The Studying Of Quantale Systems And Bicomplete Fuzzy Posets

Abstract Steven Vickers connects the topological methods with logic the-ory and builds up the topological system, also he applies the theories to study the theories of computer. Domain theories and quantale theories established in the early century70’s and80’s which can be taken as the mathematical foundations of computer science have become the common concern of mathematics and computer science researchers. Since2000, the fuzzy set theories have been applied to the quantitative domain theories, forming fuzzy domain theories. Frist, this paper will connects quantale theories with quantum space theories to the theories of topology system, defining the concept of quantale system, and the system is studied further, then combining with the method of fuzzy sets, we give the definition of fuzzy quan-tum space, proving that the dual equivalence between sober fuzzy quantum space and spatial hibateral L-Quantale. In the end, it is defined the concept of the bicom-plete fuzzy poset, also we obtain some category characters of the bicomplete fuzzy poset. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we give some basic concepts and results of the quantale theory, space quantum space theory and the fuzzy set theory which will be used throughout the thesis.Chapter Two: Quantale theory. By introducting the concept of quantale sys-tems, we define the continuous mapping and homeomorphism of quantale systems and prove that the inverse of homeomorphism mappings are also homeomorphism mappings. Then we research the spatialization of quantale systems, and establish the adjoint between the category of quantale systems and the category of quantum spaces. It is discussed that the Q-Localification of quantale systems and proved the Q-Locale about quantales are quantale systems and the Q-Localification of quantale systems are Q-Locales. In the end, we establish the adjoint between the category of quantale systems and the category of Q-Locales.Chapter Three: Fuzzy quantum space. We give the definition of fuzzy quantum space, and construct an adjunction between the category of stratified fuzzy quantum spaces and that of the opposite category of hibateral fuzzy quantales. Next we show the dual equivalence between the category of sober fuzzy quantum space and that of spatial hibateral L-Quantale Chapter Four: Bicomplete fuzzy poset and its equivalent category. Firstly, the concepts of bicomplete fuzzy poset is introduced, and then we study the simple prop-erties of approximable bimodule function. It is proved that algebra fuzzy Domain category and bicomplete fuzzy poset category are equivalent.