Algebra in the Stone-CМЊech compactification : theory and applications

610 Pages·2012·4.054 MB·English

by  Neil Hindman, Dona Strauss

De Gruyter Textbook
Hindmann / Strauss • Algebra in the Stone-Čech Compactification

Neil Hindman, Dona Strauss
Algebra in the Stone-Čech
Compactification
Theory and Applications
2nd revised and extended edition
De Gruyter
Mathematics Subject Classification 2010: Primary: 05D10, 22A15, 54D35, 54H20;
Secondary: 03E05, 20M10, 54H13.
ISBN: 978-3-11-025623-9
e-ISBN: 978-3-11-025835-6
Library of Congress Cataloging-in-Publication Data
Hindman, Neil, 1943–
Algebra in the Stone-Cech compactification : theory and applications / by Neil
Hindman, Donna Strauss. – 2nd revised and extended ed.
p. cm. – (De gruyter textbook)
Includes bibliographical references and index.
ISBN 978-3-11-025623-9 (alk. paper)
1. Stone-Cech compactification. 2. Topological semigroups. I. Strauss, Dona,
1934– II. Title.
QA611.23.H56 2012
514′.32–dc23
2011039515

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
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Preface to the First Edition
The semigroup operation defined on a discrete semigroup .S;(cid:2)/ has a natural exten-
sion, also denoted by (cid:2), to the Stone–Cˇech compactification ˇS of S. Under the
extended operation, ˇS is a compact right topological semigroup with S contained
in its topological center. That is, for each p 2 ˇS, the function (cid:2) W ˇS ! ˇS is
p
continuous and for each s 2 S, the function (cid:3) W ˇS ! ˇS is continuous, where
s
(cid:2) .q/Dq(cid:2)p and(cid:3) .q/Ds(cid:2)q.
p s
InPartIofthisbook,assumingonlythemathematicalbackgroundstandardlypro-
videdinthefirstyearofgraduateschool,wedevelopthebasicbackgroundinforma-
tion about compact right topological semigroups, the Stone–Cˇech compactification
of a discrete space, and the extension of the semigroup operation on S to ˇS. In
PartII,westudyindepththealgebraofthesemigroup.ˇS;(cid:2)/andinPartIIIpresent
some of the powerful applications of the algebra of ˇS to the part of combinatorics
knownasRamseyTheory. WeconcludeinPartIVwithconnectionswithTopological
Dynamics,ErgodicTheory,andthegeneraltheoryofsemigroupcompactifications.
The study of the semigroup .ˇS;(cid:2)/ has interested several mathematicians since it
wasfirstdefinedinthelate1950s. Asaglanceatthebibliographywillshow,alarge
numberofresearchpapershavebeendevotedtoitsproperties.
ThereareseveralreasonsforaninterestinthealgebraofˇS.
ItisintrinsicallyinterestingasbeinganaturalextensionofS whichplaysaspecial
role among semigroup compactifications of S. It is the largest possible compactifi-
cationofthiskind: IfT isacompactrighttopologicalsemigroup, ' isacontinuous
homomorphism from S to T, 'ŒS(cid:4) is dense in T, and (cid:3) is continuous for each
'.s/
s 2S,thenT isaquotientofˇS.
We believe that ˇN is interesting and challenging for its own sake, as well as for
its applications. Although it is a natural extension of the most familiar of all semi-
groups,ithasanalgebraicstructureofextraordinarycomplexity,whichisconstantly
surprising. Forexample,ˇN containsmanycopiesofthefreegroupon2cgenerators
[244]. AlgebraicquestionsaboutˇN whichsounddeceptivelysimplehaveremained
unsolvedformanyyears. Itis,forinstance,notknownwhetherˇN containsanyel-
ementsoffiniteorder,otherthanidempotents. Andthecorrespondingquestionabout
the existence of nontrivial finite groups was only very recently answered by Y. Ze-
lenyuk. (HisnegativeanswerispresentedinChapter7.)
The semigroup ˇS is also interesting because of its applications to combinatorial
numbertheoryandtotopologicaldynamics.
Algebraic properties of ˇS have been a useful tool in Ramsey theory. Results in
Ramsey Theory have a twin beauty. On the one hand they are representatives of
vi PrefacetotheFirstEdition
pure mathematics at its purest: simple statements easy for almost anyone to under-
stand(thoughnotnecessarilytoprove). Ontheotherhand,theareahasbeenwidely
applied from its beginning. In fact a perusal of the titles of several of the original
papers reveals that many of the classical results were obtained with applications in
mind. (Hilbert’sTheorem–Algebra;Schur’sTheorem–NumberTheory;Ramsey’s
Theorem–Logic;theHales–JewettTheorem–GameTheory).
The most striking example of an application of the algebraic structure of ˇS to
Ramsey Theory is perhaps provided by the Finite Sums Theorem. This theorem
says that whenever N is partitioned into finitely many classes (or in the terminol-
ogycommonwithinRamseyTheory,isfinitelycolored),thereisasequencehx i1
n nD1
with FS.hx i1 / contained in one class (or monochrome). (Here FS.hx i1 / D
n nD1 n nD1
P
¹ x W F is a finite nonempty subset of Nº.) This theorem had been an open
