Finite Structures with Few Types

200 Pages·2003·0.892 MB·English

by  Gregory Cherlin, Ehud Hrushovski

Annals of Mathematics Studies
Number 152

Finite Structures with Few Types
by
Gregory Cherlin and Ehud Hrushovski
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2003
iv Replacebyfinalversion
Copyright(cid:13)c 2003byGregoryCherlinandEhudHrushovski.
PublishedbyPrincetonUniversityPress,41WilliamStreet,
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IntheUnitedKingdom:PrincetonUniversityPress,
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TheAnnalsofMathematicsStudiesareeditedby
JohnN.MatherandEliasM.Stein
LibraryofCongressCataloging-in-PublicationData
Cherlin,GregoryL.,1948-
Finitestructureswithfewtypes/byGregoryCherlinandEhudHrushovski.
p.cm.–(Annalsofmathematicsstudies;no.152)
Includesbibliographicalreferencesandindex.
ISBN0-691-11331-0(alk.paper)–ISBN0-691-11332-7(pbk.:alk.paper)
1.Modeltheory.2.Grouptheory.I.Hrushovski,Ehud,
1959-II.Title.III.Series.
QA9.7.C482003
511.3—dc21 200202980
Thepublisheracknowledgestheauthorsforprovidingthe
camera-readycopyfromwhichthisbookwasprinted.
GC acknowledges support by NSF Grants DMS 8903006, 9121340, and 9501176. Support by MSRI, Berkeley and the
MathematicsInstitituteoftheHebrewUniversity,Jerusalemisgratefullyacknowledged.
EHacknowledgesresearchsupportbyMSRI,BerkeleyandbygrantsfromtheNSFandISF.
Printedonacid-freepaper.∞
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PrintedintheUnitedStatesofAmerica
10987654321
Contents
1. Introduction 1
1.1 TheSubject 1
1.2 Results 4
2. Basic Notions 11
2.1 FinitenessProperties 11
2.1.1 Quasifiniteness,weakorsmoothapproximability 11
2.1.2 Geometries 13
2.1.3 Coordinatization 16
2.2 Rank 18
2.2.1 Therankfunction 18
2.2.2 Geometries 20
2.2.3 Adigression 22
2.3 ImaginaryElements 23
2.4 Orthogonality 31
2.5 CanonicalProjectiveGeometries 36
3. Smooth Approximability 40
3.1 Envelopes 40
3.2 Homogeneity 42
3.3 FiniteStructures 46
3.4 OrthogonalityRevisited 50
3.5 LieCoordinatization 54
4. Finiteness Theorems 63
4.1 GeometricalFiniteness 63
4.2 Sections 67
4.3 FiniteLanguage 71
4.4 QuasifiniteAxiomatizability 75
4.5 Ziegler’sFinitenessConjecture 79
vi CONTENTS
5. Geometric Stability Generalized 82
5.1 Typeamalgamation 82
5.2 Thesizesofenvelopes 90
5.3 NonmultidimensionalExpansions 94
5.4 CanonicalBases 97
5.5 Modularity 101
5.6 LocalCharacterizationofModularity 104
5.7 ReductsofModularStructures 107
6. Definable Groups 110
6.1 GenerationandStabilizers 110
6.2 ModularGroups 114
6.3 Duality 120
6.4 RankandMeasure 124
6.5 TheSemi-DualCover 126
6.6 TheFiniteBasisProperty 134
7. Reducts 141
7.1 RecognizingGeometries 141
7.2 ForgettingConstants 149
7.3 DegenerateGeometries 153
7.4 ReductswithGroups 156
7.5 Reducts 164
8. Effectivity 170
8.1 TheHomogeneousCase 170
8.2 Effectivity 173
8.3 DimensionQuantifiers 178
8.4 RecapitulationandFurtherRemarks 183
8.4.1 Theroleoffinitesimplegroups 183
REFERENCES 187
INDEX 191
1
Introduction
1.1 THESUBJECT
In the present monograph we develop a structure theory for a class of finite
structures whose description lies on the border between model theory and
grouptheory. Modeltheoretically, westudylargefinitestructuresforafixed
finitelanguage, withaboundednumberof4-types. Ingrouptheoreticterms,
westudyallsufficientlylargefinitepermutationgroupswhichhaveabounded
numberoforbitson4-tuplesandwhicharek-closedforafixedvalueofk.The
primitive case is analyzed in [KLM; cf. Mp2]. The treatment of the general
caseinvolvestheapplicationofmodeltheoreticideasalonglinespioneeredby
Lachlan.
Weshowthatsuchstructuresfallintofinitelymanyclassesnaturallyparam-
etrized by “dimensions” in the sense of Lachlan, which approximate finitely
many infinite limit structures (a version of Lachlan’s theory of shrinking and
stretching), and weprove uniform finite axiomatizabilitymodulo appropriate
axioms of infinity (quasifinite axiomatizability). We also deal with issues of
effectivity. At our level of generality, the proofs involve the extension of the
methodsofstabilitytheory—geometries,orthogonality,modularity,definable
groups—to this somewhat unstable context. Our treatment is relatively self-
