Research On Several Problems Of Fuzzy Closed System, M - Polar Fuzzy Set And Fuzzy Soft Algebra

This paper focuses on four aspects-fuzzy closure operator, m-polar fuzzy sets, fuzzy soft algebra, fuzzy filters on residuated lattices.Chapter1is a introduction. Some basic concepts and conclusions on lattice, fuzzy topology, category and soft set is presented.In Chapter2, we define the notions of (L, M)-fuzzy closure system and (L, M)-fuzzy closure operator, and prove that the category of (L, M)-fuzzy closure system space and the category of (L, M)-fuzzy closure operator space are isomorphic. We prove that the category of (2, M)-fuzzy closure system space (resp.,(2, M)-fuzzy closure operator space) can be embedded into the category of (L, M)-fuzzy closure system space (resp.,(L, M)-fuzzy closure operator space).In Chapter3, it is proved that the notion of bipolar fuzzy set (or2-polar fuzzy set) can be identified with the notion [0,1]2-set. Based on this result we put forward the notion of m-polar fuzzy set (a generalization of bipolar fuzzy set) and illustrate how concepts which have been defined based on bipolar fuzzy sets and results which are related to these concepts can be generalized to the case of m-polar fuzzy sets, we also give examples to show how to apply m-polar fuzzy sets in real world problems.In Chapter4, the notions of anti-fuzzy soft ideals in BCI-algebras are given, some of their properties (such as union, intersection and AND) are studied. The definitions of fuzzy soft homomorphism and fuzzy soft isomorphism of fuzzy soft ideals in BCI-algebras are defined and the theorems of isomorphic image and homomorphic pre-image of fuzzy soft ideals in BCI-algebras are given. The notion of bipolar-value fuzzy soft ideal in sub-BCI-algebras is given, some of their properties are studied. Furthermore, definitions of bipolar-value fuzzy soft homomorphism and bipolar-value fuzzy soft isomorphism of bipolar-value fuzzy soft ideals in sub-BCI-algebras are defined and the theorems of isomorphic image and homomorphic pre-image of bipolar-value fuzzy soft ideals in sub-BCI-algebras are given.Chapter5mainly studies fuzzy filters and intersectional soft filters on residuated lattices. We introduce the notion of fuzzy filter and fuzzy congruence relation on resid-uated lattices, give a one-to-one correspondence from the set of all fuzzy filters and the set of all fuzzy filters in residuated lattices, and prove that an appropriate order relation can be defined on the set of all fuzzy filters (resp., the set of all fuzzy filters) such that it is a complete lattice isomorphism. Endowing a residuated lattice as a parameter set, the concept of intersectional soft filter on a residuated lattice is introduced, some charac-terizations and intersectional operations of intersectional soft filters are discussed, and the relationship between intersectional soft filters on a residuated lattice and intersectional soft congruence relations on a residuated lattice is given. In particular, it is proved that the set of all intersectional soft filters on a residuated lattice and the set of all intersectional soft congruence relation on a residuated lattice are complete lattices which are isomorphic. Finally, the concepts of image and preimage of a intersectional soft filters of residuated lattice and their properties are shown.