Simple Relation Algebras

629 Pages·2017·8.91 MB·English

by  Steven Givant

Steven Givant · Hajnal Andréka
Simple
Relation
Algebras
Simple Relation Algebras
Steven Givant • Hajnal Andre´ka
Simple Relation Algebras
123
StevenGivant HajnalAndre´ka
DepartmentofMathematics Alfre´dRe´nyiInstituteofMathematics
MillsCollege InstituteofMathematics
Oakland,CA,USA HungarianAcademyofSciences
Budapest,Hungary
ISBN978-3-319-67695-1 ISBN978-3-319-67696-8 (eBook)
DOI10.1007/978-3-319-67696-8
LibraryofCongressControlNumber:2017955544
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Preface
Thetheoryofrelationalgebrasisanabstract,finitelyaxiomatizableversionofthe
calculusofrelations,whichinturnisanalgebraictheoryofbinaryrelationsdating
back in origins to the second half of the nineteenth century. The theory was cre-
ated by Alfred Tarski in 1941 and initially developed by Tarski and his students,
J.C.C.McKinsey,BjarniJo´nsson,RogerLyndon,andDonaldMonk.
Oneoftheimportantresultsofthetheoryisthateveryrelationalgebraisasub-
directproductofrelationalgebrasthataresimpleintheclassicsenseoftheword,
thatistosay,theyhaveexactlytwoideals,theimproperidealandthetrivialideal.
In analogy with the program of classifying the finite simple groups, the program
suggestsitselfofattemptingtoclassifythesimplerelationalgebras,oratleastthe
finite,simplerelationalgebras.Thepurposeofthisbookistogiveanexpositionof
severaldifferentmethodsforconstructingsimplerelationalgebras—someofthem
older,butmostofthemnewwiththisbook—and,inparticular,todemonstratethat
theseseeminglydifferentmethodsarereallyalldifferentaspectsofonegeneralap-
proachtoconstructingsimplerelationalgebras.Severaldifferentapplicationsofthe
methodaregiven.Abroadsketchofthemethodanditsapplicationsisgiveninthe
Introduction.
Intendedaudience
Thebookwillbeofinterestnotonlytomathematicians,inparticularthoseinterested
inlogic,algebraiclogic,oruniversalalgebra,butalsotophilosophersandtheoreti-
calcomputerscientistsworkinginfieldsthatusemathematics.Forthatreason,ithas
beenwritteninacarefulanddetailedstylesoastomakethematerialaccessibleto
asbroadanaudienceaspossible.Inparticular,thebackgroundinrelationalgebras
thatisneededtoreadthisbookisgiveninanappendix.
The book contains more than 400 exercises, some of them routine to help the
readergraspthematerial,othersquitedifficult.Hintsandsolutionstosomeofmore
challengingexercisesaregiveninanappendix.
vii
viii Preface
Acknowledgements
The first author learned the theory of relation algebras from Alfred Tarski in an
inspiring course given in 1970 at the University of California at Berkeley, and
throughlatercollaborationwithTarskioveraperiodoftenyears.TheworkofBjarni
Jo´nsson, his great interest in relation algebras and, more generally, universal alge-
bra, and his open and stimulating discussions with the authors have motivated a
numberoftheresultsinthisbook.MostoftheresultsinChapter3(thoughnotthe
particulardevelopmentandpresentationadoptedthere)areduetohim,anddiscus-
sionswithhimledthefirstauthortothediscoveryoftheresultsinChapter5.Roger
Maddux’s work has also had a significant impact on the development of some of
the material in the book. In particular, one of his papers was the original stimulus
for the research that culminated in the definitions and theorems in Chapter 8 and
thefirsthalfofChapter9.ItwasMadduxwhocalledourattentiontothethesisof
MohamedElBachraoui[13],andthisthesiswasthedirectinspirationfortheresults
inChapter6.Theauthorsaredeeplyindebtedtoalltheseindividuals.
WewouldalsoliketoexpressourindebtednesstoLorettaBartoliniandherentire
productionteamatSpringerforpullingoutallstops,anddoingthebestpossiblejob
in the fastest possible way, to produce this volume. Loretta served as the editor of
this volume, and her constant encouragement, help, and support made the process
ofpublishingthisbookmuch,mucheasierthanitmighthaveotherwisebeen.Any
errorsorflawsthatremaininthevolumesare,ofcourse,ourownresponsibility.
Finally, we would like to acknowledge the mutual interests, support, and years
ofdevotedfriendshipthattheauthorshavesharedwitheachotherandwithIstva´n
Ne´meti.Withoutthese,thisbookwouldnotexist.
