Table Of ContentDevelopments 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
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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
