Novel results on G -lower L -fuzzy rough approximation operator in ( O,G ) -fuzzy rough set model 1

Abstract

Lately, Jiang and Hu (H.B. Jiang, B.Q. Hu, On $(O, G)$ -fuzzy rough sets based on overlap and grouping functions over complete lattices, Int. J. Approx. Reason. 144 (2022) 18-50.) put forward $(O, G)$ -fuzzy rough sets via overlap and grouping functions over complete lattices. Meanwhile, they showed the characterizations of $O$ -upper and $G$ -lower $L$ -fuzzy rough approximation operators in $(O, G)$ -fuzzy rough set model based on some of specific $L$ -fuzzy relations and studied the topological properties of the proposed model. Nevertheless, we discover that the partial results given by Jiang and Hu could be further optimized. So, as a replenish of the above article, in this paper, based on $G$ -lower $L$ -fuzzy rough approximation operator in $(O, G)$ -fuzzy rough set model, we further explore several new conclusions on the relationship between $G$ -lower $L$ -fuzzy rough approximation operator and different $L$ -fuzzy relations. In particular, the equivalent descriptions of relationship between $G$ -lower $L$ -fuzzy rough approximation operator and $O$ -transitive ( $O$ -Euclidean) $L$ -fuzzy relations are investigated, which are not involved in above literature and can make the theoretical results of this newly fuzzy rough set model more perfect.

Keywords

(𝔒,𝔊)-fuzzy rough set 𝔏-fuzzy relation overlap function grouping function complete lattice

1 Introduction

In 1982, Pawlak [19] put forward the theory of rough sets. Over nearly four decades, it has received all-around development both in real-life applications [14 , 30] and theoretical research [2 , 23–25]. On the other side, the theory of fuzzy sets, as a crucial mathematical tool to deal with ambiguous and vagueness matters, was proposed by Zadeh [26] in 1965. Further, in 1990, Dubois and Prade [6] raised fuzzy rough sets via the idea of combining rough sets and fuzzy sets. It is worth attention that fuzzy rough sets have made plenty of good works in terms of applications and theory since it was introduced. For example, in 2020, Zhan, Jiang and Yao [27] introduced covering-based variable precision fuzzy rough sets with Preference Ranking Organization Method for Enrichment Evaluation-Evaluation based on Distance from Average Solution methods. In 2021, Jiang and Hu [12] proposed a decision-theoretic fuzzy rough set model in hesitant fuzzy information systems and discussed its application in multi-attribute decision-making. In 2022, Zhang and Jiang [31] gave measurement, modeling, reduction of decision-theoretic multigranulation fuzzy rough sets based on three-way decisions.

For another, the axiomatic definitions of overlap and grouping functions were given by Bustince, Fernández, Mesiar et al. [4] and Bustince, Pagola, Mesiar et al. [5], respectively, in 2010 and 2012. And so over time, they have been comprehensively developed both in applications [8 , 18] and theory [1 , 28]. As a matter of fact, overlap and grouping functions more complex and flexible than usual fuzzy logical connective t-norms and t-conorms due to the requirement for associativity of them is not necessary. Recently, the research on the combination of overlap and grouping functions and rough sets has become a hot research direction and many scholars have studied the fuzzy rough set model based on overlap and grouping functions on unit closed interval [0, 1]. For instance, in 2021, Qiao [20] raised $(I_{O}, O)$ -fuzzy rough set model based on overlap functions. In 2022, Li, Yang and Qiao [16] studied $(O, G)$ -granular variable precision fuzzy rough sets derived from overlap and grouping functions. In particular, in order to make the fuzzy rough set model based on overlap and grouping functions serve to solve more practical problems involving incomparable information, it is very important to study the fuzzy rough set model based on overlap and grouping functions over a wider lattice structure than the unit closed interval [0,1]. Based on this, Jiang and Hu [13] presented $(O, G)$ -fuzzy rough sets based on overlap and grouping functions over complete lattice. Meanwhile, they discussed the characterizations of $(O, G)$ -fuzzy rough approximation operators via different kinds of $L$ -fuzzy relations and investigated the topological properties of the proposed model.

