Nine Chapters on the Semigroup Art

349 Pages·2015·1.564 MB·English

by  Alan J. Cain

Alan J. Cain
Nine Chapters on the
Semigroup Art
Lecture notes for a tour through semigroups
2015 | Porto & Lisbon
©2012–15AlanJ.Cain ([email protected])
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Contents
Preface vi
Prerequisites ◆ Acknowledgements
Chapter1|Elementarysemigrouptheory 1
Basicconceptsandexamples ◆ Generatorsandsubsemi-
groups ◆ Binaryrelations ◆ Ordersandlattices ◆ Homo-
morphisms ◆ Congruencesandquotients ◆ Generating
equivalences and congruences ◆ Subdirect products ◆
Actions ◆ Cayleygraphs ◆ Exercises ◆ Notes
Chapter2|Freesemigroups&presentations 44
Alphabetsandwords ◆ Universalproperty ◆ Propertiesof
freesemigroups ◆ Semigrouppresentations ◆ Exercises ◆
Notes
Chapter3|Structureofsemigroups 66
Green’s relations ◆ Simple and 0-simple semigroups ◆
D-classstructure◆InversesandD-classes◆Schützenberger
groups ◆ Exercises ◆ Notes
Chapter4|Regularsemigroups 89
Completely0-simplesemigroups ◆ Idealsandcompletely
0-simplesemigroups ◆ Completelysimplesemigroups ◆
Completelyregularsemigroups ◆ Leftandrightgroups ◆
Homomorphisms ◆ Exercises ◆ Notes
• iii
Chapter5|Inversesemigroups 114
Equivalentcharacterizations ◆ Vagner–Prestontheorem
◆ Thenaturalpartialorder ◆ Cliffordsemigroups ◆ Free
inversesemigroups ◆ Exercises ◆ Notes
Chapter6|Commutativesemigroups 150
Cancellative commutative semigroups ◆ Archimedean
decomposition ◆ Freecommutativesemigroups ◆ Rédei’s
theorem ◆ Exercises ◆ Notes
Chapter7|Finitesemigroups 171
Green’srelationsandideals ◆ Semidirectandwreathprod-
ucts ◆ Division ◆ Krohn–Rhodesdecompositiontheorem
◆ Exercises ◆ Notes
Chapter8|Varieties&pseudovarieties 195
Varieties◆Pseudovarieties◆Pseudovarietiesofsemigroups
andmonoids◆Freeobjectsforpseudovarieties◆Projective
limits ◆ Pro-Vsemigroups ◆ Pseudoidentities ◆ Semidirect
productsofpseudovarieties ◆ Exercises ◆ Notes
Chapter9|Automata&finitesemigroups 230
Finiteautomataandrationallanguages ◆ Syntacticsem-
igroups and monoids ◆ Eilenberg correspondence ◆
Schützenberger’stheorem ◆ Exercises ◆ Notes
Solutionstoexercises 264
Bibliography 326
Index 332
• iv
List of Tables
Table8.1 Varietiesofsemigroups 204
Table8.2 Varietiesofmonoids 204
Table8.3 Varietiesofsemigroupswithaunaryoperation−1 205
Table8.4 S-pseudovarietiesofsemigroups 222
Table8.5 M-pseudovarietiesofmonoids 223
Table9.3 Varietiesofrational∗-languages 251
Table9.4 Varietiesofrational+-languages 252
•
• v
Preface
‘Aprefaceisfrequentlyasuperior
compositiontotheworkitself…’
—IsaacD’Israeli,
‘Prefaces’.In:CuriositiesofLiterature.
• Thiscourse is a tour through selected areasof semi-
grouptheory.Thereareessentiallythreeparts:
◆ Chapters1–3studygeneralsemigroups,includingpresentationsfor
semigroupsandbasicstructuretheory.
◆ Chapters4–6examinespecialclasses:namelyregular,inverse,and
commutativesemigroups.
◆ Chapters7–9studyfinitesemigroups,theirclassificationusingpseu-
dovarieties,andconnectionswiththetheoryofautomataandregular
languages.
Thecourseisbroadratherthandeep.Itisnot intendedtobecomprehens-
ive:itdoesnottrytostudy(forinstance)structuretheoryasdeeplyas
Howie,FundamentalsofSemigroupTheory,pseudovarietiesasdeeplyas
Almeida,FiniteSemigroupsandUniversalAlgebra,orlanguagesasdeeply
asPin,VarietiesofFormalLanguages;rather,itsampleshighlightsfrom
eacharea.Itshouldbeemphasizedthatthereisverylittlethatisoriginal
in this course. It is heavily based on the treatments in these and other
standardtextbooks,asthebibliographicnotesineachchaptermakeclear.
The main novelty is in the selection and arrangement of material, the
slightlyslowerpace,andthegeneralpolicyofavoidingleavingproofsto
thereaderwhenthecorrespondingresultsarerequiredforlaterproofs.
Preface • vi
FigureP.1showsthedependenciesbetweenthechapters.Attheend
of each chapter, there are a number of exercises, intended to reinforce
