| In1990, D.Yetter found the interrelation between linear logic, the logical foundation of theoretical computer science, which was presented by Girard, and the theory of quantales.Since then, the theory of quantales has aroused great interests of many scholars and experts, especially attracts a lot of academicians from the areas of mathematics and theoretical computer science to study. So that the research and application of the theory of quantales have developed a lot in recent twenty years. In1991, L. Roman and B. Rumbos constructed a new category-quantic lattice category, which contains quantales as its objects. And they pointed out that for every orthomodular lattice L, if a&b(?)(a∧V b⊥)∨b for every a, b in L, then L is a quantic lattice. On one hand, the filters of quantales are studied by using the fuzzy set theory in this thesis, which enrich the theory of quantales. on the other hand, the fuzzification of quantic lattices theory is studied by using the fuzzy order, which not only can help study structures of quantic lattices but also can generalize, to some extent, the theory of quantales. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we recapitulate the basic concepts and results of the theories of quantales, frames, quantic lattices, L-ordered sets and categories which will be used throughout this thesis.Chapter Two:Fuzzy filters in quantales. In this chapter, firstly, the concepts of L-fuzzy filters and L-fuzzy prime filters in quantales are given and their some related properties are studies. Secondly, when L is a closed-set lattice, the equiv-alent charactizations of L-fuzzy filters and L-fuzzy prime filters are given. It is proved that the filter generated mapping of LQ is a quantic nucleus and the quan-tale FilL(Q) consisting of all L-fuzzy filters is the idempotent quantic quotient of LQ when L is a spatial frame and Q is an idempotent left semi-communicative quan-tale. Finally, L-fuzzy filiter topology in quantales is defined and it is obtained that a quantale homomorphism is continuous with regard to the corresponding L-fuzzy filiter topologies.Chapter Three:L-quantic lattices. In this chapter, firstly, based on the defi-nition of L-quantales, the definition of L-quantic lattices is given, and their some related examples are listed. It is proved that the product of L-quantic lattices is also a L-quantic lattice, and the projection ai)is a L-quantic lattice homomorphism. Secondly, the nucleus and conucleus on L-quantic lattices are studied and some results relating L-sub quantic lattices are obtained. Finally, we study some properties of L-quantic lattices, and consider the product, limit and inverse limit of this category, and give their conformations. |
