Semigroups in Complete Lattices PDF
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Developments in Mathematics Patrik Eklund Javier Gutiérrez García Ulrich Höhle Jari Kortelainen Semigroups in Complete Lattices Quantales, Modules and Related Topics Developments in Mathematics Volume 54 Series editors Krishnaswami Alladi, Gainesville, USA Pham Huu Tiep, Piscataway, USA Loring W. Tu, Medford, USA More information about this series at http://www.springer.com/series/5834 é í Patrik Eklund Javier Guti rrez Garc a (cid:129) ö Ulrich H hle Jari Kortelainen (cid:129) Semigroups in Complete Lattices Quantales, Modules and Related Topics 123 Patrik Eklund Ulrich Höhle Department ofComputing Science Schoolof Mathematics UmeåUniversity andNaturalSciences Umeå University of Wuppertal Sweden Wuppertal Germany Javier Gutiérrez García Department ofMathematics Jari Kortelainen UniversityoftheBasqueCountryUPV/EHU Department ofInformation Technology Bilbao South-Eastern Finland University Spain ofApplied Sciences Mikkeli Finland ISSN 1389-2177 ISSN 2197-795X (electronic) Developments inMathematics ISBN978-3-319-78947-7 ISBN978-3-319-78948-4 (eBook) https://doi.org/10.1007/978-3-319-78948-4 LibraryofCongressControlNumber:2018938638 MathematicsSubjectClassification(2010): 06F07,18C20,18D15,46L05 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG partofSpringerNature Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To Aija, Ana, Felix, Heidi-Tuulia and Kesia Foreword This book is about quantales, lattice-ordered semigroups structured in such a way that they generalize the locales of point-free topology and various multiplicative latticesofidealsfromthetheoriesofringsandC(cid:1)-algebras.Theformalnotionofa quantale—a complete lattice equipped with a semigroup multiplication which distributes over arbitrary joins—arose in the 1980s, and the subject has flourished and diversified in content and range of mathematical connections since that beginning, with developments appearing in a diversity of journals. The most notable and distinctive feature of this monograph is a shift from regarding quantales as the principal objects of study to a categorical emphasis that encompasses the category Sup (morphisms preserve all suprema) of complete lattices, the category of quantales with Sup-semigroup morphisms, other related categories naturally connected with quantales, and a variety of categorical con- structions that enhance quantale theory. Categorical properties of Sup such as existenceoftensorproductsandthefactitissymmetricandmonoidalclosedledthe authors to a whole new categorical perspective and approach of applying enriched category to the study of quantales. The authors have pioneered work along these lines in recent years, and now for the first time seek to bring it together in one organized, comprehensive, and mostly self-contained source. Anattractiveadditionalfeatureofthetextistheefforttomakesomeconnections andapplicationstootherareas.Oneistheintroductionandstudyofthespectrumof a C(cid:1)-algebra as a quotient of the tensor product of the quantales of closed left and right ideals and the use of quantale modules of this quantale to study algebra representations. Another rather vast direction of study arises by replacing standard objectssuchasf0;1gastherangeforcharacteristicfunctionsforsubsetsofasetor the positive reals for metric spaces with quantales, so that one gets many-valued versionsofsets,ormetricspaces,orotherstandardobjects.Thus,theauthorsseek to show how the foundations of a theory of many-valuedness can be based on quantale theory. The authors are to be commended for collecting and putting together in an attractive and accessible manner the necessary categorical back- ground, the systematic, organized presentation from their perspective of quantale vii viii Foreword theory,andfortheinterestingapplicationstoC(cid:1)-algebratheoryandtomany-valued structures. This book can serve a wide audience from being an introduction to quantale theory to providing a valuable source of reference. Baton Rouge, Louisiana Jimmie Lawson September 2017 Preface Since the mid-1980s, quantales have found an increasing interest in various areas of mathematics. This development has its origin in ideas coming from non-commutativegeometryandisdrivenbythedesiretocreateanon-commutative and non-idempotent set theory known as the theory of quantale sets. Since that time,anextensiveliteraturehasgrownandisscatteredoveradiversityofjournals. Furthermore,andasalgebraatthesametimebecameconnectedwithdevelopments using category theory, dedicated workshops and discussion fora emerged to focus onenrichedcategorytheoryinfurthersupportofstudyingquantales.The33rdLinz Seminar2012devotedtoenrichedcategorytheory,andorganizedbyU.Höhleand E.P. Klement, was typical in that respect. In that particular workshop, the general feeling emerged that there exists a necessity to have a new comprehensive and coherent representation of the field. This insight became the starting point of the cooperation between the authors leading to the present book. The foundations of many-valuedness, as enabling non-commutativity for its relatedoperators,areanotherthematicandapplicativeaspectoverlayingtheformal algebraic treatment in this book. Indeed, the book aims to provide a treatment of semigroupsincompletelatticeswithaspecialfocusonquantalesandmodulesand offers a new way to approachmany-valuedness in mathematics—in particularas it is related to algebraic, logical, and topological methodology. We give an introduction to the theory of quantales with applications to C(cid:1)-algebras and many-valued order theory from the perspective of the category of complete lattices and join-preserving maps, which also includes a brief sketch on automatainthiscontext.Thebookaimstobeself-contained.However,somebasic knowledge in category theory is desirable, and similarly knowledge in functional analysis is helpful with respect to applications. This book does not exclude the possibility of being used as a textbook. Therefore, every section is accompanied by a subsection of exercises ranging in difficultyfromatrivialtoaratheradvancedlevel.Themoreadvancedexercisesare suppliedwithvarioushintswhichcanbeconsideredasanextensionoftheprevious results.Wehopethatthebookformsausefulbasisforfurtherresearchinenriched ix x Preface category theory, non-classical logic, many-valued and non-commutative topology, and computer science. During the preparation of the book, we received support from various people. ExplicitlywewouldliketomentionthepassionatecommitmentofT.Kubiak,who collected a large variety of papers on quantales, was involved in the discussion on how to organize such a book, and made most valuable suggestions during its preparation.WealsoacknowledgethestimulatingadvicewereceivedfromM.Barr and J. Lawson at advanced stages of its development. In addition, and in fact over the past few years, from idea generation to finalizing the book, several colleagues and collaborators have been supportive and provided encouragement on various occasions. This book would not have seen daylight without the support of our working environments and people in our vicinity. Our working environments were diverse but always supportive and flexible, enabling all of us to use required time for the productionofthebook.SomedirectedfinancialsupportwasenabledbytheSpanish Government provided by MTM2015-63608-P (MINECO/FEDER), and further indirect financial support was available through various parallel projects. We also express our gratitude to Springer-Verlag, especially to Dr. Remi Lodh for his stimulating advice and efficient assistance during the preparation of the final manuscript. Lastbutcertainlynotleast,weacknowledgetheinvaluablesupportwereceived from our families and friends. Umeå, Sweden Patrik Eklund Bilbao, Spain Javier Gutiérrez García Wuppertal, Germany Ulrich Höhle Mikkeli, Finland Jari Kortelainen December 2017
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