Overview
- Provides a categorical approach to quantales and applications
- Develops the theory of modules on unital quantales
- Includes exercises and bibliographical notes
Part of the book series: Developments in Mathematics (DEVM, volume 54)
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Table of contents (3 chapters)
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Front Matter
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Back Matter
Keywords
- MSC (2010) 06F07, 18C20, 18D15, 46L05
- quantales and monoidal categories
- semi-unital quantales
- balanced and bisymmetric quantales
- tensor product of quantales
- spectrum of non-commutative C*-algebras
- Frobenius quantales and complete MV-algebras
- modules on unital quantales
- quantales and enriched category theory
- automata and free unital quantales
About this book
First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer science. This book recasts the subject within the powerful framework of categorical algebra, showcasing its versatility through applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical remarks at the end of each chapter, this self-contained book provides readers with a valuable source of references and hints for future research.
This book will appeal to researchers across mathematics and computer science with an interest in category theory, lattice theory, and many-valued logic.
Reviews
Authors and Affiliations
About the authors
Javier Gutiérrez García has been interested in many-valued structures since the late 1990s. Over recent years these investigations have led him to a deeper understanding of the theory of quantales as the basis for a coherent development of many-valued structures (cf. Fuzzy Sets and Syst. 313 43-60 (2017)).
Since the late 1980s the research work of Ulrich Höhle has been motivated by a non-idempotent extension of topos theory. A result of these activities is a non-commutative and non-idempotent theory of quantale sets which can be expressed as enriched category theory in a specific quantaloid (cf. Fuzzy Sets and Syst. 166, 1-43 (2011), Theory Appl. Categ. 25(13), 342-367 (2011)). These investigations have also led to a deeper understanding of the theory of quantales. Based on a new concept of prime elements, a characterization of semi-unital and spatial quantales by six-valued topological spaces has been achieved (cf. Order 32(3), 329-346 (2015)). This result has non-trivial applications to the general theory of C*-algebras.
Since the beginning of the 1990s the research work of Jari Kortelainen has been directed towards preorders and topologies as mathematical bases of imprecise information representation. This approach leads to the use of category theory as a suitable metalanguage. Especially, in cooperation with Patrik Eklund, his studies focus on categorical term constructions over specific categories (cf. Fuzzy Sets and Syst. 256, 211-235 (2014)) leading to term constructions over cocomplete monoidal biclosed categories (cf. Fuzzy Sets and Syst. 298, 128-157 (2016)).
Bibliographic Information
Book Title: Semigroups in Complete Lattices
Book Subtitle: Quantales, Modules and Related Topics
Authors: Patrik Eklund, Javier Gutiérrez García, Ulrich Höhle, Jari Kortelainen
Series Title: Developments in Mathematics
DOI: https://doi.org/10.1007/978-3-319-78948-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG, part of Springer Nature 2018
Hardcover ISBN: 978-3-319-78947-7Published: 19 June 2018
Softcover ISBN: 978-3-030-07687-0Published: 26 January 2019
eBook ISBN: 978-3-319-78948-4Published: 09 June 2018
Series ISSN: 1389-2177
Series E-ISSN: 2197-795X
Edition Number: 1
Number of Pages: XXI, 326
Topics: Order, Lattices, Ordered Algebraic Structures, Category Theory, Homological Algebra, Mathematical Logic and Formal Languages