Quantale's Left Half Convertible Quotient And Fuzzy Z-Quantale Structure

As a non-commutative generalization of locales, quant ales were introduced by C.J. Mulvey in 1986 with the purpose of studying the spectrum of C*-algebra, in or-der to provide a lattice-theoretic setting for studying non-commutative C*-algebra, as well as constructive foundations for quantum mechanics. Following Mulvey, vari-ous types and aspects of quantales have been considered by many researchers. Fur-thermore, quantale theory has been applied to many research fields, such as non-commutative topology, logic and theoretical computer science because of its rich order structure and algebra structure. The first part of this thesis is to further investigate quantale theory, introducing the notion of left semi-commutative nuclei and presenting a characterization of the largest left semi-commutative quotient of a quantale; The second part is to study the fuzzification of Z-quantale structure's.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 left semi-commutative quotient of a quantale. Firstly, we introduce the notion of left semi-commutative nuclei and present a characterization of the largest left semi-commutative quotient of a quantale by means of left pre-symmetric elements. Secondly, we show that the category of left semi-commutative quantales is a reflective subcategory of the category of quantales. At last, by dis-cussing the relationships between localic nuclei and left semi-commutative nuclei, we also prove that the largest localic quotient of Q is intersections among the largest left semi-commutative quotient of Q, the largest left-sided quotient of Q and the largest idempotent quotient of Q.Chapter Three:Fuzzy Z-quantales and their categorical properties. Firstly, the concept of a fuzzy Z-quantale is introduced. It is proved that the category of fuzzy Z-quantales is a reflective subcategory of the category of fuzzy ordered semigroups. Moreover, projective objects in the category of fuzzy Z-quantales are studied. It is also proved that a fuzzy Z-quantale A is E-projective if and only if it is.^-continuous, where L is complete residuated lattice.