n2F n
problem for some decades, even though several mathematicians (including Hilbert)
hadworkedonit. AlthoughitwasinitiallyprovedwithoutusingˇN, thefirstproof
givenwasoneofenormouscomplexity.
In 1975 F. Galvin and S. Glazer provided a brilliantly simple proof of the Finite
SumsTheoremusingthealgebraicstructureofˇN. Sincethistimenumerousstrong
combinatorial results have been obtained using the algebraic structure of ˇS, where
S isanarbitrarydiscretesemigroup. Intheprocess,moredetailedknowledgeofthe
algebraofˇS hasbeenobtained.
Other famous combinatorial theorems, such as van der Waerden’s Theorem or
Rado’sTheorem,haveelegantproofsbasedonthealgebraicpropertiesofˇN. These
proofs have in common with the Finite Sums Theorem the fact that they were ini-
tially established by combinatorial methods. A simple extension of the Finite Sums
Theorem was first established using the algebra of ˇN. This extension says that
wheneverN isfinitelycoloredthereexistsequenceshx i1 andhy i1 suchthat
n nD1 n nD1
FS.hx i1 /[FP.hy i1 /ismonochrome,whereFP.hy i1 /D¹Q y WF
n nD1 n nD1 n nD1 n2F n
is a finite nonempty subset of Nº. This combined additive and multiplicative result
wasfirstprovedin1975usingthealgebraicstructureofˇN anditwasnotuntil1993
thatanelementaryproofwasfound.
Other fundamental results have been established for which it seems unlikely that
elementaryproofswillbefound. AmongsuchresultsisadensityversionoftheFinite
Sums Theorem, which says roughly that the sequence hx i1 whose finite sums
n nD1
are monochrome can be chosen inductively in such a way that at each stage of the
induction the set of choices for the next term has positive upper density. Another
suchresultistheCentralSetTheorem,whichisacommongeneralizationofmanyof
thebasicresultsofRamseyTheory. Significantprogresscontinuestobemadeinthe
combinatorialapplications.
ThesemigroupˇS alsohasapplicationsintopologicaldynamics. Asemigroup S
X
ofcontinuousfunctionsactingonacompactHausdorffspaceX hasaclosurein X
(the space of functions mapping X to itself with the product topology), which is a
PrefacetotheFirstEdition vii
compact right topological semigroup. This semigroup, called the enveloping semi-
group, was first studied by R. Ellis [134]. It is always a quotient of the Stone–Cˇech
compactification ˇS, as is every semigroup compactification of S, and is, in some
importantcases,equaltoˇS. Inthisframework,thealgebraicpropertiesofˇS have
implicationsforthedynamicalbehaviorofthesystem.
Theinteractionwithtopologicaldynamicsworksbothways. Severalnotionswhich
originatedintopologicaldynamics,suchassyndeticandpiecewisesyndeticsets,are
importantindescribingthealgebraicstructureofˇS. Forexample, apointp ofˇS
isintheclosureofthesmallestideal of ˇS ifandonlyifforeveryneighborhood U
ofp,U \S ispiecewisesyndetic.
This last statement can be made more concise when one notes the particular con-
structionofˇS thatweuse. Thatis,ˇS isthesetofallultrafiltersonS,theprincipal
ultrafiltersbeingidentifiedwiththepointsofS. Underthisconstruction,anypointp
of ˇS is precisely ¹U \S W U is a neighborhood of pº. Thusp is in the closure of
thesmallestidealofˇS ifandonlyifeverymemberofp ispiecewisesyndetic.
Inthisbook,wedevelopthealgebraictheoryofˇS andpresentseveralofitscom-
binatorial applications. We assume only that the reader has had graduate courses in
algebra, analysis, and general topology as well as a familiarity with the basic facts
about ordinal and cardinal numbers. In particular we develop the basic structure of
compactrighttopologicalsemigroupsandprovideanelementaryconstructionofthe
Stone–Cˇechcompactificationofadiscretespace.
With only three exceptions, this book is self contained for those with that min-
imal background. The three cases where we appeal to non elementary results not
provedhereareTheorem6.36(duetoM.RudinandS.Shelah)whichassertstheex-
istence of a collection of 2celements of ˇN no two of which are comparable in the
Rudin–Keisler order, Theorem 12.37 (due to S. Shelah) which states that the exis-
tenceofP-pointsinˇNnN cannotbeestablishedinZFC,andTheorem20.13(dueto
H.Furstenberg)whichisanergodictheoreticresultthatweusetoderiveSzemerédi’s
Theorem.
AllofourapplicationsinvolveHausdorffspaces,sowewillbeassumingthrough-
out,exceptinChapter7,thatallhypothesizedtopologicalspacesareHausdorff.
The first five chapters are meant to provide the basic preliminary material. The
concepts and theorems given in the first three of these chapters are also available in
otherbooks. Theremainingchaptersofthebookcontainresultswhich,forthemost
part, can only be found in research papers at present, as well as several previously
unpublishedresults.
