Characterization of neighborhood operators based on neighborhood relationships

: Neighborhood relationships play a pivotal role in rough set theory, addressing the limitations of equivalence relations. This article focuses on de ﬁ ning upper and lower approximation operators using neighborhood relationships and exploring their properties in terms of serialization, inverse serialization, re ﬂ ex-ivity, symmetry, transitivity, and Euclidean relations. Furthermore, a necessary and su ﬃ cient condition for the upper approximation operator to function as a topological closure operator is derived. Overall, this research sheds light on the signi ﬁ cance of neighborhood relationships and their implications within rough set theory.


Introduction
Rough set theory was proposed by Professor Pawlak in 1982 as a new and effective tool for handling imprecise, incomplete, and inconsistent data [1][2][3].In recent years, with the development of technology, rough set theory has been widely applied in various fields such as artificial intelligence, machine learning, data mining, and pattern recognition.It has been continuously optimized, leading to the definition of many effective models.However, scholars found that the rough set of equivalence relations has its limitations.To solve these problems, scholars have defined rough sets based on neighborhood relationships.Neighborhood rough set has become an important branch of rough set.For information systems, a neighborhood relationship is to classify and distinguish the range while keeping its neighborhood range unchanged.The purpose of neighborhood relationship is to standardize the operation rules of neighborhood operators under various constraints and obtain corresponding results.
The process of human understanding various things in the world begins with establishing concepts, from which continuous thinking approximation is conducted.This process ultimately leads to rational judgments and decisions.Pawlak's rough set has expanded from two aspects: granulation and approximation.Based on human cognitive methods, it has further deepened our understanding of the profound meaning of rough sets over the course of its 40-year development.In 1990, Lin [4] proposed the concept of neighborhood rough sets, using neighborhood relationships as a new granulation method to implement domain approximation.Lin [5] put forward the concept of neighborhood relationship and neighborhood model based on interior point and closure in topology.In decision analysis, relevant scholars introduced the dominance relation rough set model for the problem of ordered classification, which is used for the case that the decision attributes contain ordered structure.Mao et al. [6] researched on attribute reduction under co-occurrence neighborhood relationships.Li and Yang [7] investigated three-way decisions with fuzzy probabilistic covering-based rough sets and their applications in credit evaluation.Qi et al. [8] obtained some neighborhood-related fuzzy coveringbased rough set models and their applications for decision-making.
Topology [9] is a very important basic subject in mathematics, which provides a strong foundation in mathematics for the study of other theories.In this article, we discuss some properties of approximation operators with the help of neighborhood relations and prove the conditions under which the topological closure operator holds.From the perspective of closure operator, we prove the relationship between closure operators and neighborhood relations.
Although many generalized rough set models have been proposed.However, the equivalence relation on the universe plays a crucial role in the Pawlak rough set model, but in practical problems, the neighborhood relationship on the universe is not equivalent.Because the application of Pawlak rough set model is limited, it must be generalized.This article studies the rough set model of the general neighborhood relationship.
This article is organized as follows: Section 2 reviews some elementary concepts on relations, defines neighborhood relation, and obtains some interesting results; Section 3 is the core of this article, wherein we introduce the properties of δ under neighborhoods relationships and obtain the necessity and sufficiency conditions for neighborhood operators to become topological closure operator under condition attribute B; and Section 4 summarizes this article.

Preliminaries
A decision information system is a binary group , where U is a non-empty domain, AT is a condition attribute, D is a decision attributes, and represents the division on the domain of discourse induced by the decision attribute ( ) , and X i represents the ith decision, ≤ ≤ i N 1 .
Definition 2.1.Given a decision information system (DIS) ∀ ⊆ B AT, and ≥ r 0, we define where represents the distance between x and y under the condition attribute B, and , where ( ) a x represents the value of x in attribute a.

∀ ∈
x U and according to equation (1), we obtain the neighborhood of sample x with respect to B: where x U : B represents a neighborhood granulation result induced by the conditional attribute B, which constitutes the covering of domain U .

Definition 2.2. [10]
U is a non-empty set, ⊆ T 2 U .T is called a topology, if it satisfies the following conditions: In a topological space, the closure of the set X is defined as follows:  Obviously, N B is reflexive  Obviously, these neighborhoods have the following relationships: The neighborhood relation N B and the neighborhood operators N Bs and N Bp can be mutually uniquely determined as follows:

B B B p s
Theorem 2.5.Let N B and M B be two neighborhood relations on U, then

1
They are called the union, intersection, inverse, and complement of N B and N S , respectively.
If X is the open neighborhood of point x, then the point x is an inner point of the set X .If the set X is an open set of topological space, then the complement X of U is a closed set.If the set X is both an open set and a closed set, then X is said to be an open close set.
is called a topological space, and each element of T is called an open set of a topological space ( ) U T , .If ⊆ X U and ∈ x X, there exists an open set ∈ A T such that ∈ ⊆ x A X , then X is called the open neighborhood of point x.
For any neighborhood relation N B on U , . It is similar to prove the converse.Similarly, (2)-(4) can be obtained.The properties of neighborhood relations can be described by the following neighborhood operators: