Semigroups and Automata: SELECTA Uno Kaljulaid (1941- 1999) (Stand Alone)

496 Pages·2006·3.4 MB·English

by  J. Peetre

SEMIGROUPS AND AUTOMATA
SELECTA
UNO KALJULAID (1941–1999)

Semigroups and Automata
SELECTA
Uno Kaljulaid (1941–1999)
Edited by
Jaak Peetre
Lund, Sweden
and
Jaan Penjam
Tallinn, Estonia
Amsterdam • Berlin • Oxford • Tokyo • Washington, DC
© 2006 The authors.
All rights reserved. No part of this book may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, without prior written permission from the publisher.
ISBN 1-58603-582-7
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CONTENTS v
Contents
Preface. vii
BiographyofUnoKaljulaid.J.Peetre xi
BibliographyofUnoKaljulaid. xxi
ChapterI. Representationsofsemigroupsandalgebras 1
1. [K69a]Onthecohomologicaldimensionofsomequasiprojectivevarieties. 3
2. [K77a]Triangularproductsofrepresentationsofsemigroupsandassociative
algebras. 15
3. [K79a] Triangular products and stability of representations. Candidate
dissertation. 19
4. [K79b]Triangularproductsandstabilityofrepresentations.(Authorreviewof
CandidatethesisinPhysico-MathematicalSciences). 101
5. [K87a]SomeremarksonShevrin’sproblem. 111
6. [K90]Transferableelementsingrouprings. 117
7. [K00]Ω-ringsandtheirflatrepresentations.CoauthorO.Sokratova 127
ChapterII. Automatatheory 141
1. Preamble.Editors 143
2. Automataandtheirdecomposition. 145
3. [K97]Ontwoalgebraicconstructionsforautomata.CoauthorJ.Penjam 183
4. [K98c]Revisitingwreathproducts,withapplicationstorepresentationsand
invariants. 203
ChapterIII. Majorization 205
1. Generalizedmajorization.CoauthorJ.Peetre 207
2. VanderWaerden’sconjectureandhyperbolicity.J.Peetre 225
3. Ongeneralizedmajorization.J.Peetre 233
ChapterIV. Combinatorics 237
1. [K88a]OnStirlingandLahnumbers. 239
2. Letter(ordraftofletter)c.1991fromUnoKaljulaidtoTorbjörnTambour. 243
3. OnFibonaccinumbersofgraphs. 245
ChapterV. HistoryofMathematics 251
1. Th.Molien,aninnovatorofalgebra. 253
2. [K87e]OntheresultsofMolienaboutinvariantsoffinitegroupsandtheir
renaissanceincontemporarymathematics. 257
3. TheodorMolien,abouthislifeandmathematicalworkasseenacenturylater.
(Abiographicalsketchandaglimpseofhiswork). 265
4. Notesonfive19thcenturyTartumathematicians(Backlund,Kneser,Lindstedt,
Molien,Weihrauch). 291
ChapterVI. PopularizationofMathematics 325
1. [K68a]and[K69b]OnthegeometricmethodsofDiophantineAnalysis. 327
vi CONTENTS
2. [K68b]LeninprizeforworkinDiophantinegeometry. 351
3. [K69c]Thehistoryofsolvingequations. 355
4. [K70]Additionalremarksongroups. 373
5. [K73a]Polynomialsandformalseries. 389
6. [K75a]OnGaloistheory. 399
7. [K75b]Theoryofautomata.CoauthorE.Tamme 413
8. [K93c]Mordell’sproblem. 427
9. [K96]Ontwodiscretemodelsinconnectionwithstructuresofmathematics
andlanguage. 447
IndexofNames 459
SubjectIndex 467
PREFACE vii
Preface
WehavethepleasuretooffertotheMathematicalPublictheSelectaoftheeminent,
lateEstonianalgebraistUnoKaljulaid.ItcontainsmainlypaperspublishedinKaljulaid’s
lifetime. Manyofthemwereoriginallywritten inRussian, a fewalso in Estonian,and
havenowbeentranslatedintoEnglish,mainly,byoneofus,J.Peetre1.
Heritage. In addition to this published material, Kaljulaid left a large number of
manuscripts in various states of completion. They are currently in the custody of the
SeniorEditor.Forinstance,thereisanalmostcompletepaperonrightordergroups,sur-
veyingthesubjectinitshistoricaldevelopment,startingwithD.Hilbert;somematerial
onPetrinets,etc.,thingsthat,apparently,occupiedKaljulaidinhislastyears.Hopefully,
partofitcanalsobemadepublic,atalaterstage,perhapsintheformofSelectaII.
LetusnowhighlightsomeofthemainitemsofthepresentVolume.
