The Remainder Kernel Mapping And Algebraic Ideals On Quantale Algebra

Quantales were introduced by Mulvey in 1986 with the purpose of studying the spectrum of non-commutative C*-algebra, in order to provide a new mathe-matical model for studying quantum mechanics. Quantales have been applied to non-commutative C*-algebra, the ideal theory of rings, linear logic and theoretic computer science. In order to describe the process semantic in theoretical computer science, the concept of a quantale module was introduced by Abramsky and Vickers in 1993. As a generalization of quantales, quantale module has been paid atten-tion to many experts. Quantale algebras (short for Q-algebras) were introduced by Solovyov as a mixture of the notions of quantales and quantale modules. Moreover, Quantale algebras have been applied to the study of topological spaces and aroused great interests of many scholars and experts. Since then, a great deal of new ideas and applications of quantale algebras has been proposed. This thesis is to further investigate the conuclei on quantale algebras and the algebraic ideal of quantale algebras. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, some basic concepts and relevant conclusions of quantale and quantale algebra which will be used throughout the thesis are given.Chapter Two:The conuclei on quantale algebras. Firstly, some properties of quantale algebra nuclei and quantale algebra quotients are discussed; Secondly, the notion of quantale algebra conuclei is introduced, some properties of quantale algebra conuclei are studied; At last, the extensions of quantale algebra conuclei to unital quantale algebra and Girard quantale algebra are discussed.Chapter Three:The algebraic ideal of quantale algebras. Firstly, the con-cept of ideal extension is introduced, some properties of ideal extension are given. A powerset quantic congrunce is constructed by the ideal extension; Secondly, the re-lationship between algebraic ideal and quantale algebra homomorphism is discussed; At last, the relationship between algebraic ideal and quantale algebra congruence is investigated. Then, we also discuss the congruence relation on the product of quantale algebra.