L-abstract Basis And Left LQ-modules

Abstract Domain theory and quantale theory have both theoretical computer science and mathmatica research background dual. They developed independently, but they are based on order theory from the viewpoint of common mathemati-cal foundation. Meanwhile, they make the close relationship with algebra, logic, category and so on. Although domain theory and quantale theory have different research objects and characteristics, they are mutual penetration and influence in some aspects, for example, Pawel Waszkiewicz popularize domain theory on Girard quantales. Since2000, fuzzy set theory has applied to the quantitative domain the-ory and many beautiful conclusions and properties are obtained. The first part of this thesis is to research L-abstract basis in the domain theory. The second part is to apply the fuzzy set theory to quantale modules, the concept of left LQ-module is given, and its properties are discussed.The arrangement of this thesis is as follows:Chapter One:Preliminaries. The basic concepts and existing results of the theories of fuzzy domain and category which will be used throughout the thesis are given.Chapter Two:L-Abstract Basis and The Fuzzy Round Ideal Completion of L-Abstract Basis. Firstly, the L-abstract basis and the fuzzy round ideal are in-troduced, and the equivalent characterization of a fuzzy round ideal is given, it is proved that the fuzzy round ideal of L-abstract basis (X,(?)) on a fuzzy domain X is isomorphic to the fuzzy domain. Secondly, the fuzzy round ideal completion of the L-abstract basis is studied, and the fuzzy round ideal completion of L-abstact basis on a fuzzy poset is a fuzzy domain. Finally, a Scott continuous retract of a fuzzy domain is a fuzzy domain.Chapter Three:The basic properties of left LQ-modules. Firstly, the concept of left LQ-module and the homomorphism of two left LQ-modules are introduced, and related examples are given. Secondly, the concept of nuclei on left LQ-modules is defined, and its properties are discussed. It is proved that the left LQ-module nuclei is one-to-one correspondence to some left LQ-module. At last, it is to further investigate fuzzy Girard bimodules and involutive left LQ-modules, we show that any fuzzy quantale (fuzzy involutive quantale)is an (involutive)subquantale of Q(M) for a suitable fuzzy Girard bimodules M. Chapter Four:The category of left LQ-modules. The concept of left LQ-submodule is given, and we establish the correspondence between left LQ-submodule and left LQ-module conuclei and prove that the category of left LQ-modules has products and is complete.