| Domain theory that established in the late 1960’s and early 1970’s was in-troduced by D.Scott.Meanwhile,it makes the close relationships with topology,algebra,category and so on.In 1965,L.A.Zadeh introduced the concept of"Fuzzy Sets",which is the symbol that fuzzy mathematics was born.Since 2000,fuzzy set theory has been applied to the quantitative domain theory,forming fuzzy domain theory.In the classical setting,open sets and closed sets of topological spaces have a good corresponding relationship by taking complement.However,in the fuzzy setting,since the table of truth values is not usually a Boolean algebra,there is no good corresponding relationship between open sets and closed sets.Recently,making use of the fiuzzy order between closed sets,Professor Zhang Dexue estab-lished a theory of fuzzy sobriety based on irreducible closed sets.Following this idea,on one hand,this paper extends bounded sobriety and k-bounded sobriety to Q-cotopological spaces.On the other hand,we develop the C-continuity of fuzzy posets via the Scott Q-cotopological spaces.Specifically speaking,this dissertation is divided into the following three parts:Chapter One:Preliminaries.In this chapter,some basic concepts and rela-tive conclusions concerned with fuzzy order and category are given.Chapter Two:Bounded sober and k-bounded sober Q-cotopological spaces.Firstly,the concepts of Q-cotopological spaces,bounded sober and k-bounded sober Q-cotopological spaces are given.Secondly,we prove that the category of all bound-ed sober Q-cotopological spaces is a reflective full subcategory of the category of all stratified Q-cotopological spaces.Then we obtain the equivalent condition that.when a commutative and integral quantale Q is linearly ordered,a topological space X is bounded sober if and only if the corresponding Q-cotopological space ωQ(X)is bounded sober,where ωQ is the Lowen functor in fuzzy topology.At last,the prop-erties of k-bounded sober Q-cotopological spaces are investigated and we present the relationships among Hausdorff,T1,sobriety,bounded sobriety and k-bounded sobriety in the setting of Q-cotopological spaces.Chapter Three:C-continuous fuzzy posets.Firstly,we introduce the concept of Scott Q-cotopological spaces and define a fuzzy auxiliary relation by using of the I-closed fuzzy sets.Secondly,the definition of C-prealgebraic fuzzy posets is given and we prove that Scott Q-cotopology is C-prealgebraic.At last,we build an adjoint between the category of irreducible complete fuzzy posets and the category of C-prealgebraic fuzzy complete posets. |
