الفهرس 
PreliminariesOnly 14 pages are availabe for public view
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Abstract
This thesis consists of six chapters distributed as follows:
Chapter 1: is the introductory chapter and contains the basic concepts and properties of relations and topological structures. It contains also the basic concepts and properties of multiset theory, rough set theory, nano topological spaces, soft set theory, soft multi topological spaces, fuzzy set theory and hesitant fuzzy sets.
In Chapter 2, two kinds of approximation operators via ideals which represent extensions of Pawlak’s approximation operator have been presented. In both kinds, the definitions of upper and lower approximations based on ideals have been given. Moreover, a new type of approximation spaces via two ideals which is called bi-ideal approximation spaces was introduced for the first time. This type of approximations was analyzed by two different methods, their properties are investigated and the relationship between these methods is proposed. The importance of these methods was its dependent on ideals as a topological tools and the two ideals represent two opinions instead of one opinion. Also, an applied example had been introduced in the chemistry field by applying the current methods to illustrate the definitions in a friendly way. A covering of a universe is a generalization of the concept of partition of the universe. Rough sets based on coverings instead of partitions have been studied in several manuscripts [115, 126, 127, 128]. Finally, we show that [Lemma 3.3, p.538] which was introduced in [1] is incorrect in general, by giving counter examples. Consequently, [Proposition 3.2, p.539] is also incorrect. Moreover, the correction form of the incorrect results in [1] is presented.
Chapter 3: the main aim of rough multiset is reducing the boundary region and increasing the accuracy measure by increasing the lower approximation and decreasing the upper approximation. So in this chapter, a new approach of rough multiset via multiset ideals is proposed to reduce the boundary region and increase the accuracy measure. The concepts of lower and upper multiset approximations via multiset ideals are introduced. In addition, some properties and results of these multiset approximations are presented. The relationships between the current multiset approximations are presented. Moreover, comparisons between the present method and the previous methods [41, 123] are presented and shown to be more general. Furthermore, the multiset topology induced by the current method is finer than the multiset topology induced by the previous methods [41, 123]. The importance of the current chapter is not only that it is reducing the boundary region and increasing the accuracy of sets which is the main aim of rough multiset, but also it is introducing an applied example in the medicine by applying the current method to illustrate the concepts in a friendly way. We point out where the errors occur in [41] and then give counter examples to confirm our claim. Finally, the correction form of this errors is presented.
The goal of Chapter 4 is to study the concept of I-nano topological spaces induced by different neighborhood. This concept based on ideals. Some important characteristics and significant properties of these spaces are presented. Moreover, the notions of j-ideal nano generalized closed sets and its properties are studied. We provide a comparative study of the present spaces and the previous one. It turn out that every nano topological space induced by different neighborhoods is an I-nano topological space induced by different neighborhoods. Afterwards, to emphasize these results some counter examples are considered. Eventually, an application from the real life problems is presented.
The main purpose of Chapter 5 is to introduce some important and basic issues of hesitant fuzzy soft multisets and present some results for hesitant fuzzy sets. We introduce the concept of mapping on hesitant fuzzy soft multisets and present some results for these mappings. The notions of inverse image and identity mapping are introduced and their basic properties are investigated. Therefore, the composition of two hesitant fuzzy soft multi mappings with the same dimension are presented. The concept of hesitant fuzzy soft multi topology is defined and some types of hesitant fuzzy soft multi mapping are presented in details such as continuity, open, closed and homeomorphism. Also, their properties and results are investigated. Furthermore, we introduce the concept of hesitant fuzzy soft multi connected space and present some of their properties in details.
In Chapter 6, some operators on the hesitant fuzzy soft msets such as ``AND”, ``OR” operators are introduced. The main results of the current branch are studied and some of its structural properties are established such as the neighborhood hesitant fuzzy soft multisets, interior hesitant fuzzy soft multisets and hesitant fuzzy soft multi basis. Therefore, we show that how to apply the concept of hesitant fuzzy soft multisets in decision-making problems.