The q-theory of Finite Semigroups

670 Pages·2009·3.248 MB·English

by  John Rhodes, Benjamin Steinberg

Springer Monographs in Mathematics
Forother titles published in this series, go to
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John Rhodes Benjamin Steinberg
The q-theory of Finite
Semigroups
123
John Rhodes Benjamin Steinberg
UniversityofCalifornia Cereleton University
Department of Mathematics School of Mathematics and Statistics
1000 Centennial Dr. 1125 Colonel by Drive
Berkeley CA 94720-3840 Ottawa ON K1S 5B6
USA Canada
[e-mail] [e-mail]
ISSN:1439-7382
ISBN:978-0-387-09780-0 e-ISBN:978-0-387-09781-7
DOI10.1007/978-0-387-09781-7
LibraryofCongressControlNumber: 2008939929
Mathematics Subject Classification (2000): 20M07, 20M35, 20F20, 20F32, 20A15, 20A26, 16Y60,
06B35, 06E15, 06A15
(cid:2)c SpringerScience+BusinessMedia,LLC2009
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In memory of Bret Tilson
Preface
When people are trying to learn mathematics for the purpose of research,
they usually start at the forefront and then go backwards as needed in order
to understand the results more fully. Yet with a mathematics book, it is not
uncommon for people to start on page one and read onwards. This book is a
research manuscript, and we heartily encourage the reader to delve in, read
what is of interest, and go back as necessary. We hope that the material at
the end of the book | the charts, tables and indices | will make this easier
going.
Not many books have appeared in recent years dedicated to state-of-the-
art Finite Semigroup Theory. There was the famous \Arbib" book in the
1960s[171] containing the lectures of Kenneth Krohn, John Rhodes and Bret
Tilson.SamuelEilenberg’streatise[85],withtwochaptersbyTilson[362,363],
appearedmorethan30yearsago.Itrevolutionizedsemigrouptheorywiththe
introductionofpseudovarietiesofsemigroupsandvarietiesoflanguages.How-
ever, the most recent book on the subject is that of Jorge Almeida [7], which
was originally published in 1992! Almeida’s book made pro(cid:12)nite methods in
semigroup theory accessible. Howard Straubing’s 1994 book [346] does touch
onsemigrouptheory,but it is moreconcernedwith applicationsto Computer
Science than with semigroups themselves.
This volume is intended both to introduce a quantized version of Eilen-
berg’s theory, by going to operators on pseudovarieties and relational mor-
phisms, and to (cid:12)ll in some of the vacuum that has accrued from years that
have passed with no new books on (cid:12)nite semigroups. Also, we have tried to
include a number of classical results that pro(cid:12)t from a recasting in a modern
language to increase accessibility.
The philosophy of the book and an explanation of the individual chapters
is covered in the Introduction, which we strongly recommend reading before
enteringinto the body of the text. Here we try to providea brief guide to the
structure of the current volume.
Let us begin with what this book does not do. We do not intend in any
way to give a basic introduction to Semigroup Theory. We have included for
viii Preface
the reader’s convenience an appendix on the basic structure theory, such as
Green’s relations and Rees’s Theorem, but it is by no means complete. The
classic treatise of A. H. Cli(cid:11)ord and G. B. Preston [68] still servesadmirably
for learning this material, as does the \Arbib" book [171]. One can also turn
to the books of J. M. Howie [139] and P. M. Higgins [133]. For the current
text, it is handyto be familiarwith alittle bit of categorytheory[185],basic
topology [158], and perhaps some combinatorial group theory [184].