Notesonthehistoricaldevelopmentaregivenattheendofeachchapter.
Let us make a few remarks about organization. Chapters are numbered consecu-
tivelythroughoutthebook,regardlessofwhichofthefourpartsofthebookcontains
them. Lemmas, theorems, corollaries, examples, questions, comments, and remarks
are numbered consecutively in one group within chapters (so that Lemma 2.4 will
viii PrefacetotheFirstEdition
be found after Theorem 2.3, for example). There is no logical distinction between a
theorem and a remark. The difference is that proofs are never included for remarks.
Exercisescomeattheendofsectionsandarenumberedconsecutivelywithinsections.
The authors would like to thank Andreas Blass, Karl Hofmann, Paul Milnes, and
Igor Protasov for much helpful correspondence and discussions. Special thanks go
to John Pym for a careful and critical reading of an early version of the manuscript.
The authors also wish to single out Igor Protasov for special thanks, as he has con-
tributedseveralnewtheoremstothebook. TheywouldliketothankArthurGrainger,
AmirMaleki,DanTang,ElaineTerry,andWenJinWoan forparticipatinginasem-
inar where much of the material in this book was presented, and David Gunderson
forpresentinglecturesbasedontheearlymaterialinthebook. Acknowledgementis
also due toourcollaborators whoseefforts are featured inthis book. These collabo-
rators include John Baker, Vitaly Bergelson, John Berglund, Andreas Blass, Dennis
Davenport,WalterDeuber,AhmedEl-Mabhouh,HillelFurstenberg,SalvadorGarcía-
Ferreira,YitzhakKatznelson,JimmieLawson,AmhaLisan,ImreLeader,HannoLef-
mann, Amir Maleki, Jan van Mill, Paul Milnes, John Pym, Petr Simon, Benjamin
Weiss,andWen-jinWoan.
The authors would like to acknowledge support of a conference on the subject of
this book in March of 1997 by DFG Sonderforschungsbereich 344, Diskrete Struk-
turen in der Mathematik, Universität Bielefeld, and they would like to thank Walter
Deuberfororganizingthisconference. BothauthorswouldliketothanktheEPSRC
(UK) for support of a visit and the first author acknowledges support received from
theNationalScienceFoundation(USA)undergrantDMS9424421duringtheprepa-
rationofthisbook.
Finally, the authors would like to thank their spouses, Audrey and Ed, for their
patiencethroughoutthewritingofthisbookaswellashospitalityextendedtoeachof
usduringvisitswiththeother.
April1998 NeilHindman,DonaStrauss
Preface to the Second Edition
Inthefourteenyearssincethepublicationofthefirstedition,researchinthealgebraic
theory of the Stone–Cˇech compactification of a discrete semigroup has been very
active. (We have added more than 165 research papers to the bibliography.) In this
editionwehaveaddedseveralofthemoreaccessibleresultsthathavebeenestablished
during this time. These include six new sections, namely 1.1.1, 4.5, 7.4, 9.5, 14.6,
and 15.6. They also include more than 50 new lemmas and theorems. In addition,
many typographical errors have been corrected, and the notes at the end of chapters
havebeenextended.
Thefirsteditionhasbeenwidelycitedinthemathematicalliterature,andwehave
beenveryconcernedwithmaintainingitsvalueasareference. Accordingly,withtwo
exceptions,anyresultsinthesecondeditionthatshareanumberwitharesultfromthe
first edition are either identical with that result or trivially imply the result from the
first edition. (The two exceptions are Lemmas 12.40 and 12.41 which are technical
lemmas replacing earlier technical lemmas which we do not feel are of independent
interest.) Newmaterialthathasbeeninsertedbeforetheoldendofchapterreceivesa
newnumberoftheformx.y.zwherethenumberx.yistheitemfromthefirstedition
that immediately precedes it. For example in between Definition 14.14 and Theo-
rem 14.15 from the first edition come a definition and eight lemmas and theorems
numbered14.14.1through14.14.9.
Insomecases,wehavehadtomoveresultsearlierinthetextbecausenewlyinserted
materialneededtheseresults. Forexample,Lemma6.6wasneededfornewmaterial
whichfitnaturallyinChapter5soitbecameLemma5.19.1. WealsokeptLemma6.6
initsplaceandsimplycitedLemma5.19.1foritsproof. Wedonot,however,repeat
definitionswhichwehavebeenforcedtomoveearlierinthetext. Inthosecases,the
numberassociatedwiththedefinitioninthefirsteditionissimplyomitted.
Wehaveaddedonemoreresult,namelyTheorem17.35.1,tothethreeresultsinthe
firsteditionwhichweusebutdonotprove.
Intheprefacetothefirstedition,wesaidthatwethoughtitunlikelythatanelemen-
taryproofofthedenseFiniteSumsTheoremwouldbefound. However,H.Towsner
hasrecentlydiscoveredsuchaproof[378].
ThefirstauthoracknowledgessupportreceivedfromtheNationalScienceFounda-
tion(USA)viagrantDMS-0852512duringthepreparationofthisrevision.
Washington/Leeds,September2001 NeilHindman,DonaStrauss