Fuzziness Of Subeffect Algebras And Related Research Of Rough Sets

Abstract In the first part of this thesis, fuzziness of effect algebras are mainly studied by connecting effect algebras with fuzzy structure (Fuzzy sets, Intuitionistic sets/Vague sets, and Soft sets), i.e.,Ⅱ-subeffect algebras, fuzzy subeffect algebras and soft effect algebras; the second part is to study rough sets theory. The main content of this paper is as follows:Chapter One:Prelimilaries. We give the concepts and results of the theories of fuzzy sets, soft sets, effect algebras, and fuzzy logic operators, which will be used throughout this thesis.Chapter Two:We mainly studied intuitionistic fuzzy effect algebrasⅡ-subeffect algebras, for short) and fuzzy subeffect algebras. First, the notions ofⅡ-subeffect al-gebras and fuzzy subeffect algebras of effect algebras are given. Characterizations of I-subeffect algebras and fuzzy subeffect algebras are obtained, respectively. Second, we show:Both the set of allⅡ-subeffect algebras and the set of all fuzzy subeffect algebras are complete lattices. The set of allⅡ-subeffect algebras (resp., the set of all fuzzy subeffect algebras) is a Hutton algebra and a complete sublattice of (ⅡE,≤) (resp., of ([0,1]E,≤)) in the case of│E│≤5. Last, we also discussⅡ-characteristic subeffect algebras and fuzzy characteristic subeffect algebras.Chapter Three:We mainly applied the ideals of soft sets to effect algebras. We define the notions of soft effect algebra, ideal of soft effect algebra, and soft ideal based on the notions of effect algebra and soft set. And we point out effect algebra, ideal of soft effect algebra, and soft ideal are generalizations of fuzzy subeffect algebra and fuzzy ideal. We also investigate relations between soft effect algebras and soft ideals. Properties of algebraic operations, of soft effect algebras, ideals of soft effect algebras, and soft ideals are discussed in detail, respectively.Chapter Four:We mainly studied the decomposition of rough sets based on the Vague equivalence relation and transformation of intuitionistic fuzzy rough set models. (αt,αf)-equivalence classes based on the Vague equivalence relation are introduced and (αt,αf)-rough sets are defined based on-equivalence classes, it is ob-tained that (αt,αf)-rough sets is a generalization ofλ-rough sets, and properties of (αt,αf)- equivalence classes and (αt,αf)-rough sets are investigated. The decom-position structures based on the Vague equivalence relation of (αt,αf)-equivalence classes,rough sets and the boundary of rough sets are obtained, respectively. Transformation of intuitionistic fuzzy rough set models containing union, intersec-tion, inverse and composition of intuitionistic fuzzy approximation spaces. As ap-plications, intuitionistic fuzzy approximation spaces induced from several kinds of kernels and closures of intuitionistic fuzzy relations are also investigated. Chapter Five:We mainly studied a general fuzzy relation based (I, J)-fuzzy rough sets by using constructive and axiomatic approaches in a general framework. In the constructive approach, by employing a pair of implicators (I, J),lower and upper approximations of fuzzy sets with respect to a fuzzy approximation space are first defined. Properties of (I, J)-fuzzy rough approximation operators are ex-amined. The connections between special types of fuzzy relations and properties of fuzzy approximation operators are established. In the axiomatic approach, ax-iomatic characterizations of (I, J)-fuzzy rough sets are given.