Semigroups and combinatorial applications PDF
Preview Semigroups and combinatorial applications
PURE AND APPLIED MATHEMATICS AWiley-InterscienceSeriesofTexts,Monographs,andTracts FoundedbyRICHARDCOURANT Editors: LIPMAN BERS, PETER HILTON, HARRY HOCHSTADT ASH—InformationTheory AUBIN—AppliedAbstractAnalysis AUBIN—AppliedFunctionalAnalysis BELLMANandWING—AnIntroductiontoInvariantImbedding BEN-ISRAELandGREVILLE—GeneralizedInverses:TheoryandApplications CARTER—SimpleGroupsofLieType CLARK—MathematicalBioeconomics COHN—DiflerenceAlgebra CURTISandREINER—RepresentationTheoryofFiniteGroupsandAssociative Algebras DAVIS—AppliedNonstandardAnalysis DUNFORDandSCHWARTZ—LinearOperators Part l—GeneralTheory Part2—SpectralTheory, SelfAdjointOperatorsin HilbertSpace Part3—SpectralOperators EHRENPRElS—FourierAnalysisinSeveralComplexVariables FRIEDMAN—Difl'erentialGames GRIFFITHSandHARRIS—AlgebraicGeometry HALE—OrdinaryDilferentialEquations HARRIS—MathematicalStructureofLanguage HENRICI—AppliedandComputationalComplexAnalysis,Volume 1 AppliedandComputationalComplexAnalysis,Volume2 HESTENES—OptimizationTheory:TheFiniteDimensionalCase HILLE—OrdinaryDifferentialEquationsintheComplexDomain HILTONandWU—ACourseinModernAlgebra HOCHSTADT—TheFunctionsofMathematicalPhysics HOCHSTADT—IntegralEquations KELLYandWEISS—GeometryandConvexity:AStudyinMathematicalMethods KOBAYASHIandNOMIZU—FoundationsofDifl'erentialGeometry,Volume 1 FoundationsofDifi'erentialGeometry,Volume2 KUIPERSandNIEDERREITER—UniformDistributionofSequences LALLEMENT—SemigroupsandCombinatorialApplications LINGENBERG—MetricPlanesandMetricVectorApplications LINZ—TheoreticalNumericalAnalysis:AnIntroductiontoAdvancedTechniques LOVELOCKandRUND—Tensors,DifferentialForms,andVariationalPrinciples MAGNUS,KARRASS,andSOLITAR—CombinatorialGroupTheory MARTIN—NonlinearOperatorsandDifferentialEquationsinBanachSpaces MELZAK—CompaniontoConcreteMathematics MELZAK—MathematicalIdeas,ModelingandApplications(VolumeIIof CompaniontoConcrete Mathematics) MEYER—IntroductiontoMathematicalFluidDynamics MORSE—VariationalAnalysis: CriticalExtremalsandSturmianExtensions NAYFEH—Perturbation Methods NAYFEHandMOOK—NonlinearOscillations ODEN andREDDY—AnIntroductiontotheMathematicalTheoryofFiniteElements PAGE—TopologicalUniformStructures PASSMAN—TheAlgebraicStructureofGroupRings PRENTER—SplinesandVariationalMethods RIBENBOIM—AlgebraicNumbers RICHTMYERandMORTON—DifferenceMethodsforInitial-ValueProblems, 2ndEdition RIVLIN—TheChebyshevPolynomials RUBIN—FourierAnalysisonGroups SAMELSON—AnIntroductiontoLinearAlgebra SIEGEL—TopicsinComplexFunctionTheory Volume l—EllipticFunctionsand UniformizationTheory Volume2—AutomorphicFunctionsandAbelianIntegrals Volume 3—AbelianFunctionsandModularFunctionsofSeveral Variables - STAKGOLD—Green’sFunctionsandBoundaryValueProblems STOKER—Difi'erentialGeometry STOKER—NonlinearVibrationsinMechanicalandElectricalSystems STOKER—WaterWaves WHITHAM—LinearandNonlinearWaves WOUK—ACourseofAppliedFunctionalAnalysis SEMIGROUPS AND COMBINATORIAL APPLICATIONS SEMIGRUUPS AND COMBINATORIAL APPLICATIONS GERARD LALLEMENT PennsylvaniaState University .A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, New York - Chichester - Brisbane - Toronto Copyright© 1979byJohnWiley&Sons,Inc. Allrightsreserved. PublishedsimultaneouslyinCanada. Reproductionortranslationofanypartofthiswork beyondthatpermittedbySections 107or 108ofthe 1976UnitedStatesCopyrightActwithoutthepermission ofthecopyrightownerisunlawful. Requestsfor permissionorfurtherinformationshouldbeaddressedto thePermissionsDepartment,JohnWiley&Sons,Inc. LibraryofCongressCataloginginPublicationData: Lallement,Gerard, 1935— Semigroupsandcombinatorialapplications (Pureandappliedmathematics) “AWiley-Interscienccpublication.” Includesbibliographicalreferencesandindex. 1. Semigroups. 2. Machinetheory. 3. Combina- torialanalysis. I. Title. QAl7l.L27 512’.2 78-23561 ISBN0-471—04379-6 PrintedintheUnitedStatesofAmerica 10987654321 PREFACE Thepurposeofthisbookistopresentthosepartsofthetheoryofsemigroups that are directly related to automata theory, algebraic linguistics, and com- binatorics. In the last 20 years a growing number ofpublications in these mathematical disciplines contained results and methods pertaining to the algebraic theory of semigroups. This has contributed to a considerable enrichment ofthistheory, enlargedits scope, and improved itspotential for becoming a major domain ofalgebra. In return, it appears that semigroup theoryprovides aconvenientgeneralframeworkforunifyingand clarifying a number oftopics in fields that seem, at first sight, unrelated. The book is intended as atextbookforgraduate students in mathematics and computer science and as a reference