格上粗糙集研究
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摘要
Z.Pawlak于1982年提出的粗糙集理论,是一种新的处理不确定知识的数学工具.本文主要利用格、Quantale上的同余关系和集值同态,分别建立格和Quantale上的粗糙集和广义粗糙集,通过对其性质讨论,进一步刻画格、Quantale的代数结构.
同余关系是代数结构中与其相容的等价关系.文中利用格上同余关系建立上、下逼近算子,对粗子集的性质进行了研究,并定义了粗理想、粗素理想等,得到了粗理想、粗素理想分别是理想、素理想的延伸概念,并讨论了上、下逼近算子之间的包含关系.对这些上、下逼近算子的格结构进行了研究,得到格中所有可定义集的集合是一个有顶交结构和完备集域,有最小元的格中所有可定义理想的集合是有顶的代数交结构.讨论了格中上、下逼近算子的乘积、商与集合乘积、商的上、下逼近算子之间的关系.并考虑了上、下逼近算子的格同态问题.
格中有一类特殊的并同余关系,是由理想所确定的,当格满足分配时是同余关系.在格中利用这类特殊同余关系建立上、下逼近算子,对其具有特殊的性质进行讨论.利用理想的相关性质,对上、下逼近算子的包含关系进行了讨论.并对这类特殊同余关系的可定义集进行了讨论,得到格中的理想关于它本身所确定的并同余是可定义的,同时格中包含某个理想的所有理想的集合等于这个理想所确定的并同余关系的可定义理想的集合,且这个集合是代数格.同时对一些特殊的理想,如下集、核理想等的上、下逼近算子进行了讨论.
在代数系统上将粗糙集推广到两个论域的情形,需要把同余关系进行推广.文中的第三章和第四章,分别在格、Quantale中提出了集值同态的定义,并分别利用集值同态概念给出了广义上、下逼近算子,讨论了广义的粗子集、粗理想等性质.在格中还定义了关于理想的特殊集值同态,得到特殊的广义粗糙集,对其性质进行讨论,并举例说明在形式概念分析方面的应用.
导子的定义来自解析理论,导子是定义在代数系统上的函数,对导子的研究有助于进一步研究该代数系统的代数性质和结构.在第五章中给出了Quantale上导子的定义,并定义了简单导子,恒等导子,零导子等,并探讨了它们的性质.研究了Quantale中(严格)左、右、双侧元关于导子的像及Quantale上一个导子的所有不动点集合的具体结构.证明了满足一定条件的Quantale其上导子的集合构成一个Quantale.核映射在Quantale理论中很重要,通过预核映射与核映射之间的关系,进一步研究了Quantale上导子与核映射之间的关系.
Rough set theory was introduced by Z.Pawlak (1982), which is a new math-ematical approach to deal with uncertain information. In this paper, the authordiscusses the properties of rough sets and generalized rough sets which are con-structed by congruences and set-valued homomorphisms respectively in latticesand quantales, then the algebraic structure of lattices and quantales is explored indetail.
A congruence relation is an equivalence relation consistent with the algebraicstructure. In the paper, the upper and lower approximations are constructed bycongruence relations. The concepts of rough ideal and rough prime ideal are de-fined which are the extended notion of ideal and prime ideal respectively, and theinclusion relations of the upper and lower approximations are discussed. The lat-tice structures of the lower and upper approximations are studied. We obtain theresults that the collection of all definable sets in a lattice is a topped intersectionstructure and also a complete field of sets. And the collection of all definable idealsin a lattice with a bottom element is a topped algebraic intersection structure. Wealso discuss the relations among products、quotients of the lower、upper approx-imations and the lower、upper approximations of products、quotients. And theproblems of homomorphism of the lower and upper approximations are studied.
There is a special class of join congruence relations determined by ideals oflattice, it is a congruence relation if the lattice is distributive. We study thespecial properties of the upper and lower approximations constructed by means ofthese special congruence relations on a lattice. By using the properties of ideals,we analyze the inclusion relation of the lower and upper approximations. Thedefinable sets with respect to the special congruences are studied. We obtain thatthe collection of definable ideals with respect to the special congruence determinedby an ideal is that of all ideals which contained this ideal, and the set is an algebraiclattice. The lower and upper approximations of some particular ideals are discussedsuch as lower set and kernel ideal, etc.
In an algebraic system, we consider the rough set in the case of two universes,the congruence relation needs to be generalized. Chapter3and chapter4intro-duce the notion of set-valued homomorphism of lattices and quantales respectively.We probe into the properties of generalized rough subsets and generalized roughideals. We also propose the special set-valued homomorphisms with respect to ideals of lattice, the properties of the generalized rough sets constructed by themare discussed. And the example in the application of formal concept analysis isgiven.
The notion of derivation is a function on an algebraic system introduced fromthe analytic theory, is helpful to the research of structure and property in al-gebraic systems. Chapter5, the concept of derivation on quantales is defined,simple derivation、identity derivation and zero derivation are proposed, the prop-erties of derivations are discussed in detail. The image of derivation of (strict)left、right、two sided elements of quantales are studied. The structure of the col-lection of all fixed points of a derivation is explored. The collection of all derivationson a quantale which satisfies some conditions is a quantale. The quantic nuclei playan important role in quantale theory, by using the relation between pre-quanticnuclei and quantic nuclei, the relations between derivation and quantic nuclei arestudied.