Nevertheless, we find that Propositions 3.12, 3.13, 3.16, 3.17 given by Jiang and Hu [13] can be further optimized. Thus, to replenish this research gap on a theoretical level, this study is devoted to exploring some new conclusions on the relationship between $G$ -lower $L$ -fuzzy rough approximation operators in $(O, G)$ -fuzzy rough set model and different $L$ -fuzzy relations. Particularly, under given conditions, the equivalent descriptions of the relationship between $G$ -lower $L$ -fuzzy rough approximation operators in $(O, G)$ -fuzzy rough set model and $O$ -transitive ( $O$ -Euclidean) $L$ -fuzzy relations are shown (more details can be seen from Propositions 3.1 and 3.3 in below Section 3), which make the theoretical results of $(O, G)$ -fuzzy rough set model more perfect and can be better applied to solve practical problems involving indetermination and inaccuracy information.

In the rest of this paper is organized as below. In Section 2, we sort out a few basic notions and needed results in this paper. In Section 3, we give some new results on the relationship between $G$ -lower $L$ -fuzzy rough approximation operator and different $L$ -fuzzy relations. Particularly, under given conditions, the equivalent descriptions of relationship between $G$ -lower $L$ -fuzzy rough approximation operator and $O$ -transitive ( $O$ -Euclidean) $L$ -fuzzy relations are respectively discussed. In Section 4, we summarize this research.

2 Preliminaries

This section reviews several fundamental concepts and vital results that need to be applied later.

We first give a few basic concepts of lattice theory. A poset $L$ is a nonempty set equipped with a partial order ≤ [9]. A complete lattice is a poset in which every subset has a sup and an inf [9]. In the ensuing discussion, we use the notation Ξ to denote a nonempty index set and the symbol $L$ to denote a complete lattice with the greatest element $1_{L}$ and smallest element $0_{L}$ [13].

Definition 2.1. ([13, 20]) A binary operator $O : L \times L ⟶ L$ is referred to as an overlap function on $L$ if, for each $a, b, c \in L$ and ${c_{i} : i \in Ξ} \subseteq L$ , it satisfies the following conditions:

$O (a, b)$ = $O (b, a)$ ;

$O (a, b) = 0_{L}$ iff $a = 0_{L}$ or $b = 0_{L}$ ;

$O (a, b) = 1_{L}$ iff $a = 1_{L}$ and $b = 1_{L}$ ;

$O (a, b) \leq O (a, c)$ whenever b ≤ c;

$O (a, ⋁_{i \in Ξ} c_{i})$ = $⋁_{i \in Ξ} O (a, c_{i})$ ;

$O (a, ⋀_{i \in Ξ} c_{i})$ = $⋀_{i \in Ξ} O (a, c_{i})$ .

If $O (a, 1_{L}) = a$ for all $a \in L$ , then we say that the overlap function $O$ has $1_{L}$ as neutral element [13].

Definition 2.2. ([13]) A binary operator $G : L \times L ⟶ L$ is referred to as a grouping function on the complete lattice $L$ if, for each $a, b, c \in L$ and ${c_{i} : i \in Ξ} \subseteq L$ , it satisfies the following conditions:

$G (a, b)$ = $G (b, a)$ ;

$G (a, b) = 0_{L}$ iff $a = 0_{L}$ and $b = 0_{L}$ ;

$G (a, b) = 1_{L}$ iff $a = 1_{L}$ or $b = 1_{L}$ ;

$G (a, b) \leq G (a, c)$ whenever b ≤ c;

$G (a, ⋁_{i \in Ξ} c_{i})$ = $⋁_{i \in Ξ} G (a, c_{i})$ ;

$G (a, ⋀_{i \in Ξ} c_{i})$ = $⋀_{i \in Ξ} G (a, c_{i})$ .

If $G (a, 0_{L}) = a$ for all $a \in L$ , then we say that the grouping function $G$ has $0_{L}$ as neutral element [13].

Definition 2.3. ([17]) A function $N : L ⟶ L$ is referred to as an $L$ -fuzzy negation if, for any $a, b \in L$ , it satisfies the following conditions:

$N (0_{L}) = 1_{L}$ and $N (1_{L}) = 0_{L}$ ;

$N (b) \leq N (a)$ whenever a ≤ b.