Contents. We offer here the English translation of Kaljulaid’s 1979 Tartu/Minsk
Candidatethesis[K79a],whichoriginallywastypewritteninRussianandmanufactured
in notso many copies. The thesis was devotedto representationtheoryin the spirit of
his thesis advisor B. I. Plotkin: representationsof semigroupsand algebras, especially
extensionto this situation, and applicationof the notionof triangularproductof repre-
sentationsforgroupsintroducedbyPlotkin.Weincludealsotwosummariesofthethesis
[K77a]and[K79b].
Throughrepresentationtheory,Kaljulaidbecamealsointerestedinautomatatheory,
whichatalaterphasebecamehismainareaofinterest.
Anotherfieldofresearchconcernscombinatorics.
BesidesbeinganoutstandingandmostdedicatedmathematicianUnoKaljulaidwas
also very much interested in the history of mathematics. In particular, he took a vivid
interestinthelifeandworkofthegreat19thcenturyDorpat-TartualgebraistTh.Molien
(seeChapterV). PerhapshesawinMolienakindredsoul,asneitherofthetwogotquite
therecognitionfromtheirAlmaMater, whichtheyforsuredeserved;inMolien’scase,
hehadtogointovoluntaryexileinTomsk,Siberia.
Kaljulaid was also very interested in the teaching and exposition, or populariza-
tion of mathematics; he had several outstanding research students. Some of his more
popular-scientificpapers were published in an Estonian language journal Matemaatika
ja Kaasaeg (Mathematics and Our Age). Amongst there is a whole series of papers
about algebraic matters, culminating in a brilliant, elementary – although partly rather
philosophical–essaydevotedtoGaloistheory[K75a]. Anothersuchseriesishisexcel-
lentessayofDiophantineGeometry[K68a,69b],invariousinstallments,followedbyhis
éloge [K68b]to another of his teachers Yu. I. Manin. We believe that the inclusion of
thesepapersherewillmaketheVolumemoreinterestingforbeginners,andperhapseven
contributetoattractingyoungpeopletomathematics,inEstoniaandelsewhere.
1LateronreferredtoasSeniorEditor.
viii PREFACE
Presentation. ThepapersintheVolumeareassembledinchaptersaccordingtothe
theme.
Importantmattersornotionshaveoften,withsomeconsequence,beensetinitalics,
sometimesupontheirfirstappearance,orelsewheretheyaredefined.
RatherrarequotesinotherlanguagesthanEnglishareusuallyfollowedbyatrans-
lationwithinparentheses.
References to items of Uno Kaljulaid come in the form [Kx], where x (a year) is
taken modulo1900, andrefer to the bibliography. Referencesto other mathematicians
come in the form [y], where y runs through 1,2,3..., independentlyin each separate
paper.
Incase ofbookstranslatedintoRussian, theRussian translationisoftenindicated,
alongwiththeoriginalforthebenefitoftheReadersreadingRussianorhavingaccessto
theRussian book. IntransliteratingtheCyrillic into Englishwe use, with someconse-
quence,thesysteminMathematicalReviews,assetforthonp.1–2ofthebook[1].
Some facts about Estonia and Estonian mathematics. Itshouldperhaps
alsoberecalledherethatEstoniaisthenorthernmostofthethreeBalticRepublics,fac-
ing the Finnish Gulf in the north, borderingto Latvia in the south andto Russia in the
East. Its population is about 1.3 million, most of them Estonians, many living in the
capitalTallinn; there is also a largeRussian speaking minority. The Estoniansspeak a
languagesomewhataffinedtoFinnishandnotatallrelatedtothelanguageoftheirsouth-
ern neighborsthe Latvians and the Lithuanians. Estonianswere mentioned already by
theRomanwriterTacitus(c.55–117)whospokeofthemastheAestorumgentes. How-
ever,aroundthebeginningofthe13thcenturytheEstonianswerestillamongthosefew
peopleinEuropewhohadnotacceptedChristianity. Inadevastatingwar(1208–1227),
againstGerman,SwedishandDanishCrusaders,thenewreligionwasforceduponthem.
