Matroidal structures on S-approximation spaces

Abstract

S-approximation spaces generalize models of two-universe by defining a decision mapping, which in fact is an extension of the approximation operator in a two-universe model. In this paper, we study matroidal structures induced by S-approximation spaces. Firstly, we define the upper and lower approximation numbers based on the knowledge and decision mappings, and point out that the upper approximation numbers (resp. lower approximation numbers) of $S_{\mathrm{min}}$-approximation spaces (resp. $S_{\mathrm{max}}$-approximation spaces) are nondecreasing submodular functions. The S-matroids are defined according to the induced submodular functions. Secondly, we investigate the relation between rank functions of S-matroids and approximation number functions of S-approximation spaces, and depict S-matroids in terms of circuits. Finally, we study the operations of S-approximation spaces, and answer the question whether the intersection, union and complement of S-approximation spaces can induce S-matroids.