Condition for neighborhoods in covering based rough sets to form a partition

TL;DR

This paper establishes a direct necessary and sufficient condition for neighborhoods in covering-based rough sets to form a partition, focusing on the covering's structure rather than all neighborhoods, and introduces new concepts like uniform blocks, repeat degree, and excluded number.

Contribution

It provides a novel, direct criterion based on the covering itself for when neighborhoods form a partition, advancing understanding in covering-based rough set theory.

Findings

Identifies the influence of reducible elements on neighborhoods.

Introduces the concepts of uniform block, repeat degree, and excluded number.

Derives a necessary and sufficient condition for neighborhoods to form a partition.

Abstract

Neighborhood is an important concept in covering based rough sets. That under what condition neighborhoods form a partition is a meaningful issue induced by this concept. Many scholars have paid attention to this issue and presented some necessary and sufficient conditions. However, there exists one common trait among these conditions, that is they are established on the basis of all neighborhoods have been obtained. In this paper, we provide a necessary and sufficient condition directly based on the covering itself. First, we investigate the influence of that there are reducible elements in the covering on neighborhoods. Second, we propose the definition of uniform block and obtain a sufficient condition from it. Third, we propose the definitions of repeat degree and excluded number. By means of the two concepts, we obtain a necessary and sufficient condition for neighborhoods to form a partition. In a word, we have gained a deeper and more direct understanding of the essence over that neighborhoods form a partition.