TheworkofAndre´kawassupportedbytheAlfre´dRe´nyiInstituteofMathemat-
ics,HungarianAcademyofSciences,underthefollowinggrants:T43242,T30314,
T35192 and COST Project 274. The typing of the manuscript for the monograph
wassupportedbyagrantfromMillsCollege.
Oakland,CA,USA StevenGivant
Budapest,Hungary HajnalAndre´ka
Introduction
Thecalculusofrelationsisamathematicaltheoryofbinaryrelationsthatisoverone
hundredandfiftyyearsold,datingbacktothemiddleofthenineteenthcentury.The
firstpaperonthesubjectwaswrittenbyDeMorgan[11].Aproperfoundationfor
thetheorywasformulatedbyPeircein[45],afterseveralearlierattempts.Schro¨der
systematicallydevelopedPeirce’stheoryintheextensivemonograph[50].In1903,
Russell[48]couldwrite
Thesubjectofsymboliclogicisformedbythreeparts:thecalculusofpropositions,the
calculusofclasses,andthecalculusofrelations.
Theobjectsofstudyofthecalculusofrelationsarealgebrasofbinaryrelations,
consistingofauniverseofbinaryrelationsonsomebaseset,togetherwiththeopera-
tionsofunion,intersection,complement,relationalcomposition,relationaladdition
(thedualofrelationalcomposition),andrelationalinverse(orconverse).Thereare
alsodistinguishedconstants:theemptyrelation,theuniversalrelation,theidentity
relation, and the diversity relation on the base set. (Some of these operations and
constants—forinstance,theoperationofrelationaladdition—aredefinableinterms
oftheothersandmaythereforenotbeincludedinthelistofprimitivenotions.)
The calculus of relations was given an axiomatic foundation by Tarski, in the
spirit of general algebra, starting with the 1941 paper [55]. The original axioma-
tization was not equational, but Tarski realized that an equational axiomatization
was possible (see the remark at the top of p.87 in op.cit.). Within a short time he
producedaveryelegantone(theoneusedin[58])andsucceededinshowing(some-
timeintheperiod1942–44)thatthissimple,finitelyaxiomatized,equationaltheory
providesasufficientlyrichformalismforthedevelopmentofallofclassicalmathe-
matics.(Thistheoremanditsconsequenceswereeventuallypublishedin[58].)
Tarskiposedtwofundamentalquestionsin[55]regardinghisfoundationofthe
calculusofrelations(orthetheoryofrelationalgebras,asiteventuallycametobe
called).Thefirstconcernedcompleteness:iseverytrueequationderivablefromthe
setofaxioms?Thesecondconcernedtheexistenceofageneralrepresentationtheo-
rem:iseveryabstractmodelofthetheoryisomorphictoaconcretemodelofbinary
relations?Success eluded himinhispursuitoftheanswerstothesequestions,but
ix
x Introduction
alongthewayheandhiscollaboratorsMcKinseyandJo´nssonwereabletoestablish
someimportantalgebraicpropertiesofrelationalgebras.Firstofall,everyrelation
algebra is semi-simple in the sense that it is a subdirect product of simple relation
algebras(see[29]andTheorems4.10and4.14in[31]forstatementsofcloselyre-
latedresults).Second,everyrelationalgebracanbeextendedtoacompleteatomic
relationalgebra,andtheextensionissimpleorintegraljustincasetheoriginalalge-
braissimpleorintegral(seeTheorem4.21in[31]).(Arelationalgebraissaidtobe
integralifthecompositionoftwonon-zeroelementsisalwaysnon-zero.)Thesetwo
theoremsprovidedthefoundationforfuturealgebraicinvestigations.Theyindicate
clearly the important role that simple relation algebras, particularly those that are
completeandatomic,playinthetheory.
In 1950, Lyndon [33] proved that the answer to both of Tarski’s questions is
negative.HegaveanexampleofatrueequationthatisnotderivablefromTarski’s
axioms,andhegaveanexampleofafiniterelationalgebrathatisnotrepresentable
asanalgebraofbinaryrelations.Thisoriginalexampleofanon-representablerela-
tionalgebrawasnoteasytounderstand.In[35],buildingonearlierworkofJo´nsson
[26],Lyndonshowedhowtoconstructrelationalgebrasascomplexalgebras(alge-
brasofallsubsets,orcomplexes)ofarbitraryprojectivegeometries,andprovedthat
manyofthealgebrasconstructedfromprojectivelinesarenotrepresentable.Monk
usedthesealgebrasin[41]todemonstratethatnofinitesetofpostulatesissufficient
toaxiomatizetheclassofrepresentablerelationalgebras.Asaconsequence,nofi-
nitesetofpostulatesisstrongenoughtoderivealltrueequationsinthecalculusof
relations.