This is a book about (cid:12)nite semigroups. Except for free semigroups,
free groups and pro(cid:12)nite semigroups, the reader should not expect to en-
counter in(cid:12)nite semigroups. Readers interested in the Prime Decomposition
Theorem for in(cid:12)nite semigroups should consult the relevant papers on the
topic[3,50{52,88,125,214,215,276,277,279].Outsideof abriefforaytoprove
Schu(cid:127)tzenberger’sTheorem,wedonotenterintoFormalLanguageTheoryand
Automata Theory.FormalLanguageTheoryis an important aspect of Finite
SemigroupTheory,andmuchofthemotivationforproblemsinthe(cid:12)eldderive
fromit. But there isalreadyasizableliteratureof excellentbooksdevotedto
this facet (for instance, [85,177,224,229,346]). Nonetheless, it would be an
important task in the future to reinterpret the results of this book from the
language theoretic point of view. Another important aspect of Finite Semi-
group Theory that we touch upon only slightly is the (cid:12)nite basis problem.
Mark Sapir, Marcel Jackson and Mikhail Volkov, among others, have done
important work on this subject [77,142{144,146,159,246,306{309,372,374].
The book is divided into four parts. The (cid:12)rst part, entitled The q-
operator and Pseudovarieties of Relational Morphisms, is the heart
of q-theory.Chapter 1is foundationalmaterial,and much of it maybe famil-
iartothereader.However,attheendofthechaptertheimportantnotionsof
division of relational morphisms and relational morphism between relational
morphismsareintroduced.Chapter2introducestheclassesofrelationalmor-
phisms that are of central importance in this volume: continuously closed
classesandpseudovarietiesofrelationalmorphisms.Weintroducetheoperator
qanditsimageischaracterized.Maximalandminimalmodels,meaningmax-
imalandminimum classesde(cid:12)ningagivenoperator,arestudied,andwegive
examples showing that all the commonly occurring operators in Finite Semi-
groupTheory(cid:12)tintoourframework.TheDerivedandKernelSemigroupoids
are presented and their basic properties explored in order to introduce the
pseudovarieties of relational morphisms V and V , whose images under q
D K
arethe operatorsV ( ) and V ( ),thanksto theDerivedSemigroupoid
(cid:3) (cid:0) (cid:3)(cid:3) (cid:0)
and Kernel Semigroupoid Theorems.
The last chapter of Part I, Chapter 3, develops the equational theory for
pseudovarieties of relational morphisms. Because the development parallels
Reiterman’sTheoremandusesheavilypro(cid:12)nitesemigroups,wehaveprovided
a brief introduction to the classical theory of pseudoidentities and pro(cid:12)nite
Preface ix
and pro-V semigroups. Then we establish the generalization of Reiterman’s
Theorem to pseudovarieties of relational morphisms and give examples of
pseudoidentity bases for the pseudovarieties of relational morphisms associ-
ated to the most commonly studied operators: semidirect products, Mal’cev
productsandjoins.Inevitablesubstitutions areintroducedandstudied, lead-
ingtoabasistheoremforthecompositionoftwopseudovarietiesofrelational
morphisms. Classical basis theorems from the literature are deduced as con-
sequences.We discusswhenthebasistheoremforthe semidirectproduct[27]
applies and provide a new basis theorem coveringthe general case.
ThesecondpartofthebookiscalledComplexity in Finite Semigroup
Theory.Chapter4providesacomprehensivelookatgroupcomplexitystart-
ingfromtheverybeginningwithaproofofthePrimeDecompositionTheorem
and ending with such advanced topics as Ash’s Theorem [33], the Ribes and
ZalesskiiTheorem[300],Rhodes’sPresentationLemma[43,334],Tilson’s2J-
class Theorem [360] and Henckell’s Aperiodic Pointlikes Theorem [121,129].
Also, some important and hard-to-(cid:12)nd results from the literature are fea-
tured, including Graham’s Theorem [107] on the idempotent-generated sub-
semigroup of a Rees matrix semigroup. Using the language and viewpoint
of q-theory, a simpli(cid:12)ed presentation of the computability of complexity for
semigroups in DS is presented [170,269,368]. A large part of this chapter is
dedicatedtoanupdatedpresentationoftheresultsinthe\Arbib"book[171]
concerningMal’cevproductsandsubdirectlyindecomposablesemigroups.We
do not make any attempt to duplicate the material covered in Tilson’s chap-
ters in Eilenberg’s book [362,363],and so as a consequence the Fundamental
Lemmaof Complexityis not provedhere. Few of the resultsin Chapter 4are
new, although some of the proofs are novel.
