Filter Convergence On Partially Ordered Sets And Order Convergence On Fuzzy Partially Ordered Sets

Domain theory,emerged in the 1970s,with double backgrounds of theoretical computer science and pure mathematic research,in order to provide a mathemat-ic foundation for the semantics of programming language.As its abundant order structures and topology structures,domain theory became a common focus for com-puter science researchers and mathematic researchers and filter convergence and net convergence on continuous poset are important research orientations in domain the-ory.Filter convergence and net convergence are the key which connect algebraic properties with topological properties in continuous posets.In 1965,the concept of fuzzy set was proposed by L.A.Zadeh,which marks the birth of fuzzy math-.ematics.Since 2000,the fuzzy domain theory has been formed with fuzzy poset theory applied to the research of qualitative domain theory.As a generalization of classical posets,the problems of convergence in fuzzy posets are concerned by some researchers.Two kinds of filter convergence spaces on posets were discussed and fuzzy order convergence on fuzzy posets was studies in this paper.The main content are arranged as follows:Chapter One:Preliminaries.In this chapter,some basic concepts and rele-vant conclusions about lattice theory,category and fuzzy domain are given.Chapter Two:Filter convergence space on posets.Firstly,we study the filter convergence structure ?2 on posets and introduce the concept of strong meet S2 continuous poset.We prove that the convergence space(P,?2)satisfies Merge-niceness property if and only if P is strong meet S2 continuous when P is a meet-semilattice.Secondly,we introduce a filter convergence structure ?c2,and prove the convergence spaces(P,?2)and(P,?C2)are equivalence when P is a bound complete poset.At last,we prove that the category CONV2 is not Cartesian closed and give a necesarry condition of closed full subcategories of CONV2 being Cartesian closed.Chapter Three:Order convergence on fuzzy posets.Firstly,the concept of S*-double continuous fuzzy poset is introduced and the sufficient and necessary condition of S*-double continuous fuzzy poset to be double continuous fuzzy poset is given.Secondly,a new definition message of fuzzy order topology is established and its related properties are researched.Finally,we proved that the fuzzy order convergence of L-filter on fuzzy poset P is topological if and only if P is S*-double continuous.