The last stronghold of the Estonians, the Castle of Valjala on the island of Saaremaa,
wasconqueredbyaCrusader’sarmy,comingfromPärnuandmarchingoverthefrozen
archipelago, in February, 1227. Then the Estonians became united, together with the
Latvians,inastateruledbytheOrderoftheBrethrenoftheSword,laterknownasthe
TeutonicOrder,whilethenativepopulationcametolive,forcenturies,inserfdom. The
rule of the Order lasted until mid 16th century. At later times, Estonia was governed,
alternatingly,bySwedes, Poles, andRussians. Thesituationoftheindigenousdeterio-
rated ever more and was particularly low towards the end of the 18th century, farmers
were freely sold to the highest bidding landowner; one could even draw a parallel to
the Belgian Congo at a much later epoch. However, in the mid of the 19th century a
nationalawakeningtookplace. Afterhardstruggles,theEstoniansmanagedtoforman
independentcountryoftheir ownin 1918–20,in the aftermathof World War, whenall
empirescollapsed, theRussian oneincluded. Inthe adventofthe Molotov-Ribbentrop
treatyinAugust,1939itwasannexedbytheSovietUnioninJune,1940,andregained
itsindependencein1991,duringthefalloftheSovietempire.
Formoredetailsabouttheabove,andalsosomeinformationaboutmathematicsin
Estonia until 1940, with a tradition going back to the Academia Gustaviana in Tartu,
foundedbytheSwedishKingGustavusAdolphusin1632,closeddownin1656,when
the city was captured by the Russians, and then followed by the Academia Carolina
PREFACE ix
(1690–1710)2,werefertoanarticlebyÜloLumiste,nestorofEstonianmathematicians,
in the book [2]. After a long interregnum the university was reopened in 1802, under
the auspices of czar Alexander I; Estonia was now a part of the Russian Empire, the
university’s official name being Kaiserliche Universität zu Dorpat, as the language of
teachingwasGerman.
Acknowledgements. The appearanceof the presentcompilationwould nothave
been possible without the generous assistance of a large number of friends and col-
leagues, students, secretaries, librarians, family members, etc. – from Novosibirsk in
the East to Iowa in the West. To all of them we express here our sincere thanks. The
followinglistofnames(inalphabeticorder)comprisesprobablyonlyafractionofall:
Gert Almkvist, Marianne Blauert, Leonid Bokut, Kerstin Brandt,
Michael Cwikel, Martina Eicheldinger, Miroslav Engliš, Jan Gus-
tavsson,LászlóFilep,EilaRitvaJansson,MargrethJohnsson,Kalle
Kaarli, Dan and Christer Kiselman, Andi Kivinukk, Richard Koch,
Petr Krylov, Ruvim Lipyanskiˇı, Indrek Martinson, Caroline Myr-
berg,AleksandrNikolskii,Inga-BrittPeetre,Jakob-SebastianPeetre,
MonikaPerkmann,Ann-ChristinPersson,UlfPersson,ProfessorPa-
ter Anders Piltz O.P., Boris Plotkin, Olga Sokratova, Sven Spanne,
GunnarSparr,MichaelDavidSpivak,AnnikaTallinn,HellisTamm,
Marje Tamm, Enn Tamme, Erki Tammiksaar, Gunnar Traustason,
MichaelTsfasman,VictorUfanrovski,AleksandrZubkov.
Amongstinstitutions,wementioninparticularthefollowing:
EestiLoodusuurijateSelts(EstonianNaturalists’Society,Tartu,Es-
tonia);VerlagHeyn(Klagenfurt,Austria).
Wehavehadaninvaluableaidfrommanylibraries,amongstothers:
MathematicallibrariesofLund,andtheoneofUppsala(namedthe
Beurling library); Lund University, Giesen, and Heidelberg; the li-
brary of the Mittag-Leffler Institute; the library of the Institute of
CyberneticsatTallinnUniversityofTechnology.
Finally,weexpressourgreatesteemforthegenerosityofoursponsors,theRoyalPhysio-
graphicSocietyofLund,takingoverallcostsofpublicationandtheEuropeanUnion’s
Fifth Framework Programme project IST-2001-37592 (eVikings II) that partially sup-
portedtheeditingofthisbookandtherelatedvisitsofJaanPenjamtoLund.
The Editors
References
[1] A.,J.Lohwater.Russian-EnglishDictionaryofthemathematicalsciences.AmericanMathematicalSoci-
ety,Paris,1961.
[2] Ü.LumisteandJ.Peetre.EdgarKrahn,Acentenaryvolume1894–1961.IOSPress,Providence,Rhode
Island,1994.
2Probably,fewmathematicians areawareofthatthefirstevertoteachaboutNewton’scosmologywas
theSwedeSvenDimberginTartu[3].
x PREFACE
[3] Ü.LumisteandH.Piirimäe. Newton’s Principiainthecurricula oftheUniversity ofTartu(Dorpat)in
theearly1690’s.In:R.Vihalemm(ed.),Estonianstudiesinthehistoryandphilosophyofscience.Kluwer
AcademicPublishers,Dordrecht,Boston,NewYorkandLondon,2001,1–18.Swedishtranslation,based
onenlaged1981Estonianversion:J.Peetre–S.Rodhe,Normat(toappear).