book for researchers interested in associative structures. For the first halfofit (Chapters 1 to 4) the only pre- requisite is a rudimentary knowledge ofelementary algebra (definition ofa group, ring, field, lattice, etc.). In the second half, which analyzes the inter- play between semigroups, automata, languages, and combinatorics, it often has been necessary to call upon other branches ofmathematics (e.g. logic, probability theory). In such instances, motivated by the desire of keeping thepresentationwithinreasonablelimits,wegivetheappropriatedefinitions or state the results in the form needed in the text, and indicate the proper references for further detail. At the end ofeach chapter the reader will find a number of exercises with references for solutions, as well as additional reading references and historical comments. Chapters 1 to4andtheportionofChapter5coveringfreesemigroupsand monoids contain the basic concepts in the theory of algebraic semigroups (Green’s relations, the Suschkewitsch—Rees theorem, the prime decomposi- tion theorem for transformation semigroups). Free semigroups, languages, and codes are introduced in Chapter 5, where we also briefly discuss decid- ‘ability and algorithms, since these are recurring themes in the applications. InChapter6wepresenttheelementsofautomatatheory, includingKleene’s theorem, and the recent variety theorem ofEilenberg. The emphasis in this V vi PREFACE chapterisontransformationmonoidsandsyntacticmonoidsofrecognizable languages. The star operation on languages and how it affects syntactic monoidsaretheobjectivesofChapter7.Itisshown,inparticular,thatdeeper results on languages expressible with the use ofthe star operation depend crucially on the study oflanguages generated by prefix codes. This study is the purpose of Chapter 8, where we present the basic results in an area closelyrelatedtopermutationgroupsandcombinatorics.Passingtocontext- free languages (called here algebraic languages) in Chapter 9, we outline their most immediate properties, show their relationship with (algebraic) power-seriesinnoncommutingvariables,andspecifythepositionofrational languages in this larger class. In the final two chapters we explore further connections between semigroups and combinatorial questions related successively to the Burnside problem and to MacMahon’s master theorem. This book does not pretend to cover all of the existing applications of semigroup theory. I have selected a number oftopics with two related im- peratives in mind. First, I wanted to emphasize the generic aspect ofsemi- group theory, since its specific aspect (i.e. the study ofsemigroups per se) has already been presented in several books [6.g. the two volumes ofA. H. Clifford and G. B. Preston (1961, 1967)]. Second, I wanted to show ex- plicitly how the study of semigroups, viewed as members of a generic structure, inserts itselfinto the study ofmore complex branches ofmath- ematics like automata theory, the theory of formal languages, and com- binatorics in general. Most ofthe materialpresented here emerged slowlyfrom several second- year graduatecourses I gave to students ofthe Mathematics and Computer ScienceDepartmentsatthePennsylvaniaStateUniversityduringtheperiod 1970—1975, and also fromnumerous talks inflicted upon my colleagues and studentsinvariousseminarsduringthesameperiodoftime. Ihavebenefited greatly from many ofmylisteners’ criticisms and suggestions, and I express mygratitude to all ofthem. During the academic year 1975—1976 J. F. Perrot invited me to become a member of a team of lecturers in a third cycle course at the University Pierre and Marie Curie (Paris VI). A large number oftopics contained in Chapters2, 3, and 5 to 9werecoveredinthiscourse. Thisprovidedmewith the best possible opportunity for completing the final draft of this book, and I wish to thank J. F. Perrot, J. Berstel, D. Perrin, and J. Sakarovitch for contributing to many valuable discussions and making many useful suggestions in a very friendly atmosphere. My former student M. Keenan meticulously read the entire manuscript, and I thank him deeplyfor his constanthelp in improvingthe presentation. I am also most grateful to J. F. Perrot, D. Perrin, and G. Jacob for their numerouscomments and suggestionsconcerning Chapters 7, 8, and 10. PREFACE vii M. P. Schiitzenberger introduced me to automata, languages, and com- binatorics almost 15 years ago. This book, in which I attempt to give a reflection ofhis pioneering work and ofthe stimulating effect ofhis ideas, is a small token ofmy gratitude and personal indebtedness. GERARD LALLEMENT UniversityPark, Pennsylvania January1979
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