Z-Quantale Structure And The Projection Object In The Fuzzy Quantale Category

Quantales were introduced by C.J.Mulvey in 1986 with the purpose of studying the spectrum of C*-algebra, in order to provide a lattice-theoretic setting for study-ing non-commutative C*-algebra, as well as constructive foundations for quantum mechanics. In 1990, D.Yetter found the interaction between linear logic, the logical foundation of theoretical computer science, which was present by Girard, and the theory of quantales. Since then, the theory of quantales has aroused great interests of many scholars and experts, and a great deal of new ideas and applications of quantales has been proposed. This thesis is to further investigate Z-quantale and the projective objects in the category of fuzzy quantales. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, some basic concepts and relevant conclusions used in this paper are given.Chapter Two:The structure of Z-quantales. Firstly, we introduce the notion of meet Z-quantales. We prove that the family of all Z-closed sets of a meet Z-quantale, when ordered by inclusion, is a Frame. Secondly, the relations between nuclei, quotients and congruences on meet Z-quantales are investigated. At last, we show that the category of quantales is a reflective subcategory of the category of Z-quantales.Chapter Three:The projective objects in the category of fuzzy quantales. Firstly, the concept of fuzzy weakly (?)-stable completely distributive lattices is in-troduced. It is proved that the family of all down sets of a fuzzy ordered semigroup with a appropriate operation (?) is a fuzzy weakly (?)-stable completely distributive lattice. A necessary and sufficient condition for a fuzzy completely distributive lat-tice to be a fuzzy weakly (?)-stable completely distributive lattice is given. Moreover, projective objects in the category of fuzzy quantales are studied. It is also proved that the E-projective objects in the category of fuzzy quantales are exactly the fuzzy weakly (?)-stable completely distributive lattices.Chapter Four:The E*-projective objects in the category of fuzzy quantales. Firstly, the concept of fuzzy prequantales is introduced. We show that the category of quantales is a reflective subcategory of the category of prequantales. Secondly, E*-projective objects in the category of fuzzy quantales are studied. At last, it is proved that Q is a E*-projective object in the category of fuzzy quantales if and only if Q has a coalgebra structure.