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Indium AmmllHlod Algebra u 7 dgm Applled Alg@bra Garrett Birkhoff DepartmentofMathematics HarvardUniversity and Thomas C. Bartee DivisionofEngineering andAppliedPhysics HarvardUniversity McGraw-Hill BookCompany NewYork,St.Louis,SanFrancisco,Diisuldorf, London,Mexico,Panama,Sydney,Toronto ThisbookwassetinModernbyTheMaplePressCompmy, andboundbyTheMaplePressCompany.Thedesignerwas MarshaCohen;thedrawingsweredonebyJosephBuchner.The~editors wereDonaldK.PrentissandMaureenMcMahon.SallyR.Ellyson supervisedtheproduction. CorrectedThirdPrinting,1970 ' Mod-mAppliedAlgebra Copyright©1970byMcGraw-Hill,Inc.Allrightsreserved. PrintedintheUnitedStatesofAmerica.Nopartofthis publicationmaybereproduced,storedinaretrievalsystem, ortransmitted,inanyformorbyanymeans,electronic, mechanical,photocopying,recording,orotherwise,without thepriorwrittenpermissionofthepublisher‘ libraryofCanons;CatalogCardNumber70—88879 ISBN 07-005381-2‘ 567890 MAMM 7987654321 Presses The name “modern algebra” refers to the study of algebraic systems (groups,rings,Booleanalgebras,etc.)whoseelementsaretypicallynon- numer’ical. By contrast, “classical” algebrais basically concernedwith algebraic equations or systems of equations whose symbols stand for realorcomplexnumbers,Overthepast40years,“modern”algebrahas beensteadilyreplacing“classical”algebrainAmericancollegecurricula. Thepast20yearshaveseenanenormousexpansioninseveralnew areas of technology. These new areas include digital computing, data communication,andradarandsonarsystems.Workineachoftheseareas reliesheavilyonmodernalgebra.Thisfacthasmadethestudyofmodern algebraimportantto appliedmathematicians, engineers, and scientists whousedigitalcomputersorwhoworkintheotherareasoftechnology mentionedabove. Thisbookattemptstopresentthoseideasandtechniquesofmodern algebrawhichhaveprovedmostusefulfortheseareas.Althoughsome separation is inevitable between the discussion of mathematical prin- vi Preface ciplesandtheirapplications,wehavetriedtoweavetogethertheunder- lyingideasandthetechnicalalgorithmswhicharebasedonthem. Thebookbeginsbypresentingandillustratingthenotionsofset, function, mathematical induction, binary relation, and graph. Discus- sions are also included ofBoolean algebras, monoids, morphisms, and otherbasicalgebraicconcepts.Thismaterialconstitutestwochapters. Inthenextchapterweintroducetheconceptsoffinite-statemachine andTuringmachine, alongwithrepresentationschemesforthesetypes ofmachines,andsystematictechniquesforreducingthenumberofstates infinite-statemachines.Chapter4providesanintroductiontothedigital computerprogramminglanguagecalledALGOLandtoitssyntax. Chapter5introducestheaxiomaticapproachwhichissocharacter- isticofmodernalgebra.Booleanalgebrasaredefinedformallybyappro— priatepostulates,andthepropertiesofBooleanarededucedfromthese postulates.Booleanalgebraisthenrelatedtologic,gatingnetworks,and ALGOL programming, after which the canonical form for Boolean polynomialsisderived.Chapter6centersaroundtheconceptofoptimiza- tion, beginning with schemes for finding paths of least cost through networks. Thentechniquesarepresentedfordescribingthegatingnet- worksusedindigitalcomputers,andforsimplifyingthem(andthereby reducing their cost) using Boolean polynomials. Finally, a method is described for realizing an arbitrary finitestate machine by means of gatesandflip-flop(memory)elements. Chapter7presentsanaxiomatictreatmentofmonoidsandgroups. It covers muchmore about monoids andsomewhat less about groups thandomostbooksonmodernalgebra.Chapter8appliessomeofthese ideastodatacommunicationssystemsinwhichnoisemayleadtoerrors in the messages transmitted. Using the standard “binary symmetric model” ofcommunicationstheoryto describetheprobabilityof error, this chapterdescribestechniquesforgenerating, coding, and decoding groupcodessoastooptimizetheirefficiencyindetectingandcorrecting errors. Thisisfollowedbyachapteronlattices,whichshowshowfar- reaching generalizations of Boolean algebra can be derived from the studyofpartialorderingrelations. . Chapter10dealswithringsandfields.Thischapteremphasizesthe variouskindsofringswhichcanariseinapplications,relatesmorphisms toideals, and discussesuniquefactorizationand Gaussianelimination. Chapter 11 studies polynomial rings and applies polynomials to the constructionandanalysisoferror-correctinganderror-detectingcodes. FiniteorGaloisfieldsarestudiedinChapter12andareusedthere to derive aspecial class of codes called Bose-Chaudhuri-Hoquenghem codes.Chapter13thenintroducesdifferenceequationsandaparticular classofcodeswhicharebasedondifferenceequationsandareusedwith an‘ relation ‘ in ' " andradarsystems. Preface vii Chapter 14, thefinalchapter, thenpassesfromthefinitesystems whichhavebeenstudieduptothispointtoinfinitesystems.Afteritis shownthattherealnumbersareuncountable,theconceptofcomputability isintroducedandrelatedtotheconceptofaTuringmachine. Finally, thenotionofmachinecomputabilityisrelatedtoideasofmathematical linguisticsandtotheclassificationofprogramminglanguages. Anumberofdifferentcoursescanbedevelopedfromthematerial inthisbook,whichcontainsmanystarredsectionswhichcanbeomitted iftimerequires.AtHarvard,thematerialispresentedintwohalf-year courses:AppliedMathematics106,primarilyanadvancedundergraduate course, and Applied Mathematics 206, agraduate course. No specific prerequisites are listed for Applied Mathematics 106. Because of the basicnatureofthematerialcovered,thesecoursesareprerequisitesfor a number of other courses Applied Mathematics 206 also includes a shortintroductiontorealandcomplexmatrices,withemphasisonspecial propertiesofmatricesarisinginvariousapplications;aboutone-thirdof thetermisdevotedtothismaterial. TheauthorsoweagreatdealtoJohnLipson,whocarefullyproofread twodraftsofthebookinmanuscriptform,whocooperatedinwriting ourexpositiononALGOL, andwhowroteourALGOLprograms.They alsoowemuchtoAspiWadia,whoalsoproofreadthemanuscript,and wroteoutcarefullywordedsolutionsformanyoftheexercises. Finally, theywerehelpedbycriticismsandsuggestionsfrommanyfriends and colleagues, including especially Donald Anderson, Marshall Hall, Sesumo Kuno, Donald MacLaren, Albert Meyer, Werner Rheinboldt, HartleyRogers,DavidSchneider,andHaoWang. Cambridge,Massachusetts GarrettBirkhofi April11,1969 ThomasC.Bartee