Some Researches On Idempotent, Involutive And Girard Quantale

In 1986, C. J. Mulvey firstly introduced the concept of a quantale with the purpose of studying the spectrum of the noncommutative C~*—algebra. From then on, many mathematicians and logicists paid close attention to the theory of quantales. In 1992, C. J. Mulvey and J. W. Pelletier put forward a new concept--involutive quantale, basing on the theory of the quantale and C~*—algebra.And in 1993, S. Abramsky and S. Vickers proposed the concept of a quantale module. There are abundant contents in the structure of quantales, and so is the relative structure. The aim of this paper is to study some properties of the relative structure of quantale. At the same time, we also study carefully and deeply the limit and the inverse limit of the category of Girard quantales.The followings are the main contents of this paper:Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and results of the theory of quantales and the category which are used in the whole paper.Chapter Two The properties of the relative structure of quantales. First of all, the relationship between algebraic quantales and spatial quantales is studied, under the circumstance of idempotent quantales. And the sufficient condition that quantales and their subquantales are spatial is obtained. Meenwhile, some examples of subquantales and quantic quotients are given. Next, quantale matrices are considered. We mainly discuss some properties of nilpotent matrices on idem-potent and right-sided quantales. At the same time, we obtain some important conclusion that the sufficient and necessary condition of nilpotent matrices on this quantales and the characteristic theorem of nilpotent index of nilpotent matrices. Finally, in the view of the definition of the conuclei of quantale modules, we find the correspondence between the conuclei of quantale modules and submodules over quantales. Then, the definition and properties of dual bimodule of quantale are given.Chapter Three Involutive quantale and the direct limit of the category of involutive quantales. Firstly, we introduce the definition of the relational involutive...