The Categories Of Continuous Quantale And Double Quantale Module

Since the concept of quantale was introduced by C. J. Mulvey and J. W. Pelletier in 1986, many mathematicians and logicists have paid attention to the theory of quantale. The research of these subjects has related to several reseach 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, and so is the relative of quantale. We study carefully and deeply some properties of continuous quantale, double quantale module and the category of double quantale module. The arrangement of this paper is as follows:Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and results of the theory of quantale and that of category which are used in the whole paper.Chapter Two Some properties of continuous quantale. Firstly, the definitions of semicontinuous quantale and continuous quantale are gived, based on the concepts of semicontinuous quantale and continuous quantale, we discuss the relation of regular quantale, semicontinuous quantale and continuous quantale, obtain that semicontinuous quantale is continuous under some conditions. Secondly, the concepts of the ideal of continous quantales is gived. we discuss some properties of the ideal of continous quantales. At last, we study some category properties of continuous quantale, we prove that the category of continuous quantale is pointed and connected, the each projection of the category of continous quantales is retract, and the category of continuous quantale has products.Chapter Three Some properties of double quantale module. Firstly, the definitions of double quantale module is gived, we discuss some some properties of double quantale module. Secondly, based on the properties of double quantale module, we discuss some relation of submodules and ideals of double quantale module. At last, we present the concepts of nucleus and congruence of the double quantale module, and study some properties of them. we proved that the surjective homomorphisic image of an double quantale module is isomorphic to the image of some double quantale module nucleus. And an one-to-one correspondence between the set of all double quantale module nuclei on an double quantale module and the set of all the congruence relations on it can be obtained. They are isomorphic complete lattices.Chapter four Some properties of the category of double quantale modules. Firstly, we prove that the category of double quantale modules is a pointed and connected, we discuss the equalizer, the coequalizer, the product, the collective pullback in the category of double quantale modules, and obtained the concrete structures of them. we prove that the each projection of the category of double quantale modules is retract, and the category of double quantales modules has kernel and cokernel. Secondly, we give the constructure of the limit of the category of double quantale modules, so the category of double quantale modules is completed. At last, we talk about some properties of the inverse systems of the category of double quantale modules, we construct the inverse limit of the category of double quantale modules, introducing the definition of a mapping between two inverse systems, we can get the limit mapping in the category of double quantale modules.