Involutive Quantale And Its Categorical Properties

Since the concept of quantale was introduced by C.J.Mulvey in 1986, many mathematicians and logicists pay close attention to the theory of quantales. Basing on the theory of quantales and the theory of C* -algebras, C.J.Mulvey and J.W.Pelletier advanced a new concept-involutive quantale, in 1992. The aim of this paper is to study some basic properties of involutive quantales. According to the researches concerning the congruence relation and the homomorphism of involutive quantales, we can discuss the lift on power involutive quantales further. At last, we also study carefully and deeply the limit and the inverse limit of the category of involutive quantales. 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 quantales and the theory of category which are used in the whole paper.Chapter Two Involutive Quantales. In this part, some basic concepts of involutive quantales are defined, such as involutive quantales, homomorphisms, sub-involutive quantales, ideals, nucleis and some special elements of an involutive quantale. Meanwhile, we would like to give some examples of involutive quantales and discuss a few properties about involtions. Then, we study some properties about sub-involutive quantales and ideals of involutive quantales, and construct some types of them.Chaper Three The Congruence Relation of an Involutive Quantale. Firstly, we give the definition of the congruence relation of an involutive quantale and argue about the relationship between the congruence relation and the homomorphism. Secondly, we give an equivalent item that an equivalent relation on an involutive quantale is a congruence relation. Then, we also discuss the congruence relation on the product of involutive quantales and a kind of congruence relation which is induced by an ideal of an involutive quantale. And then, in view of the definition of a power involutive quantale, we find that the homomorphism and the congruence...