A Study Of Prequantale Property And The Ideals In Quantale

Ever since 1986 when C.J.Mulvey, the mathematician, put forward the concept "Quantale", quantale theory has been a focus of study among mathematicians and logicians. In a few decades a large number of new ideals and applications of quantale theory have been continually disclosed, but its application is confined by the combinability of algebraic calculation in the structure of quantale. Thus the structure of prequantale comes into being. The research into the inner structure of prequantale is the basis of this theory. This article presents some basic concepts in prequantale and discusses its properties. It not only lays the foundation for the understanding of prequantale structure but enriches and develops the contents of quantale theory. The arrangement of this article is as follows:Chapter One Preliminary knowledge In this chapter the author gives the main concepts and results of quantale theory, category theory used in the whole article.Chapter Two Prequantale and its categorical properties This part presents some basic concepts in prequantale, such as subprequantale, quotient prequantale, nucleus, conucleus and some special elements in prequantale, and discusses the relevant properties of prequantale, and gives the equivalent description of subprequantale and quotient prequantale and clarifies the relation between peculiar prequantale quotient and peculiar elements. In the 4th section of this chapter, the author puts forward the concepts of prequantale morphism and prequantale category, researches into the peculiar morphism and peculiar object in the category of prequantale, then proves the limitation of prequantale category. Meanwhile a necessary and sufficient condition which the left adjoint of an adjoint pair is a morphism of prequantale is obtained.Chapter Three Prequantale congruence and its application This chapter the author puts forward the concept of prequantale congruence, discusses the relevant properties of prequantale congruence, finds out the concrete structure of prequantale quotient corresponding to prequantale congruence, and obtains a conclusion that all prequantale quotients, all nuclei and all prequantale congruences of a prequantale are isomorphism. This chapter presents the relations between prequantale congruence and prequantale morphism and proves the conclusion that it is ratherdifficult to prove only by means of category theory and knowledge of prequantale that prequantale category has coequalizers.Chapter Four Ideals of the quantale In this chapter the author gives the concept of (left, right) ideal in quantale. By doing deep research into the structure of this ideal, we obtain the concrete structure of the (left right) ideal generated by a given set, also study the relation between(left right)ideal of a quantale and the ideal of the lattice, introduces a natural quantale structure in the set of all (left, right) ideals of a quantale, and validates its serial properties.