rough set model on the basis of overlap and grouping functions and the QL-operators (and also the QL-implications) derived from them from the theoretical viewpoint. The main results of this work are listed as follows.

As a generalization of the Pawlak’s rough sets, we propose the concept of (G_O, O)-fuzzy rough set model and utilize QL-operators derived from overlap and grouping functions and fuzzy negations (and also the derived QL-implications) to define the G_O-lower fuzzy rough approximation operator in (G_O, O)-fuzzy rough sets. Furthermore, we show the basic properties of G_O-lower and O-upper fuzzy rough approximation operators, especially for G_O-lower fuzzy rough approximation operator.

We give the characterizations of G_O-lower fuzzy rough approximation operator in (G_O, O)-fuzzy rough set model by common classes of fuzzy relations, such as serial, reflexive, symmetric, O-transitive and O-Euclidean ones.

We discuss the topological properties of G_O-lower fuzzy rough approximation operator in (G_O, O)-fuzzy rough sets. In particular, it is proved that the set of fixed points of G_O-lower fuzzy rough approximation operator in (G_O, O)-fuzzy rough sets forms an Alexandroff topology.

As our future work, for one thing, we plan to investigate axiomatic characterizations of (G_O, O)-fuzzy rough sets. For another, we intend to consider the real-life applications of (G_O, O)-fuzzy rough set model.

Acknowledgments

The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly.

This work was supported by the National Natural Science Foundation of China (62166037 and 11901465), the Science and Technology Program of Gansu Province (20JR10RA101), the China Postdoctoral Science Foundation (2021M692561), the Scientific Research Fund for Young Teachers of Northwest Normal University (NWNU-LKQN-18-28) and the Doctoral Research Fund of Northwest Normal University (6014/0002020202).

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On ( G O,O )-fuzzy rough sets derived from overlap and grouping functions

Abstract

Keywords

1 Introduction and motivation

1.1 A short introduction of overlap and grouping functions

1.2 A brief review on the development of rough sets

1.3 The motivation of our research

5 Topological properties of (G O , O)-fuzzy rough sets

6 Conclusions

Acknowledgments

References

5 Topological properties of (G_O, O)-fuzzy rough sets