An $L$ -fuzzy negation is said to be an $L$ -fuzzy involutive negation [11], if it satisfies

$N (N (a)) = a$ .

Moreover, if vartriangle and ▿ are two binary operations on $L$ , then they are called dual to each other with respect to an $L$ -fuzzy involutive negation $N$ , if for every $a, b \in L$ , $N (a ▵ b) = N (a) ▿ N (b)$ [13].

In the following discussion, it is worth noting that the symbol $N$ always stands for the $L$ -fuzzy involutive negation unless otherwise stated.

Let $U$ be a nonempty universe. A mapping $A : U ⟶ L$ is known as an $L$ -fuzzy set on $U$ [13]. The family of all $L$ -fuzzy sets on $U$ is represented as $L^{U}$ [13]. The $L$ -fuzzy subset of $U$ with constant value a is expressed by $a_{U}$ [11]. For each $a \in L$ and $U^{'} ⫋ U$ , let $a_{U^{'}}$ denotes the special $L$ -fuzzy set written as $a_{U^{'}} (x) = a$ for every $x \in U^{'}$ and $a_{U^{'}} (x) = 0$ for every $x \notin U^{'}$ [11]. Furthermore, $A^{N} (x) = N (A (x))$ for any $A \in L^{U}$ and $x \in U$ .

For every $A, B \in L^{U}$ , the notation $A ⪯ B$ means that $A (x) \leq B (x)$ for every $x \in U$ [11].

Let $U$ and $V$ are two nonempty universes. Then an $L$ fuzzy set $R : U \times V ⟶ L$ is said to be an $L$ -fuzzy relation from $U$ to $V$ [13]. If $U = V$ , then $R$ is called an $L$ -fuzzy relation on $U$ [13].

Definition 2.4. ([13]) Let $R$ be an $L$ -fuzzy relation on $U$ . Then $R$ is referred to as:

reflexive, if $R (x, x) = 1_{L}$ for every $x \in U$ ;

$O$ -transitive, if $O (R (x, y), R (y, z)) \leq R (x, z)$ for every $x, y, z \in U$ ;

$O$ -Euclidean, if $O (R (x, y), R (x, z)) \leq R (y, z)$ for every $x, y, z \in U$ .

For each $L$ -fuzzy relation $R$ on $U$ and $z \in U$ , there is a unique $L$ -fuzzy set $[z]_{R} : U ⟶ L$ , which is given by $[z]_{R} (x) = R (x, z)$ for any $x \in U$ [21, 22]. Moreover, for all $L$ -fuzzy relation $R$ on $U$ , the $L$ -fuzzy relation $R^{- 1}$ on $U$ is defined, for every $x, y \in U$ , by $R^{- 1} (x, y) = R (y, x)$ [3, 13].

Suppose $U, V$ are two nonempty universes and $R$ is an $L$ -fuzzy relation from $U$ to $V$ , then the triple $(U, V, R)$ is referred to as an $L$ -fuzzy approximation space [13]. When $U = V$ and $R$ is an $L$ -fuzzy relation on $U$ , we say $(U, R)$ an $L$ -fuzzy approximation space [13].

Definition 2.5. ([11, 13]) Suppose $(U, V, R)$ is an $L$ -fuzzy approximation space. Then the two mappings ${\underline{R}}_{G}, {\bar{R}}_{O} : L^{V} ⟶ L^{U}$ given, for any $A \in L^{V}$ and $x \in U$ , by: ${\underline{R}}_{G} (A) (x) = ⋀_{y \in V} G (N (R (x, y)), A (y)),$ (1) and ${\bar{R}}_{O} (A) (x) = ⋁_{y \in V} O (R (x, y), A (y)),$ (2) are referred to as $G$ -lower and $O$ -upper $L$ -fuzzy rough approximation operators of $(U, V, R)$ , respectively. In addition, the pair $({\underline{R}}_{G} (A), {\bar{R}}_{O} (A))$ is referred to as the $(O, G)$ -fuzzy rough sets of $A$ with regard to $(U, V, R)$ .

Lemma 2.6. ([15]) Let $N : L ⟶ L$ be an $L$ -fuzzy involutive negation on complete lattice $L$ . Then, for all ${a_{i} : i \in Ξ} \subseteq L$ , the following statements hold.

