Neighborhood systems are used to approximate graphs as finite topological structures. Throughout this article, we construct new types of eight neighborhoods for vertices of an arbitrary graph, say, j-adhesion neighborhoods. Both notions of Allam et al. and Yao will be extended via j-adhesion neighborhoods. We investigate new types of j-lower approximations and jupper approximations for any subgraph of a given graph. Then, the accuracy of these approximations will be calculated. Moreover, a comparison between accuracy measures and boundary regions for different kinds of approximations will be discussed. To generate j-adhesion neighborhoods and rough sets on graphs, some algorithms will be introduced. Finally, a sample of a chemical example for Walczak will be introduced to illustrate our proposed methods.

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Covering-based rough sets are important generalizations of the classical rough sets of Pawlak. In this paper, by means of j-neighborhoods, complementary j-neighborhoods and j-adhesions, we build some new different types of j-covering approximations based rough sets and study related properties. Also, we explore the relationships between the considered j-covering approximations and investigate the properties of them. Using different neighborhoods, some different general topologies are generated as topologies induced from a binary relation. Finally, an interesting application of the new types of covering-based rough sets to the rheumatic fever is given.

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of j -adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

<abstract> <p>The essential aim of the current work is to enhance the application aspects of Pawlak rough sets. Using the notion of a <italic>j</italic>-neighborhood space and the related concept of <italic>θβ</italic>-open sets, different methods for generalizing Pawlak rough sets are proposed and their characteristics will be examined. Moreover, in the context of ideal notion, novel generalizations of Pawlak's models and some of their generalizations are presented. Comparisons between the suggested methods and the previous approximations are calculated. Finally, an application from real-life problems is proposed to explain the importance of our decision-making methods.</p> </abstract>

As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to define the family of complementary fuzzy β -neighborhoods and thus three kinds of covering-based multigranulation ( ℐ , T )-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation ( ℐ , T )-fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.

Recently, the concept of a soft rough fuzzy covering (briefly, SRFC) by means of soft neighborhoods was defined and their properties were studied by Zhan’s model. As a generalization of Zhan’s method and in order to increase the lower approximation and decrease the upper approximation, the present work aims to define the complementary soft neighborhood and hence three types of soft rough fuzzy covering models (briefly, 1-SRFC, 2-SRFC, and 3-SRFC) are proposed. We discuss their axiomatic properties. According to these results, we investigate three types of fuzzy soft measure degrees (briefly, 1-SMD, 2-SMD, and 3-SMD). Also, three kinds of ψ -soft rough fuzzy coverings (briefly, 1- ψ -SRFC, 2- ψ -SRFC, and 3- ψ -SRFC) and three kinds of D -soft rough fuzzy coverings (briefly, 1- D -SRFC, 2- D -SRFC, and 3- D -SRFC) are discussed and some of their properties are studied. Finally, the relationships among these three models and Zhan’s model are presented.

The rough set principle was proposed as a methodology to cope with vagueness or uncertainty of data in the information systems. Day by day, this theory has proven its efficiency in handling and modeling many real-life problems. To contribute to this area, we present new topological approaches as a generalization of Pawlak’s theory by using j-adhesion neighborhoods and elucidate the relationship between them and some other types of approximations with the aid of examples. Topologically, we give another generalized rough approximation using near open sets. Also, we generate generalized approximations created from the topological models of j-adhesion approximations. Eventually, we compare the approaches given herein with previous ones to obtain a more affirmative solution for decision-making problems.