Quasi-atomic relations based rough set model and convex geometry

Numerous studies have extensively examined the correlation between convex structures and covering rough set models. However, limited attention has been devoted to investigating the relationship between convex structures and rough set models based on relations. In this paper, we aim to integrate convex geometry with rough sets based on relations. Firstly, a novel class of binary relation known as quasi-atomic relation is introduced. We show that each ordinal convex geometry can be reformulated by rough sets based on quasi-atomic relations. Subsequently, we illustrate that any convex geometry can be embedded in an ordinal convex geometry using the rough set approach. Secondly, we develop a new model called the multi-valued rough set model which serves as a valuable tool for investigating convex geometry. A novel representation of any convex geometry is provided by multi-valued rough sets based on the family of quasi-atomic relations. Furthermore, we demonstrate that any convex geometry can be decomposed into a union of ordinal convex geometries. Lastly, We develop a method to determine whether an element belongs to a convex closure of a given set. In addition, we incorporate terminologies from rough set theory into the study of convex geometry by defining concepts such as rough convex sets and degree of convexity which are utilized to analyze the structure of convex geometry. In summary, this paper offers an efficient method to integrate rough sets with convex geometry, which extends the theory and application of rough sets.