concepts introduced in the chapter, and also to explore some related
topicsthatarenotcoveredinthemaintext.Themostimportantexercises
are marked with a star ✴ . Solutions are supplied for all exercises. At
theendofeachchapterarebibliographicnotes,whichgivesourcesand
suggestionsforfurtherreading.
Warningsagainstpotentialmisunderstandingsaremarked(likethis)
witha‘dangerousbend’symbol,asperBourbakiorKnuth.
Importantobservationsthatarenotpotentialmisunderstandingsperse
aremarkedwithan‘exclamation’symbol(likethis).
Thiscoursewasoriginallydeliveredtomaster’sstudentsattheUni-
versitiesofPortoandSantiagodeCompostella.Thecoursewascovered
during 56 hours of classes, which included lectures and discussions of
theexercises.Revisionshaveincreasedthelengthofthenotes,andabout
70hoursofclasstimewouldnowberequiredtocoverthemfully.
Thesenoteswereheavilyrevisedin2013–15.Mostofthemaintextis
nowstable,butChapter8willbefurtherrevised,andfurtherexercises
willbeadded.Theauthorwelcomesanycorrections,observations,or
constructivecriticisms;pleasesendthemtotheemailaddressonthe
copyrightpage.Atpresent,theindexislimitedtonamesand‘named
results’only.
Prerequisites
Therearefewformalprerequisites:generalmathematical
maturityisthemainone.Anunderstandingofthemostbasicconcepts
fromelementarygrouptheoryisassumed,suchasthedefinitionofgroups,
cosets,andfactorgroups.Someknowledgeoflinearalgebrawillhelpwith
understanding certain examples, but is not vital. For Chapters 1 and 5,
knowledgeofthebasicdefinitionsofgraphtheoryisassumed.Somebasic
topologyisnecessarytoappreciatepartofChapter8fully(althoughmost
Prerequisites • vii
ofthechaptercanbeunderstoodwithoutit,andtherelevantsectionscan
simplybeskipped),andsomebackgroundinuniversalalgebraisuseful,
butnotessential.ForChapter9,someexperiencewithformallanguage
theoryandautomataisuseful,butagainnotessential.
Acknowledgements
AttilaEgri-Nagymadevaluablesuggestionsandindicated
variouserrors.SomeexercisesweresuggestedbyVictorMaltcev.Nick
Ham pointed out a typo. Many improvements are due to the students
whotookthefirstversionofthiscourse:MiguelCouto,XabierGarcía,
andJorgeSoares.Theimperfectionsthatremainaremyresponsibility.
Thetitlealludesto九章算術,NineChaptersontheMathematicalArt.
A. J. C.
•
Acknowledgements • viii
Chapter 1
Elementary semigroup theory
Chapter 2
Free semigroups & presentations
Chapter 3
Structure of semigroups
Chapter 4
Regular semigroups
Chapter 5
Inverse semigroups
Chapter 6
Commutative semigroups
Chapter 7
Finite semigroups
Chapter 8
Varieties & pseudovarieties
Chapter 9
Automata & finite semigroups
FIGUREP.1
Chartofthedependenciesbetweenthechapters.Dottedarrowsindicatethat
thedependencyisonlyintheexercises,notinthemaintext.
Acknowledgements • ix
1
Elementary
semigroup theory
‘Iusetheword“elementary”inthesense
inwhichprofessionalmathematiciansuseit…’
—G.H.Hardy,
AMathematician’sApology,§21.
•Abinaryoperation∘onaset𝑆isamap∘ ∶ 𝑆×𝑆 → 𝑆.
This operation is associative if 𝑥 ∘ (𝑦 ∘ 𝑧) = (𝑥 ∘ 𝑦) ∘ 𝑧 for all elements
𝑥,𝑦,𝑧 ∈ 𝑆.Asemigroupisanon-emptysetequippedwithanassociative
binaryoperation.
Semigroups are therefore one of the most basic types of algebraic
structure.Wecouldweakenthedefinitionfurtherbyremovingtheas-
sociativityconditionandrequiringonlyabinaryoperationonaset.A
structurethatsatisfiesthisweakerconditioniscalledamagmaorgroup-
oid.(These‘groupoids’aredifferentfromthecategory-theoreticnotionof
groupoid.)
Ontheotherhand,wecanstrengthenthedefinitionbyrequiringan
identityandinverses.Structuressatisfyingthisstrongerconditionareof
coursegroups.However,therearemanymoresemigroupsthangroups.
Forinstance,thereare5essentiallydifferentgroupswith8elements(the
cyclicgroup𝐶 ,thedirectproducts𝐶 ×𝐶 and𝐶 ×𝐶 ×𝐶 ,thedihed-
8 4 2 2 2 2
ralgroup𝐷 ,andthequaterniongroup𝑄 ),butthereare3684030417
4 8
different(non-isomorphic)semigroupswith8elements.
Elementarysemigrouptheory • 1