Chapter 5 introduces our general scheme for de(cid:12)ning the complexity hi-
erarchy associated to a single operator or a pair of operators on the lattice
ofpseudovarieties.Wediscussmanyexamplesfromtheliteratureandpresent
some new ones as well. The focus of the chapter is on two-sided complex-
ity. The decomposition theory for maximal proper surmorphisms [293,296]is
included for the convenience of the reader and because the language of pseu-
dovarieties of relational morphisms sheds some new light on these results.
Theremainderofthechaptertakesthe (cid:12)rststepsingeneralizingresultsfrom
group complexity to two-sided complexity. The Ideal Theorem is proved and
the two-sided complexity of the full transformation monoid is computed.
PartIIIofthebook,The Algebraic Lattice of Semigroup Pseudova-
rieties, provides a study of lattice theoretic aspects of the algebraic lattice
of pseudovarieties of semigroups. Chapter 6 is essentially a condensed survey
of algebraic and continuous lattices [97]. It is intended to establish notation
and to introduce ideas that may not be familiar to a semigroup audience.
Some examples related to q-theory are provided. For instance, it is shown
that any continuous lattice of countable weight is the (cid:12)xed-point lattice of
an idempotent continuous operator on the lattice of pseudovarieties of semi-
groups.Chapter7isdedicatedtotheabstractspectraltheoryofthelatticeof
x Preface
pseudovarietiesofsemigroups(andtoamuchlesserdegreethelatticesofpseu-
dovarieties of relational morphisms and continuous operators). In particular,
a multitude of new results about join irreducibility are obtained. Semigroups
S with the property that S V W implies S V or S W are studied,
2 _ 2 2
as well as their exclusion pseudovarieties.We introduce severalnovelfamilies
of such semigroups, leading to the following application: if H is a pseudova-
riety of groups containing a non-nilpotent group, then the pseudovariety H
of semigroups whose subgroups belong to H is (cid:12)nite join irreducible. This
generalizes results of Margolis, Sapir and Weil [195]. Also, the classi(cid:12)cation
of Kova(cid:19)cs-Newman semigroups that we began in an earlier paper [287] is
completed.
The(cid:12)nalpartofthebook,entitledQuantales, Idempotent Semirings,
Matrix Algebras and the Triangular Product, consists of twochapters.
Chapter 8 introduces our slight weakening of the notion of a quantale and
develops the general algebraic theory of these objects. We also study the
equivalence between Boolean bialgebras and pro(cid:12)nite semigroups, leading to
a bialgebra structure on the Boolean algebra of regular languages over a (cid:12)-
nitealphabet.Chapter9containsalmostentirelynewmaterial.Viewing(cid:12)nite
quantalesas idempotent semirings,we develop a decomposition theory for (cid:12)-
nitesemiringsingeneralviathetriangularproductofPlotkin[240],following
the model of groupcomplexity. The centerpieceof the chapteris formed by a
pair of results we refer to as the Triangular Decomposition Theorem and the
Ideal Decomposition Theorem. They basically give semiring analoguesof the
resultsofMunnandPonizovski(cid:21)(cid:16)onsemigroupalgebrasover(cid:12)elds[210,245]by
embeddingthesemigroupalgebra(overasemiring)ofa(cid:12)nitesemigroupinside
aniteratedtriangularproductofmatrixalgebrasoverthegroupalgebrasofits
Schu(cid:127)tzenberger subgroups. Applying these theorems in the context of idem-
potent semirings yields the decomposition half of the Prime Decomposition
Theoremforidempotent semirings.Alargesegmentofthechapterisdevoted
toprovingthatmatrixalgebrasoverthepowersemigroupofagroupareirre-
ducible with respecttothe triangularproduct. Themoralofthe storyisthat
muchmoreofringtheoryworksforsemiringsthanonemightexpect.Finally,
Chapter 9 ends with applications of the decomposition theory of idempotent
semirings to computing the group complexity of power semigroups.
