On The Application Of Fuzzy Soft Set In Some Algebra Structures And Topologies

Molodtsov introduced the concept of soft set which can be used as generic mathematical tool for dealing with uncertainty. This paper focuses on some prob-lems about fuzzy soft sets applies to some algebraical structure and topological structure.The main contents are as follows:Chapter1is a introduction. Some basic concepts and conclusions in lattice theory,fuzzy sets, fuzzy soft sets, fuzzy algebras, fuzzy topology as well as categorical theory is presented.Chapter2mainly applies fuzzy soft sets to group structure, Lie algebras and incline algebras. The notions of anti-fuzzy soft subgroups and pseudo fuzzy soft sub-groups are firstly introduced, and quotient group induced by fuzzy soft subgroups are given. With the help of the concept of fuzzy soft sets, the notions and fuzzy soft homomorphism of fuzzy soft Lie subalgebras in Lie algebras are defined, and some of their properties such as union, intersection and AND are studied. The theorem of fuzzy soft homomorphic pre-image of fuzzy soft Lie subalgebra and a counterexample which a fuzzy soft Lie subalgebra need be a fuzzy soft Lie subalgebra under homo-morphism of fuzzy soft Lie subalgebra are given. Finally, the notions of fuzzy soft subinclines in incline algebras are given, some of their properties are studied. Fur-thermore, definitions of fuzzy soft homomorphism and fuzzy soft isomomorphism of fuzzy soft subinclines in incline algebras are defined and the theorems of isomomor-phic image and homomorphic pre-image of fuzzy soft subinclines in incline algebras are given, it is proven that the category of fuzzy soft subinclines in incline algebras is topological over the category of incline algebras.Chapter3mainly applies fuzzy soft sets to topological constructs. Firstly, the relationship between (L, M)-fuzzy (E,K)-soft closure systems and closure (L, M)-fuzzy (E,K)-soft neighborhood systems is investigated, then the relationship be-tween the idea of (L, M)-fuzzy topological space introduced by Kubiak and Sostak and the concept of (L, M)-fuzzy neighborhood system (which is a generalization of the notion of L-fuzzy neighborhood system of F.-G. Shi) is further studied. As an application of the obtained results, we introduce a notion of (L, M)-fuzzy topolog-ical group. This (L, M)-fuzzy topological group generalizes the notion of I-fuzzy topological group introduced by C.-H. Yan et al. Thus, we are able to produce various results of (L, M)-fuzzy topological groups, including some results on quo-tient groups. We show that our category of (L, M)-fuzzy topological groups is also topological over the category of groups.Chapter4mainly introduces open morphisms in categorical interior operators. Let C be a category and a fixed class M of C-monomorphisms such that(ε, M) a proper stable factorization system. IN(C, M),CL(C, M) and NO(C, M) denote the set of all categorical closure operators on C with respect to M, the set of all categor-ical neighborhood operators on C with respect to M and the set of all categorical interior operators on C with respect to M. When it satisfies some conditions and appropriate order relations can be defined on IN(C,M),CL(C,M) and NO(C, M), it can be proven they are isomorphism between complete classes.Finally, the main work of this thesis is summarized, and the future work is given.