Quantale Modules And The Categories Of Involutive Left Q-modules

The concept of quantale module was proposed by S. Abramsky and S. Vickers with the purpose of providing a new algebraic framework for operational, denotational and axiomatic semantics in1993. Because quantale modules can be seen as algebras of finitely observable properties, and there are abundant contents in the structure of quantale modules. That is why many mathematicians and logicists pay close attention to the theory of quantale modules and in twenty years a great deal of news ideals and applications of quantale modules have been proposed. Nuclei and conuclei, ideals and congruences are important tools to research the theory of quantale modules. The first one of this thesis is to investigate their properies, and then discuss their relationships, respectively. which enrich the theory of quantale modules. The second part is to study categorical properties of involutive left Q-modules for the categorical point of view. The strucure of this thesis is organized as follows:Chapter One:Preliminaries knowledge. In this chapter, we give some prelimi-naries that related with quantale theory and category theory which we will be used throughout this paper.Chapter Two:Nuclei and conuclei on quantale modules. Firstly, some properties of conuclei on left Q-module are investigated and the equality characterization for a map to be a conucleus is given. Secondly, the relationships among Dual bimodule, Girard bimodule and involutive left Q-module are discussed, and at the same time, some properties and equivalent characterizations of them are given respectively. It is proved that every bimodule can be embedded into a girard bimodule, and every quantale module can be embedded into a involutive left Q-module if the quantale is involutive. It is also proved that nuclei and conuclei on Girard bimodule and involutive left Q-module are in one-to-one correspondence.Chapter Three:Ideals and congruences on quantale modules. Definitions for heterogeneous ideals and congruences on quantale module M are given, and the respective complete lattices Idl(M) and Cong(M) are presented. We construct the smallest and the greast ideals having the same complete lattice part. The same is establised for congruences. The notions of kernel of a congruence and the congruence induced by ideal are introduced to describe a Galois connection between Idl{M) and Cong(M).Chapter Four:The category of involutive left Q-modules. In this chapter, the categorical properties of involutive left Q-modules is dealt with. First of all, the product and equalizer of this category are considered, and their conformations are given, and it is also proved that the category of involutive left Q-modules does not have null object. Secondly, the structure of limit in the category of involutive left Q-modules is given, and at the same time, the completeness and pullback of the category of involutive left Q-modules are obtained. Then, the concept of the inverse limit of the category of involutive left Q-modules, and the definition of a mapping between two inverse systems are given, so we can get the limit mapping of the category of involutive left Q-modules.