Properties of approximation operators over 1-neighborhood systems from the perspective of special granules

As generalizations of Pawlak-neighborhood systems, 1-neighborhood systems with symmetry or transitivity are closely related to both partition spaces and covering spaces. In this article, we analyze the properties of a single covering-based approximation operator on symmetric or transitive 1-neighborhood systems. We also investigate the relationships between different covering-based approximation operators on them. Theoretically, we illuminate some necessary and sufficient conditions for 1-neighborhood systems being symmetric, transitive, or partitions with one or two approximation operators. To reduce potential computation complexity owing to these equivalent characterizations, objects dealt by approximation operators in this work are three particular kinds of granules, namely, points of universes, elements of 1-neighborhood systems, and cores of 1-neighborhood systems. As experimental results indicate, this study outdoes some related works in terms of computational efficiency, establishing the advantages of computing on these granules. Furthermore, our research has resulted in a solution to a problem posed by Yun et al. (Axiomatization and conditions for neighborhoods in a covering to form a partition. Information Sciences 181(2011)1735-1740).