BeforethenegativeresultsofLyndonwerediscovered,Tarskiestablishedapar-
tialrepresentationtheoremfortheclassofallrelationalgebras,andsomefullrep-
resentationtheoremsforrestrictedclassesofrelationalgebras.Forinstance,inThe-
orem 4.29 of [31] it is shown that every atomic relation algebra with functional
atoms—atomsthatsatisfyacertaincharacteristicequationalpropertyoffunctions—
isrepresentable.Anumberofothersuchtheoremshaveappearedovertheyears,for
instance in [31, 58, 27, 37, 49, 14], and [12]. Tarski several times expressed the
opinionthatthenegativeresultsofLyndonenhancedtheinterestofrepresentation
theoremsforspecializedclassesofrelationalgebras.
Through the work of Tarski, Jo´nsson, Lyndon, Monk, and others (including a
number of their students), there has gradually arisen a general algebraic theory of
relation algebras, similar in spirit to group theory and the theory of Boolean al-
gebras, with many applications to logic, computer science, and other domains of
inquiry(see[24,38],or[18]and[19]).
Simplerelation algebras
Oneofthefundamentalproblemsofanyalgebraictheoryistheanalysisofitsmod-
els,andinparticulartheanalysisofthebasicalgebrasthatformthebuildingblocks
for the models of the theory. In group theory, for instance, manifestations of this
Introduction xi
analysis include the classification of finite, simple groups and the classification of
finiteabeliangroups.InBooleanalgebra,onemightmentionthetheoremthatevery
Booleanalgebraisasubdirectproductofthetwo-elementBooleanalgebra(which
is the only simple Boolean algebra), and every finite Boolean algebra is a direct
productofthetwo-elementBooleanalgebra.Thebasicbuildingblocksofrelation
algebrasaresimplerelationalgebras,particularlyatomicones.Severalquiteinter-
estingclassesofsimple,atomicrelationalgebrashavebeenstudiedovertheyears.
For example, McKinsey showed that the complex algebra of a group is always a
simple, atomic, representable relation algebra (see [29] and [31]). As was already
mentioned,Lyndonshowedthatthecomplexalgebraofaprojectivegeometryisa
simple, atomic relation algebra (see [26] and [35]). Maddux [36] proved that the
complexalgebraofamodularlatticewithzeroisasimple,atomicrelationalgebra.
Inanalogywithgrouptheory,theprogramnaturallysuggestsitselfofclassifying
allsimplerelationalgebras,oratleastallfinite,simplerelationalgebras.Thispro-
grammightbeinterpretedinseveraldifferentways,butthegoalistocomeupwith
awell-definedclassof“basic”simplerelationalgebras,andafinitelistofconstruc-
tiontechniquessuchthateveryfinite,simplerelationalgebraisobtainablefromthe
basic examples by applying one or more of the construction techniques, perhaps
insomespecificorder.Asaconcreteexampleofaninterpretationofthisprogram,
considertheclassoffinite,integralrelationalgebras,thatistosay,relationalgebras
offinitecardinalitygreaterthanone,suchthattherelativeproductoftwoelements
is zero if and only if one of the elements is zero. All such algebras are known to
besimple.Onemayaskifthereisalistoffinitelymanyconstructiontechniquesso
thateveryfinite,simplerelationalgebracanbeconstructedfromtheclassoffinite
integral relation algebras using one or more of these techniques. In other words,
theprogrammaybeviewedasreducingtheproblemofconstructingandanalyzing
all finite, simple relation algebras to that of constructing and analyzing all finite,
integralrelationalgebras.
This book can be viewed as a contribution to this program. It presents a fairly
generalmethodfordevelopingtoolstoconstructandanalyzesimplerelationalge-
bras,itgivessomeexamplesofhowthisgeneralmethodcanleadtoconstructions
thatareinteresting,beautiful,anduseful,andhowtheseconstructionscan,inturn,
be used to prove significant theorems. In each of these theorems, the method is
indeed used to reduce the result for simple relation algebras to the corresponding
resultforintegralrelationalgebras.
The intuitive idea behind the method is rather easy to understand from the per-
spectiveofanalyzingsimplerelationalgebras. SupposetheBooleanpartofasimple
relationalgebra—theuniverseofthealgebratogetherwiththeBooleanoperations—
hasbeendecomposedintothedirectproductofcomponent (orfactor)Booleanal-
gebras. The extra-Boolean operations of converse and relative multiplication are
distributive, in each coordinate, over arbitrary Boolean sums. Therefore, their be-
havioronthewholeuniverseiscompletelydeterminedbytheirbehavioronandbe-
tweentheindividualcomponentsoftheBooleandecomposition.Suchasemiproduct
decomposition—aBooleandirectproductdecomposition,togetherwithcompletely
distributive extra-Boolean operators (of converse and relative multiplication)—is