$N (⋁_{i \in Ξ} a_{i}) = ⋀_{i \in Ξ} N (a_{i})$ .

$N (⋀_{i \in Ξ} a_{i}) = ⋁_{i \in Ξ} N (a_{i})$ .

3 New results

This part mainly explores some new results on between $G$ -lower $L$ -fuzzy rough approximation operator in $(O, G)$ -fuzzy rough set model and different $L$ -fuzzy relations. In particular, under given conditions, the equivalent descriptions of relationship between $G$ -lower $L$ -fuzzy rough approximation operator and $O$ -transitive ( $O$ -Euclidean) $L$ -fuzzy relations are respectively showed.

Proposition 3.1. Let $(U, R)$ be an $L$ -fuzzy approximation space, $O$ and $G$ be dual with regard to $N$ . If $G$ has $0_{L}$ as the neutral element, then the following statements are equivalent.

$R$ is an $O$ -transitive $L$ -fuzzy relation.

${\underline{R}}_{G} ({(1_{L})}_{U - {x}}) ⪯ {\underline{R}}_{G} ({\underline{R}}_{G} ({(1_{L})}_{U - {x}}))$ for any $x \in U$ .

Proof. (1) ⇒ (2) It can be get from Proposition 3.13 in [13].

(2) ⇒ (1) For every $x, y, z \in U$ , from item (2) of Lemma 2.6, one obtains that

$\begin{matrix} N (⋁_{z \in U} O (R (y, z), R (z, x))) \\ = ⋀_{z \in U} N (O (R (y, z), R (z, x))) \\ = ⋀_{z \in U} G (N (R (y, z)), N (R (z, x))) \\ = ⋀_{z \in U} G (N (R (y, z)), G (N (R (z, x)), 0_{L})) \\ = ⋀_{z \in U} G (N (R (y, z)), ⋀_{z^{'} \in U} G (N (R (z, z^{'})), {(1_{L})}_{U - {x}} (z^{'}))) \\ = ⋀_{z \in U} G (N (R (y, z)), {\underline{R}}_{G} ({(1_{L})}_{U - {x}}) (z)) \\ = {\underline{R}}_{G} ({\underline{R}}_{G} ({(1_{L})}_{U - {x}})) (y) \\ \geq {\underline{R}}_{G} ({(1_{L})}_{U - {x}}) (y) \\ = ⋀_{z \in U} G (N (R (y, z)), {(1_{L})}_{U - {x}} (z)) \\ = G (N (R (y, x)), 0_{L}) \\ = N (R (y, x)), \end{matrix}$

that is, $N (⋁_{R \in U} O (R (y, z), R (z, x))) \geq N (R (y, x))$ . Furthermore, from item ( $N$ 2) of Definition 2.3 and item ( $N$ 3), we achieve that $⋁_{R \in U} O (R (y, z), R (z, x)) \leq R (y, x)$ , that is, $O (R (y, z), R (z, x)) \leq R (y, x)$ for any $x, y, z \in U$ . Thus, $R$ is an $O$ -transitive $L$ -fuzzy relation.

Proposition 3.2. Let $(U, R)$ be an $L$ -fuzzy approximation space, $O$ and $G$ be dual with regard to $N$ . Then the following statements hold.

$R$ is an $O$ -transitive $L$ -fuzzy relation.

${\underline{R}}_{G} ([x]_{R^{N}}) ⪯ {\underline{R}}_{G} ({\underline{R}}_{G} ([x]_{R^{N}}))$ for any $x \in U$ .

Then (1) ⇒ (2), and if

G

has

0_{L}

as the neutral element and

R

is a reflexive

L

-fuzzy relation, then

{\underline{R}}_{G} ([x]_{R^{N}}) = {\underline{R}}_{G} ({\underline{R}}_{G} ([x]_{R^{N}}))

for each

x \in U

Proof. (1) ⇒ (2) It can be get from Proposition 3.12 in [13].