There is not a strict logical sequence for reading the various parts of the
book(hopefully,thedependencygraphisatleastacyclic!).Allreadersshould
be familiar with the material in Chapter 1 to proceed. The basic de(cid:12)nitions
and terminology introduced in Chapter 2, at least as far as to the end of
Section 2.3, are needed for Chapters 3{8, although speci(cid:12)c results are rarely
required.ThenewmaterialinChapter3ishighlytechnicalandisnotrequired
formostoftherestofthebook,exceptforabriefreappearanceinChapter7;
the reader already familiar with pseudoidentities and pro(cid:12)nite semigroups
may skip it entirely on a (cid:12)rst reading. Much of Part II requires only Chap-
ter 1 and the language of Chapter 2, not the results. The key exceptions are
the Derived and Kernel Semigroupoid Theorems, which are used repeatedly.
Preface xi
Part III requires Part I, but Chapter 6 can be read entirely independently
of Part II. Chapter 7 occasionally appeals to results from Chapters 4 and 5.
Chapter8dependsonPartIandChapter6,whereasChapter9dependsonly
on a basic knowledge of de(cid:12)nitions from Chapters 1 and 4, familiarity with
the Fundamental Lemma of Complexity and on a small part of Chapter 8,
namely quantic nuclei and some examples.
We have interspersed throughout the text a large number of exercises,
someroutineand othersmoredi(cid:14)cult. The exercisesforman integralpartof
the subject matter,andthe readerisencouragedto solveasmanyof them as
possible.
The current volume has greatly bene(cid:12)tted from the comments and criti-
cisms of our colleagues and students. The following mathematicians deserve
special thanks for their hard workand thoughtfulness: KarlAuinger, Bridget
Brimacombe, Attila Egri-Nagy, Karl Hofmann, Gabor Horvath, Mark Kam-
bites, Jimmie Lawson, Stuart Margolis,Chrystopher Nehaniv, Boris and Eu-
gene Plotkin, Luis Ribes, Kimmo Rosenthal and Mikhail Volkov.The anony-
mous reviewers should also be acknowledged for their careful reading of the
text. All errorsand inaccuraciesthat remainarethe soleresponsibilityof the
authors.
The (cid:12)rst author would like to thank U.C. Berkeley’s generous retirement
funds. The second author has been funded by an NSERC Discovery Grant
during the course of this research. Both authors gratefully acknowledge sup-
port fromFCT through the Centro de Matema(cid:19)tica da Universidade do Porto
and both have received partial support from the FCT and POCTI approved
project POCTI/32817/MAT/2000in participation with the European Com-
munity Fund FEDER. Some of this work was done while the second au-
thor was at the University of Porto, Portugal. The second author would also
like to thank his colleagues at the University of Porto, in particular Jorge
Almeida, Manuel Delgado and Pedro Silva, for their friendship and collabo-
ration throughout the years and for making Porto an enjoyable environment
to work in.
The (cid:12)rst author would like to thank his wonderful wife, Laura Morland,
for everything,especially hereditorial skills. The second author is gratefulto
Ariane Masuda for her love and support through the long process of bring-
ing this book to fruition, as well as for her careful reading of the appendix.
Obrigado por tudo.
The inspirational spark for q-theory itself might never have been ignited,
were it not for the superb mathematics library at \Chevaleret" (Paris VI{
CNRS{ParisVII),andwethankitslibrariansfortheirdisponibilit(cid:19)e. However,
none of this work would have been possible if not for the many excellent
caf(cid:19)es that graciously allowed us to occupy their tables for overly excessive
time periods. Our special appreciation goes to the Paris caf(cid:19)es: Les Monts