子效应代数的模糊化与粗糙集相关研究
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摘要
本文第一部分将效应代数与模糊结构(模糊集、直觉模糊集/Vague集及软集)相结合,主要研究子效应代数的模糊化,即：直觉模糊子效应代数(简称Ⅱ-子效应代数)、模糊子效应代数和软效应代数；第二部分主要研究粗糙集相关理论.全文主要内容如下：
第一章预备知识.给出了本文将要用到的有关模糊集、软集、效应代数以及模糊逻辑算子的一些概念和结论.
第二章主要研究Ⅱ-子效应代数以及模糊子效应代数.首先,给出Ⅱ-子效应代数和模糊子效应代数的概念,以及给出它们的等价刻画.其次,证明：所有Ⅱ-子效应代数以及所有模糊子效应代数都构成一个完备格.当|E|≤5时,所有Ⅱ-子效应代数(相应的,所有模糊子效应代数)是一个Hutton代数且是(ⅡE,≤)(相应的,([0,1]E,≤))的一个完备子格.也讨论其他一些相关的性质.最后,研究Ⅱ-可特征化的子效应代数和模糊可特征化的子效应代数.
第三章主要将软集思想应用到效应代数中.给出软效应代数、软效应代数的理想、以及基于效应代数和理想的软理想的概念,阐明了软效应代数、软效应代数的理想、以及基于效应代数和理想的软理想是模糊子效应代数和模糊理想的推广.分别详细讨论它们之间的一些代数运算的性质.
第四章主要研究基于Vague等价关系的粗糙集分解及直觉模糊近似空间的变换.提出基于Vague等价关系的(αt,αf)-等价类,并在(αt,αf)-等价类基础上定义了(αt,αf)-粗糙集,得到(αt,αf)-粗糙集是λ-粗糙集的推广,研究了(αt,αf)-等价类和(αt,αf)-粗糙集的性质.分别得到(αt,αf)-等价类、粗糙集以及粗糙集的边界基于Vague等价关系的分解结构.给出直觉模糊近似空间的并、交、逆以及合成.作为直觉模糊近似空间的并、交、逆的应用,也考察了由直觉模糊关系的各种核与闭包诱导的直觉模糊近似空间.
第五章主要利用构造性方法以及公理化方法在一个更广阔的框架下研究基于一般模糊关系的(I,J)-模糊粗糙集.在构造性方法中,在任意一个模糊近似空间上给出(I,J)-模糊粗糙集的概念并且讨论(I,J)-模糊粗糙近似算子的性质.建立了特殊的模糊关系与模糊近似算子的性质之间的联系.在公理化方法中,给出了(I，J)-模糊粗糙近似算子的公理化刻画.
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.
第一章预备知识.给出了本文将要用到的有关模糊集、软集、效应代数以及模糊逻辑算子的一些概念和结论.
第二章主要研究Ⅱ-子效应代数以及模糊子效应代数.首先,给出Ⅱ-子效应代数和模糊子效应代数的概念,以及给出它们的等价刻画.其次,证明：所有Ⅱ-子效应代数以及所有模糊子效应代数都构成一个完备格.当|E|≤5时,所有Ⅱ-子效应代数(相应的,所有模糊子效应代数)是一个Hutton代数且是(ⅡE,≤)(相应的,([0,1]E,≤))的一个完备子格.也讨论其他一些相关的性质.最后,研究Ⅱ-可特征化的子效应代数和模糊可特征化的子效应代数.
第三章主要将软集思想应用到效应代数中.给出软效应代数、软效应代数的理想、以及基于效应代数和理想的软理想的概念,阐明了软效应代数、软效应代数的理想、以及基于效应代数和理想的软理想是模糊子效应代数和模糊理想的推广.分别详细讨论它们之间的一些代数运算的性质.
第四章主要研究基于Vague等价关系的粗糙集分解及直觉模糊近似空间的变换.提出基于Vague等价关系的(αt,αf)-等价类,并在(αt,αf)-等价类基础上定义了(αt,αf)-粗糙集,得到(αt,αf)-粗糙集是λ-粗糙集的推广,研究了(αt,αf)-等价类和(αt,αf)-粗糙集的性质.分别得到(αt,αf)-等价类、粗糙集以及粗糙集的边界基于Vague等价关系的分解结构.给出直觉模糊近似空间的并、交、逆以及合成.作为直觉模糊近似空间的并、交、逆的应用,也考察了由直觉模糊关系的各种核与闭包诱导的直觉模糊近似空间.
第五章主要利用构造性方法以及公理化方法在一个更广阔的框架下研究基于一般模糊关系的(I,J)-模糊粗糙集.在构造性方法中,在任意一个模糊近似空间上给出(I,J)-模糊粗糙集的概念并且讨论(I,J)-模糊粗糙近似算子的性质.建立了特殊的模糊关系与模糊近似算子的性质之间的联系.在公理化方法中,给出了(I，J)-模糊粗糙近似算子的公理化刻画.
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.
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