Now, we prove that, for any $x \in U$ , ${\underline{R}}_{G} ({\underline{R}}_{G} ([x]_{R^{N}})) ⪯ {\underline{R}}_{G} ([x]_{R^{N}})$ under the conditions of that $G$ is a grouping function having $0_{L}$ as the neutral element and $R$ is a reflexive $L$ -fuzzy relation.

For every fixed $x \in U$ , from item ( $N$ 1) of Definition 2.3, we obtain that

$\begin{matrix} {\underline{R}}_{G} ([x]_{R^{N}}) (z) \\ = ⋀_{y \in U} G (N (R (z, y)), [x]_{R^{N}} (y)) \\ = ⋀_{y \in U} G (N (R (z, y)), R^{N} (y, x)) \\ = ⋀_{y \in U} G (N (R (z, y)), N (R (y, x))) \\ = ⋀_{y \in U} G (N (R (z, y)), G (0_{L}, N (R (y, x)))) \\ = ⋀_{y \in U} G (N (R (z, y)), G (N (1_{L}), N (R (y, x)))) \\ = ⋀_{y \in U} G (N (R (z, y)), G (N (R (y, y)), N (R (y, x)))) \\ \geq ⋀_{y \in U} G (N (R (z, y)), ⋀_{z^{'} \in U} G (N (R (y, z^{'})), N (R (z^{'}, x))) \\ = ⋀_{y \in U} G (N (R (z, y)), ⋀_{z^{'} \in U} G (N (R (y, z^{'})), R^{N} (z^{'}, x))) \\ = ⋀_{y \in U} G (N (R (z, y)), ⋀_{z^{'} \in U} G (N (R (y, z^{'})), [x]_{R^{N}} (z^{'}))) \\ = ⋀_{y \in U} G (N (R (z, y)), {\underline{R}}_{G} ([x]_{R^{N}}) (y)) \\ = {\underline{R}}_{G} ({\underline{R}}_{G} ([x]_{R^{N}})) (z) \end{matrix}$

for any $z \in U$ .

In conclusion, we have that ${\underline{R}}_{G} ([x]_{R^{N}}) = {\underline{R}}_{G} ({\underline{R}}_{G} ([x]_{R^{N}}))$ for each $x \in U$ .

Proposition 3.3. Let $(U, R)$ be an $L$ -fuzzy approximation space, $O$ and $G$ be dual with regard to $N$ . If $G$ has $0_{L}$ as the neutral element, then the following statements are equivalent.

$R$ is an $O$ -Euclidean $L$ -fuzzy relation.

${\underline{R}}_{G} ({(1_{L})}_{U - {x}}) ⪯ {\underline{R^{- 1}}}_{G} ({\underline{R}}_{G} ({(1_{L})}_{U - {x}}))$ for every $x \in U$ .

Proof. (1) ⇒ (2) It can be get from Proposition 3.17 in [13].

(2) ⇒ (1) For any $x, y, z \in U$ , from item (2) of Lemma 2.6, we have that

$\begin{matrix} N (⋁_{z \in U} O (R (z, y), R (z, x))) \\ = ⋀_{z \in U} N (O (R (z, y), R (z, x))) \\ = ⋀_{z \in U} G (N (R (z, y)), N (R (z, x))) \\ = ⋀_{z \in U} G (N (R (z, y)), G (N (R (z, x)), 0_{L})) \\ = ⋀_{z \in U} G (N (R^{- 1} (y, z)), ⋀_{z^{'} \in U} G (N (R (z, z^{'})), {(1_{L})}_{U - {x}} (z^{'}))) \\ = ⋀_{z \in U} G (N (R^{- 1} (y, z)), {\underline{R}}_{G} ({(1_{L})}_{U - {x}}) (z)) \\ = {\underline{R^{- 1}}}_{G} ({\underline{R}}_{G} ({(1_{L})}_{U - {x}})) (y) \\ \geq {\underline{R}}_{G} ({(1_{L})}_{U - {x}}) (y) \\ = ⋀_{z \in U} G (N (R (y, z)), {(1_{L})}_{U - {x}} (z)) \\ = ⋀_{z \in U} G (N (R (y, x)), 0_{L}) \\ = N (R (y, x)), \end{matrix}$

that is, $N (⋁_{z \in U} O (R (z, y), R (z, x))) \geq N (R (y, x))$ . Furthermore, from item ( $N$ 2) of Definition 2.3 and item ( $N$ 3), we achieve that $⋁_{z \in U} O (R (z, y), R (z, x)) \leq R (y, x)$ , that is, $O (R (z, y), R (z, x)) \leq R (y, x)$ for any $x, y, z \in U$ . Thus, $R$ is an $O$ -Euclidean $L$ -fuzzy relation.

