Condition for neighbourhoods induced by a covering to be equal to the covering itself
Hua Yao, William Zhu · International Journal of Granular Computing Rough Sets and Intelligent Systems · 2014
It is a meaningful issue that under what condition neighbourhoods induced by a covering are equal to the covering itself. In this paper, firstly, through defining some concepts, such as repeat degree and invariable covering, we present a necessary and sufficient condition for this issue. Secondly, we concentrate on the inverse issue of computing neighbourhoods by a covering, namely for a given arbitrary covering, whether or not there exists another covering such that the neighbourhoods induced by it is just the former covering. We present a necessary and sufficient condition for this issue as well. In a word, through the study on the two fundamental issues induced by neighbourhoods, we have gained a deeper understanding of the relationship between neighbourhoods and the covering which induces the neighbourhoods.