New soft rough approximations via ideals and its applications

Theories of soft sets and rough sets are two different approaches to analyzing vagueness. A possible fusion of rough sets and soft sets was proposed in 2011. At this time the concept of soft rough sets was introduced, where parametrized subsets of a universal set are basic building blocks for lower and upper approximations of a subset. The main purpose of soft rough sets is to reduce the soft boundary region by increasing the lower approximation and decreasing the upper approximation. In this paper, we present two new approaches for soft rough sets that is related to the notion of ideals. The main characteristics of these recent approaches are explained and interpreted through the use of suitable propositions and examples. These recent approaches satisfy most of the conditions of well known properties of Pawlak's model. Comparisons between our methods and previous ones are introduced. In addition, we prove that our approaches produce a smaller boundary region and greater value of accuracy than the corresponding defined definitions. Furthermore, two new styles of approximation spaces related to two distinct ideals, called soft bi-ideal approximation spaces, are introduced and studied. Analysis of the fulfilled and the non-fulfilled properties is presented, and many examples to ensure and explain the advantages and the disadvantages between our styles and the previous ones are provided.