Covering rough sets based on neighborhoods: An approach without using neighborhoods

TL;DR

This paper explores covering rough sets using neighborhoods, demonstrating that upper approximations can be defined without neighborhoods and introducing operations and homomorphisms to analyze coverings.

Contribution

It presents a neighborhood-free approach to covering rough sets and introduces operations and homomorphisms for analyzing coverings.

Findings

Upper approximation based on neighborhoods can be equivalently defined without neighborhoods.

Introduces unary and composition operations on coverings.

Defines homomorphisms to relate covering approximation spaces.

Abstract

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.