Proposition 3.4. Let $(U, R)$ be an $L$ -fuzzy approximation space, $O$ and $G$ be dual with regard to $N$ . Then the following statements hold.

$R$ is an $O$ -Euclidean $L$ -fuzzy relation.

${\underline{R}}_{G} ([x]_{R^{N}}) ⪯ {\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x]_{R^{N}}))$ for any $x \in U$ .

Then (refp18i1) ⇒ (refp18i2), and if

G

has

0_{L}

as the neutral element and

R

is a reflexive

L

-fuzzy relation, then

{\underline{R}}_{G} ([x]_{R^{N}}) = {\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x]_{R^{N}}))

for any

x \in U

Proof. (refp18i1) ⇒ (refp18i2) It can be get from Proposition 3.16 in [13].

Next, for each $x \in U$ , we prove ${\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x]_{R^{N}})) ⪯ {\underline{R}}_{G} ([x]_{R^{N}})$ under the conditions of that If $G$ is a grouping function having $0_{L}$ as the neutral element and $R$ is a reflexive $L$ -fuzzy relation.

For any fixed $x \in U$ , from item ( $N$ ) of Definition 2.3, one has that

$\begin{matrix} {\underline{R}}_{G} ([x]_{R^{N}}) (z) \\ = ⋀_{y \in U} G (N (R (z, y)), [x]_{R^{N}} (y)) \\ = ⋀_{y \in U} G (N (R (z, y)), R^{N} (y, x)) \\ = ⋀_{y \in U} G (N (R (z, y)), N (R (y, x))) \\ = ⋀_{y \in U} G (N (R (z, y)), G (0_{L}, N (R (y, x)))) \\ = ⋀_{y \in U} G (N (R (z, y)), G (N (1_{L}), N (R (y, x)))) \\ = ⋀_{y \in U} G (N (R (z, y)), G (N (R (y, y)), N (R (y, x)))) \\ \geq ⋀_{y \in U} G (N (R (z, y)), ⋀_{z^{'} \in U} G (N (R (z^{'}, y)), N (R (z^{'}, x)))) \\ = ⋀_{y \in U} G (N (R (z, y)), ⋀_{z^{'} \in U} G (N (R^{- 1} (y, z^{'})), R^{N} (z^{'}, x))) \\ = ⋀_{y \in U} G (N (R (z, y)), ⋀_{z^{'} \in U} G (N (R^{- 1} (y, z^{'})), [x]_{R^{N}} (z^{'}))) \\ = ⋀_{y \in U} G (N (R (z, y)), {\underline{R^{- 1}}}_{G} ([x]_{R^{N}}) (y)) \\ = {\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x]_{R^{N}})) (z) \end{matrix}$

for any $z \in U$ .

In conclusion, we obtain that ${\underline{R}}_{G} ([x]_{R^{N}}) = {\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x]_{R^{N}}))$ for any $x \in U$ .

Next, we give an example to illustrate the reasonability of Propositions 3.2. and 3.4.

Example 3.5. Consider the complete lattice $L = [0, 1],$ and the $L$ -fuzzy negation $N$ on [0, 1] is defined by $N (a) = 1 - a$ for every a ∈ [0, 1] . Let $U = {x_{1}, x_{2}, x_{3}}$ and $L$ -fuzzy relation $R$ on $U$ as $R = [\begin{matrix} 1 & 0.9 & 0.5 \\ 0.7 & 1 & 0.6 \\ 0.3 & 0.5 & 1 \end{matrix}] .$ Then, $R$ is a reflexive $L$ -fuzzy relation on $U .$

If we take a grouping function as $G = G_{p}$ 1 and p = 1, then for any $A \in L^{V},$ $G$ -lower $L$ -fuzzy rough approximation operator of $A$ is rewritten as, for any $x \in U$ , ${\underline{R}}_{G} (A) (x) = ⋀_{y \in V} {1 - R (x, y) (1 - A (y))} .$ Then we obtain

${\underline{R}}_{G} ([x_{1}]_{R^{N}}) = \frac{0}{x_{1}} + \frac{0.300}{x_{2}} + \frac{0.650}{x_{3}},$ ${\underline{R}}_{G} ({\underline{R}}_{G} ([x_{1}]_{R^{N}})) = \frac{0}{x_{1}} + \frac{0.300}{x_{2}} + \frac{0.650}{x_{3}};$

${\underline{R}}_{G} ([x_{2}]_{R^{N}}) = \frac{0.100}{x_{1}} + \frac{0}{x_{2}} + \frac{0.500}{x_{3}},$ ${\underline{R}}_{G} ({\underline{R}}_{G} ([x_{2}]_{R^{N}})) = \frac{0.100}{x_{1}} + \frac{0}{x_{2}} + \frac{0.500}{x_{3}};$

${\underline{R}}_{G} ([x_{3}]_{R^{N}}) = \frac{0.460}{x_{1}} + \frac{0.400}{x_{2}} + \frac{0}{x_{3}},$ ${\underline{R}}_{G} ({\underline{R}}_{G} ([x_{3}]_{R^{N}})) = \frac{0.460}{x_{1}} + \frac{0.400}{x_{2}} + \frac{0}{x_{3}} .$

Thus, we verify that ${\underline{R}}_{G} ([x_{i}]_{R^{N}}) ⪰ {\underline{R}}_{G} ({\underline{R}}_{G} ([x_{i}]_{R^{N}}))$ for each $x_{i} \in U (i = 1, 2, 3),$ which is consistent with the result of Proposition 3.

The same way, we have

${\underline{R}}_{G} ([x_{1}]_{R^{N}}) = \frac{0}{x_{1}} + \frac{0.300}{x_{2}} + \frac{0.650}{x_{3}},$ ${\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x_{1}]_{R^{N}})) = \frac{0}{x_{1}} + \frac{0.100}{x_{2}} + \frac{0.500}{x_{3}};$

${\underline{R}}_{G} ([x_{2}]_{R^{N}}) = \frac{0.100}{x_{1}} + \frac{0}{x_{2}} + \frac{0.500}{x_{3}},$ ${\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x_{2}]_{R^{N}})) = \frac{0.100}{x_{1}} + \frac{0}{x_{2}} + \frac{0.400}{x_{3}};$

${\underline{R}}_{G} ([x_{3}]_{R^{N}}) = \frac{0.460}{x_{1}} + \frac{0.400}{x_{2}} + \frac{0}{x_{3}},$ ${\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x_{3}]_{R^{N}})) = \frac{0.460}{x_{1}} + \frac{0.400}{x_{2}} + \frac{0}{x_{3}} .$

Therefore, we verify that ${\underline{R}}_{G} ({\underline{R^{- 1}}}_{G} ([x_{i}]_{R^{N}})) ⪯ {\underline{R}}_{G} ([x_{i}]_{R^{N}})$ for each $x_{i} \in U (i = 1, 2, 3),$ which is consistent with the result of Proposition 3.4.

4 Conclusions

In this study, we mainly obtained some new results on between $G$ -lower $L$ -fuzzy rough approximation operator in $(O, G)$ -fuzzy rough set model and different $L$ -fuzzy relations. In particular, under given conditions, the equivalent descriptions of the relationship between $G$ -lower $L$ -fuzzy rough approximation operator and $O$ -transitive ( $O$ -Euclidean) $L$ -fuzzy relations are respectively showed by Propositions 3.1 and 3.3. These new results help us to further study axiomatic characterizations of $G$ -lower $L$ -fuzzy rough approximation operator in $(O, G)$ -fuzzy rough set model, and the descriptions of the correlation between $G$ -lower $L$ -fuzzy rough approximation operator and $O$ -transitive ( $O$ -Euclidean) $L$ -fuzzy relations. Moreover, they not only help to improve the theoretical results of this model, but also make it to better serve the practical application problems involving pattern recognition, decision-making and classification.

Acknowledgements

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.

Footnotes

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

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