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.

This paper was devoted to defining new soft closure operators via soft relations and soft ideals, and consequently new soft topologies. The resulting space is a soft ideal approximation. Many of the well known topological concepts were given in the soft set-topology. Particularly, it introduced the notations of soft accumulation points, soft continuous functions, soft separation axioms, and soft connectedness. Counterexamples were introduced to interpret the right implications. Also, a practical application of the new soft approximations was explained by an example of a real-life problem.

Various types of double fuzzy continuity of fuzzy multifunctions are introduced in this paper. These types are double fuzzy upper and lower $\alpha$- $\mathrm{ð}$-continuous, almost $\alpha$- $\mathrm{ð}$-continuous, weakly $\alpha$- $\mathrm{ð}$-continuous and almost weakly $\alpha$- $\mathrm{ð}$-continuous multifunctions. Double fuzzy ideals are playing the main role in defining these types of continuous multifunctions. All implications associated with these types are ensured; also, many examples are introduced to illustrate these implications, and to explain the advantages of these new types of continuity for some previous definitions.