This dissertation is typeset by software LATEX2ε.
同余关系是代数结构中与其相容的等价关系.文中利用格上同余关系建立上、下逼近算子,对粗子集的性质进行了研究,并定义了粗理想、粗素理想等,得到了粗理想、粗素理想分别是理想、素理想的延伸概念,并讨论了上、下逼近算子之间的包含关系.对这些上、下逼近算子的格结构进行了研究,得到格中所有可定义集的集合是一个有顶交结构和完备集域,有最小元的格中所有可定义理想的集合是有顶的代数交结构.讨论了格中上、下逼近算子的乘积、商与集合乘积、商的上、下逼近算子之间的关系.并考虑了上、下逼近算子的格同态问题.
格中有一类特殊的并同余关系,是由理想所确定的,当格满足分配时是同余关系.在格中利用这类特殊同余关系建立上、下逼近算子,对其具有特殊的性质进行讨论.利用理想的相关性质,对上、下逼近算子的包含关系进行了讨论.并对这类特殊同余关系的可定义集进行了讨论,得到格中的理想关于它本身所确定的并同余是可定义的,同时格中包含某个理想的所有理想的集合等于这个理想所确定的并同余关系的可定义理想的集合,且这个集合是代数格.同时对一些特殊的理想,如下集、核理想等的上、下逼近算子进行了讨论.
在代数系统上将粗糙集推广到两个论域的情形,需要把同余关系进行推广.文中的第三章和第四章,分别在格、Quantale中提出了集值同态的定义,并分别利用集值同态概念给出了广义上、下逼近算子,讨论了广义的粗子集、粗理想等性质.在格中还定义了关于理想的特殊集值同态,得到特殊的广义粗糙集,对其性质进行讨论,并举例说明在形式概念分析方面的应用.
导子的定义来自解析理论,导子是定义在代数系统上的函数,对导子的研究有助于进一步研究该代数系统的代数性质和结构.在第五章中给出了Quantale上导子的定义,并定义了简单导子,恒等导子,零导子等,并探讨了它们的性质.研究了Quantale中(严格)左、右、双侧元关于导子的像及Quantale上一个导子的所有不动点集合的具体结构.证明了满足一定条件的Quantale其上导子的集合构成一个Quantale.核映射在Quantale理论中很重要,通过预核映射与核映射之间的关系,进一步研究了Quantale上导子与核映射之间的关系.
Rough set theory was introduced by Z.Pawlak (1982), which is a new math-ematical approach to deal with uncertain information. In this paper, the authordiscusses the properties of rough sets and generalized rough sets which are con-structed by congruences and set-valued homomorphisms respectively in latticesand quantales, then the algebraic structure of lattices and quantales is explored indetail.
A congruence relation is an equivalence relation consistent with the algebraicstructure. In the paper, the upper and lower approximations are constructed bycongruence relations. The concepts of rough ideal and rough prime ideal are de-fined which are the extended notion of ideal and prime ideal respectively, and theinclusion relations of the upper and lower approximations are discussed. The lat-tice structures of the lower and upper approximations are studied. We obtain theresults that the collection of all definable sets in a lattice is a topped intersectionstructure and also a complete field of sets. And the collection of all definable idealsin a lattice with a bottom element is a topped algebraic intersection structure. Wealso discuss the relations among products、quotients of the lower、upper approx-imations and the lower、upper approximations of products、quotients. And theproblems of homomorphism of the lower and upper approximations are studied.
There is a special class of join congruence relations determined by ideals oflattice, it is a congruence relation if the lattice is distributive. We study thespecial properties of the upper and lower approximations constructed by means ofthese special congruence relations on a lattice. By using the properties of ideals,we analyze the inclusion relation of the lower and upper approximations. Thedefinable sets with respect to the special congruences are studied. We obtain thatthe collection of definable ideals with respect to the special congruence determinedby an ideal is that of all ideals which contained this ideal, and the set is an algebraiclattice. The lower and upper approximations of some particular ideals are discussedsuch as lower set and kernel ideal, etc.
In an algebraic system, we consider the rough set in the case of two universes,the congruence relation needs to be generalized. Chapter3and chapter4intro-duce the notion of set-valued homomorphism of lattices and quantales respectively.We probe into the properties of generalized rough subsets and generalized roughideals. We also propose the special set-valued homomorphisms with respect to ideals of lattice, the properties of the generalized rough sets constructed by themare discussed. And the example in the application of formal concept analysis isgiven.
The notion of derivation is a function on an algebraic system introduced fromthe analytic theory, is helpful to the research of structure and property in al-gebraic systems. Chapter5, the concept of derivation on quantales is defined,simple derivation、identity derivation and zero derivation are proposed, the prop-erties of derivations are discussed in detail. The image of derivation of (strict)left、right、two sided elements of quantales are studied. The structure of the col-lection of all fixed points of a derivation is explored. The collection of all derivationson a quantale which satisfies some conditions is a quantale. The quantic nuclei playan important role in quantale theory, by using the relation between pre-quanticnuclei and quantic nuclei, the relations between derivation and quantic nuclei arestudied.
This dissertation is typeset by software LATEX2ε.
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