Publisher's Synopsis
In this text quantales are studied from the perspective of a category theorist. Also studied are frames. Frames are particular examples of quantales and many of the constructions in the theory of quantales are motivated by attempts to generalize ideas from frame theory to the more general setting provided by quantales.;The book is written for an audience interested in any of the varied examples and as such does not require knowledge of advanced category theory.;After an initial chapter of preliminaries about sup-lattices and frames to give the reader the requisite background, the basic properties of quantales and their morphisms, as well as examples are described. The first three chapters constitute the basic knowledge of quantales needed for consideration of applications. Chapter 4 considers quantales and the ideal theory of rings including discussion of algeraic and coherent quantales, algebraic de Morgan's laws, irredundant prime decompositions, componental nuclei and sheaves of ideals. In chapter 5, the class of idempotent right-sided quantales is studied with any eye towards applications to the theory of closed ideals of C*-algebras and a study of Rosicky's notion of quantum frame. The final chapter considers quantales in the context of Girard's linear logic with a discussion of Girard quantales and their properties and the phase semantics of linear logic.
