Upper Rough Ideals Of Quantales And Fuzzy Ideals Of Quantic Lattices

Quantale was first introduced by C. J.Mulvey in1986to study non-commutative C*-algebra and provide a new mathematical model for quantum me-chanics. Since quantale theory provides a powerful tool in studying the ideal theory of ring, noncommutative C*-algebra, linear logic and theoretical computer science, it has been studied by lots of academicians from the areas of mathematics and theoret-ical computer science, and in recent twenty years, the research and applications of quantales have developed a lot. Since the ideals of quantales are important tools to research quantale theory, the rough set theory is used to study ideals of quantales in this thesis. Orthomodular lattices and quantales have the same properties, and they are important algebraic systems in studying quantum mechanics. However, they are different algebraic systems. 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∨b⊥)∧b for every a, b in L. then L is a quantic lattice. The research on ideals of quantic lattices not only can help study structures of quantic lattices but also can enrich the content of the quantale theory and orthomodular lattice theory. This thesis mainly includes four parts:Chapter One:Preliminaries knowledge. In this chapter, we give some basic concepts and results of the quantale theory and the fuzzy set theory which will be used throughout the thesis.Chapter Two:Upper rough ideals of Quantales. Firstly, the concepts of upper approximation and lower approximation of quantales are given by using the rough set theory, and some properties of upper approximation and lower approximation are studied. Secondly, we give the concepts of upper rough subquantale. upper rough ideal and upper rough quasi-ideal of a quantale. study the properties of these mathe-matical structures and discuss the relatonships between them, we prove that:ideals must be upper rough ideals:an upper rough ideal whose upper approximation equals lower approximation must be an ideal; upper rough (left, right)ideals must be upper rough quasi-ideals; an upper rough quasi-ideal which satisfies left(right)absorption law must be an upper rough left(right)ideal.Chapter Three: Fuzzy ideals of quantic lattices. Firstly, some basic properties of quantic lattices are given. Secondly, the concepts of ideal and ideal coclosure operator on a quantic lattice are given. Some properties of them are studied, and the relationship between quantic lattice homomorphisms and ideals is discussed. It is proved that in a quantic lattice, a subset is an ideal if and only if it is the image of an ideal coclosure operator. Finally, the concepts of fuzzy ideal and fuzzy subqunatic lattice are given, and some properties of fuzzy ideals of quantic lattices are studied.Chapter Four:Prime fuzzy ideals of quantic lattices. Firstly, the concept of prime fuzzy ideal on a quantic lattice is given, and some properties of prime fuzzy ideals of quantic lattices are studied. There are characterizations of prime fuzzy ideals of quantic lattices. Secondly, we introduce the concept of weakly prime fuzzy ideal on a quantic lattice, and the relationships between prime fuzzy ideals and weakly prime fuzzy ideals are discussed based on the concepts. We characterize weakly prime fuzzy ideals of quantic lattices by their level ideals and fuzzy points. Finally, the concept of fuzzy congruence on a quantic lattice is given, and some properties of fuzzy congruence on a quantic lattice are studied.