The Categories Of Involutive Quantale And 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. Based on the theory of quantale and the theory of C~*-algebra, in 1992, C. J. Mulvey and J. W. Pelletier advanced a new concept— involutive quantale. As an assosiciated constructure of quantale, quantale module has been paid attention to by many experts. We study carefully and deeply some properties of involutive quantale and quantale module, the category of involutive quantales and the category of quantale modules. 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 involutive quantale. Firstly, combining the special elements in quantale (such as symmetric element, left (right) pre-symmetric element, left-sided (righr-sided, two-sided) element, localic element, regular element etc.), the relation of << and (?), compact and supercompact elements in complete lattice, and the involutive operation * in involutive quantale, we discussed the relation of them. Secondly, involutive quantic nuclei are studied and it is proved that the surjective homomorphisic image of an involutive quantale is isomorphic to the image of some involutive quantic nucleus. And an one-to-one correspondence between the set of all involutive quantic nuclei on an involutive quantale and the set of all the congruence relations on it can be obtained. They are isomorphic complete lattices.Chapter Three Some properties of the category of involutive quantales. About the category of involutive quantales, we go on to study the properties of it further. Firstly, the free object generated by a set is given. Making use of the free object, we discuss the adjoint relation of the category of involutive quantales and that of sets. Also, we obtain the characterization of monomorphism, monosource and regular epimorphism in the category of involutive quantales. Secondly, based on the concept of involutive conucleus, the characterization of the subobject of an involutivequantale is got. Thereafter we prove that the category of involutive quantales is well-powered. Thirdly, we study the coequalizer, the intersection, the collective pullback in the category of involutive quantales and obtained the concrete structures of them. Fourthly, we talk about the projective objects in the category of involutive quantales, study their properties and obtain the relation of the category of involutive quantales and that of a few connected categories.Chapter Four The category of quantale modules. In the final chapter, based on the basic properties of quantale module, we discuss the relation of quantale ideals and submodules. Also, we prove the quantale module nucleus has similiar properties to quantic nucleus and involutive quantic nucleus. About the the category of quantale modules, we study mainly some special morphisms and objects in it. For example, the special morphisms of constant morphism, coconstant morphism, zero morphism etc. and the special objects of initial object and terminal object etc. are characterized. According to these, we know that the category of quantale modules is a pointed and connected category. Then, the structure of the equalizer is given. And, we prove that the category of quantale module has products. After that, we construct the limit and the inverse limit of the category of quantaie modules. At last, introducing the definition of a mapping between two inverse systems, we can get the limit mapping in the category of quantale modules.