Researches On Several Classes Of Quantale Structures And Quantale Matrix

The concept of quantale was introduced by C. J. Mulvey in 1986 with the purpose of studying the spectrum of C*-algebra, as well as constructive foundations for quantum mechanics. The research of these subjects has related to several research area such as non-commutative C*-algebra, the ideal theory of rings, linear logic and theoretic computer science. There are abundant contents in the structure of quantales, because quantale can be regard as the generalization of frame. This paper analyses and studied the relations among nucleus, quantic quotient and quantale congruence, and the structures of simple quantale and trivial quantale, non-trivial quantale, the ideals of quantale. The main content is as follows.Chapter One Preliminary knowledge. In this chapter the author gives the main concepts and results of quantale theory used in the whole article.Chapter Two Simple quantale and quantic quotient. In this chapter the author discusses the relations among nucleus, quantic quotient and quantale congruence, obtains the equivalent condition that a kind of quantale is a simple quantale, gives the quotient characterization of simple quantale and a new definition of simple quantale.Chapter Three The ideals of quantale. In first part of this chapter, the author studies subquantale and its properties. Firstly, we give two ways to construct subquantales, study the properties of subquantales , and in the meantime we discuss the prime element of quantale by subquantale. The second part, we give the definition of the trivial quantale by the map , and give its concrete characterization; We study the subquantale of finite quantale, and prove the existence of non-trivial subquantale; In three part we study the ideal of quantale, and give the concrete construction of the ideal generated by an arbitary subset. Furthermore we study the relation between the ideal and the map, and obtain the equivalent condition that a map is an ideal conucleus.Chapter Four Quantale Matrix. In this chapter, we give the concepts of quantale matrix and the deterninant of quantale matrix , and study the properties of quantale matrix and the deterninant of quantale matrix. In the meantime, we also give the defintions of the inverse and the generalized inverse of quantale matrix,...