A necessary and sufficient condition for two relations to induce the same definable set family

In Pawlak rough sets, the structure of the definable set families is simple and clear, but in generalizing rough sets, the structure of the definable set families is a bit more complex. There has been much research work focusing on this topic. However, as a fundamental issue in relation based rough sets, under what condition two relations induce the same definable set family has not been discussed. In this paper, based on the concept of the closure of relations, we present a necessary and sufficient condition for two relations to induce the same definable set family.

Applications of repeat degree on coverings of neighborhoods

In covering based rough sets, the neighborhood of an element is the intersection of all the covering blocks containing the element. All the neighborhoods form a new covering called a covering of neighborhoods. In the course of studying under what condition a covering of neighborhoods is a partition, the concept of repeat degree is proposed, with the help of which the issue is addressed. This paper studies further the application of repeat degree on coverings of neighborhoods. First, we investigate under what condition a covering of neighborhoods is the reduct of the covering inducing it. As a preparation for addressing this issue, we give a necessary and sufficient condition for a subset of a set family to be the reduct of the set family. Then we study under what condition two coverings induce a same relation and a same covering of neighborhoods. Finally, we give the method of calculating the covering according to repeat degree.

Condition for neighborhoods induced by a covering to be equal to the covering itself

It is a meaningful issue that under what condition neighborhoods induced by a covering are equal to the covering itself. A necessary and sufficient condition for this issue has been provided by some scholars. In this paper, through a counter-example, we firstly point out the necessary and sufficient condition is false. Second, we present a necessary and sufficient condition for this issue. Third, we concentrate on the inverse issue of computing neighborhoods by a covering, namely giving an arbitrary covering, whether or not there exists another covering such that the neighborhoods 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 neighborhoods, we have gained a deeper understanding of the relationship between neighborhoods and the covering which induce the neighborhoods.

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

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.