The Generalization Of The Auxiliary Order And The Stratified Theory

Based on the L-fuzzy set and its cuts,the auxiliary order,the closure operator and the classification of the L-fuzzy sets are discussed in this thesis.The order in classical Domain theory is just binary,so it is difficult to embody the difference among quantities.An ideal framework for the quantitative Domain is proposed by introducing L-fuzzy preordered sets in[2].A new viewpoint is developed by considering the quantitative Domain as L-fuzzy preordered sets.Based on L-fuzzy preordered sets,the L-fuzzy Domain theory is set up in[1].The L-fuzzy auxiliary order in this thesis generalizes the auxiliary order in classical Domain theory and supplements the L-fuzzy Domain theory to make it especially perfect about the discussion of order.The concept of closure operator appears in many subjects,such as algebra,topology, logic and so on.In Chapter three,we investigate the closure operator in the view of the L-fuzzy mapping and study the related contents of closure operator in those subjects through the new viewpoints.The cut,which is the crisp set,is the bridge between the classical mathematics and the fuzzy mathematics.Using cuts as a criterion of classification of L^X to show the essential difference of L-fuzzy sets provides the theoretical basis for the study of algebraic structures, such as fuzzy groups and fuzzy circles.The main results are:In Chapter one,concepts and results about lattice and L-fuzzy set and its cuts are discussed.In Chapter two,based on the L-fuzzy poset introduced by[2],the L-fuzzy auxiliary order is defined and an isomorphism about the L-fuzzy auxiliary order is given.Under the framework of[1],the definitions of L-fuzzy auxiliary order and the interpolation property of L-fuzzy auxiliary order are given.Then the relations among L-fuzzy auxiliary order,L-fuzzy approximating auxiliary order,匚_a and(?)_a are discussed. In Chapter three,based on the L-fuzzy mapping introduced by[1]and the image of L-fuzzy set introduced by[2],the L-fuzzy closure operator induced by the L-fuzzy mapping is given and its equivalent characterizations are obtained.When L is a finite distributive lattice, the algebraic L-fuzzy closure operator and the equivalent characterizations of the L-fuzzy closure system induced by the algebraic L-fuzzy closure operator are discussed.In Chapter four,according to the theory of using cuts as an essential criterion of classification of L^X,we define a new equivalent relation on fuzzy power by A_a,different from the cut in the paper[17].Then we obtain a necessary and sufficient condition under which two fuzzy sets are equivalent and give an example to explain the equivalence of them.In the end, we define the functions from A(X) to {0,1} by the minimal family and these functions are isotone.There is an isomorphism from these functions onto A_(L).