Algebraic Structure, Topology And Its Application Of Uncertainty Theory Based On Soft Set

In recent years, along with the occurrence of a large number of uncertainty in the real world, researchers from various areas, such as mathematics, quantum physics, computer science, information science, decision analysis and artificial intelligence, have shown heightened interests in modeling and manipulating uncertainties. As those traditional mathematical tools which are based on crisp sets and Boolean logic deal with problems with uncertainties by ignoring the inherent uncertainties in concepts and collected data, it often results in inaccurate even unreasonable conclusions. In order to overcome these drawbacks, a great number of new mathematical tools (such as fuzzy set theory with emphasis on the idea of gradualness, rough set theory focusing on the idea of granularity, etc.) have been put forward recently. To cope with uncertainty in a more general scheme, Molotsov proposed soft set theory with the idea of parameterization in 1999. As a newly-emerging area of interdisciplinary researches, soft set theory has received much attention from a number of researchers all over the world, and an increasing number of high-quality works on soft sets and related topics have been published over past few years. Given all this, this dissertation mainly investigates the algebraic and topological structures of soft sets theory and considers also an application of the results in decision making. Details are as follows:Chapter 1 reviews some preliminaries about several commonly used classes of lattice theory, algebra, topology, soft sets, fuzzy sets, as well as categorical theory, which are necessary for the rest chapters.Chapter 2, for the limitation and shortage of soft set relation defined by Babitha, first introduces the concept of Z-soft set relation and discusses the basic properties of Z-soft set relation in detail by defining the operations such as the intersection, union, complement, inverse and so on. In the second section of this chapter, Z-soft set relation from the point of algebraic structures is further investigated. It is proved that all the Z-soft set relations based on a soft set give rise to distributive lattices, commutative semirings with monoids and topological rough algebras, and so on. We then discuss the topological structures of Z-soft set relations in the last section of this chapter. Concretely, the notion of Z-soft set relation topologies,Z-soft set relation neighborhoods, Z-anti-reflexive interior operators and Z-reflexive closure operators are introduced. Moreover, the internal relations between Z-anti-reflexive interior operators and Z-reflexive closure operators are established. It is also proved that there exist at most six different Z-soft set relations over a soft set by the relationships mentioned above.Chapter 3 begins with the introduction of the category denoted by SFun of soft sets and soft functions. Then, the constructs of equalizers, products and so on are given. It is found that SFun is not only a topological category over Set, but also Cartesian closed. In the second section of this chapter, the category denoted by SRel of soft sets and Z-soft set relations are considered. The zero objects, biproducts, semi-additive structures, injective objects, projective objects, injective hulls and projective covers of SRel are studied. The last section discusses the essential connections of SFun and SRel by means of adjoint situations.Chapter 4 first introduces the concept of M-soft rough fuzzy sets by means of MS-approximation spaces, which not only overcomes the limitation that soft sets must be full, but also provides a new method for describing the intrinsic character of every object in universe. Then in the next section, one characterizes two fuzzy topology spaces by M-soft defined sets. And various relationships associated with the M-lower and M-upper soft rough approximations are investigated by generalizing the relations associated with lower and upper approximations defined by Feng to M-soft rough fuzzy sets in the last section of this chapter.Chapter 5 further investigates the basic properties and structural characteriza-tions of soft rough fuzzy sets. Although M-soft rough fuzzy sets weaken the condition of soft rough fuzzy sets defined by Feng and generalize it to full soft sets. In fact, there still exist some limitations. To further overcome these drawbacks, we first introduce the concept of Z-soft rough fuzzy sets, then study their basic properties, meanwhile give some examples to illustrate the rationality of our results. Then in the second sec-tion, one compares Z-soft rough fuzzy sets with rough fuzzy sets, soft rough sets, soft rough fuzzy sets defined by Feng and Meng. It is proved that Z-soft rough fuzzy sets are the generalization of rough fuzzy sets; Z-lower soft rough approximation operators are the generalization of the low soft rough approximation operators; Z-soft rough fuzzy sets are more accurate than soft rough fuzzy sets defined by Feng and Meng. In the next section, the algebraic properties of Z-soft rough fuzzy sets are studied. It is shown that Z-soft rough fuzzy sets give rise to lattices, commutative semigroups with monoids, commutative idempotent hemirings, quasi Boolean algebras and stone algebras.Finally, in the last section of this chapter, we put forward an algorithm for the approach to the decision making based on Z-soft rough fuzzy sets, and offer an example about recruit to illustrate the application of Z-soft rough fuzzy sets in...