| Quantales were firstly introduced by C.J.Mulvey in 1984 where he proposed using quantales to study the foundations of quantum logic and non-commutative C*algebra.Its purpose is to provide a new format description for non-commutative C*algebra and to provide a new mathematical model for quantum mechanics.Quantales can be seen not only as a generalization of locales,but also as a generalization of quantum logic.Because quantales have a rich order structure and algebraic structure,and are closely related to linear logic and theoretic computer science,they have received close attention from many scholars in the fields of mathematics,logic,and theoretic computer science.The first part of this thesis is establishing a more general structure compared with quantales,which called partial quantales.The second part is to study Q-ordered semigroups and Q-quantales from the perspective of Q-semiclosure semigroups.The structure of this thesis is organized in the following:Chapter One:Preliminaries.In this chapter,we give some basic conclusions about Q-ordered sets and category theory.Chapter Two:The category of partial quantales.We firstly introduce the concept of partial quantales and basic properties.Then we use coverages and sites to define the partial quantale congruence and give a method for constructing partial quantales.Finally,we discuss the relationship between congruence and coverage which satisfies Johnstone’s meet-stable.As well as we prove that the category of quantales is the reflective subcategory of the category of partial quantales.Chapter Three:The category of Q-semiclosure semigroups.Firstly we give the concept of Q-semiclosure semigroups and the equivalent condition of Qsemiclosure semigroups.Secondly,we give three different classes of Q-semiclosure semigroups,i.e.To Q-semiclosure semigroups,conditionally complete Q-semiclosure semigroups and complete Q-semiclosure semigroups,as well as study the relationships among them.Finally we study Q-ordered semigroups and Q-quantales from the perspective of Q-semiclosure semigroups.And we prove an important result of Q-semiclosure semigroups,that is,the category of complete Q-semiclosure semigroups is isomorphic to the category of Q-quantales. |