contained, although knowledge of the model theoretic background provides
considerable motivation for the results and their proofs. The reader who is
moreinterestedinthestatementofpreciseresultsthaninthemodeltheoretic
backgroundwillfindtheminthenextsection.
On the model theoretic side, this work has two sources. Lachlan worked
out the theory originally in the context of stable structures which are homo-
geneousforafiniterelationallanguage[La],emphasizingtheparametrization
by numerical invariants. Zilber, on the other hand, investigated totally cate-
goricalstructuresanddevelopedatheoryoffiniteapproximationscalled“en-
velopes,” inhisworkontheproblemsoffiniteaxiomatizability. Theclassof
ℵ -categorical, ℵ -stable structures provides a broad model theoretic context
0 0
towhichbothaspectsofthetheoryarerelevant. Thetheorywasworkedoutat
thislevelin[CHL],includingtheappropriatetheoryofenvelopes. Thesewere
used in particular to show that the corresponding theories are not finitely ax-
2 INTRODUCTION
iomatizable,byZilber’smethod. Thebasictoolusedin[CHL],inaccordance
withShelah’sgeneralapproachtostabilitytheoryandgeometricalrefinements
duetoZilber,wasa“coordinatization”ofanarbitrarystructureintheclassby
atreeofstandardcoordinategeometries(affineorprojectiveoverfinitefields,
ordegenerate. Otherclassicalgeometriesinvolvingquadraticformswerecon-
spicuousonlybytheirabsenceatthispoint.
The more delicate issue of finite axiomatizability modulo appropriate “ax-
iomsofinfinity,”whichiscloselyconnectedwithotherfinitenessproblemsas
wellasproblemsofeffectivity,tooksometimetoresolve. In[AZ1]Ahlbrandt
andZieglerisolatedtherelevantcombinatorialpropertyofthecoordinatizing
geometries, whichwerefertohereas“geometricalfiniteness,” anduseditto
prove quasifinite axiomatizability in the case of a single coordinatizing geo-
metry. Thecaseofℵ -stable,ℵ -categoricalstructuresingeneralwastreated
0 0
in[HrTC].
The class of smoothly approximable structures was introduced by Lachlan
asanaturalgeneralizationoftheclassofℵ -categoricalℵ -stablestructures,in
0 0
essencetakingthetheoryofenvelopesasadefinition. Smoothlyapproximable
structuresareℵ -categoricalstructureswhichcanbewellapproximatedbyfi-
0
nitestructuresinasensetobegivenpreciselyin§2.1.Oneoftheachievements
ofthestructuretheoryforℵ -categoricalℵ -stabletheorieswastheproofthat
0 0
they are smoothly approximable in Lachlan’s sense. While this was useful
modeltheoretically,Lachlan’spointwasthatindealingwiththemodeltheory
of large finite structures, one should also look at the reverse direction, from
smoothapproximabilitytothestructuretheory. Weshowhere,confirmingthis
notveryexplicitlyformulatedconjectureofLachlan,thatthebulkofthestruc-
turetheoryappliestosmoothlyapproximablestructures, oreven, asstatedat
the outset, to sufficiently large finite structures with a fixed finite language,
havingaboundednumberof4-types.
Lachlan’s project was launched by Kantor, Liebeck, and Macpherson in
[KLM] with the classification of the primitive smoothly approximable struc-
turesintermsofvariousmoreorlessclassicalgeometries(theleastclassical
beingthe“quadratic”geometryincharacteristic2,describedin§2.1.2). These
turnupinprojective,linear,andaffineflavors,andintheaffinecasethereare
someadditionalnonprimitivestructuresthatplaynorolein[KLM]butwillbe
needed here (“affine duality,” §2.3). Bearing in mind that any ℵ -categorical
0
structure can be analyzed to some degree in terms of its primitive sections,
the results of [KLM] furnish a rough coordinatization theorem for smoothly
approximable structures. This must be massaged a bit to give the sort of co-
ordinatization that has been exploited previously in an ω-stable context. We
willrefertoastructureas“Liecoordinatizable”ifitisbi-interpretablewitha
structurewhichhasanicecoordinatizationofthetypeintroducedbelow. Lie
coordinatizabilitywillprovetobeequivalenttosmoothapproximability,inone
directionlargelybecauseof[KLM],andintheotherbytheanalogofZilber’s
THESUBJECT 3
theory of envelopes in this context. One tends to work with Lie coordinatiz-
abilityasthebasictechnicalnotioninthesubject. Theanalysisin[KLM]was
infactcarriedoutforprimitivestructureswithaboundonthenumberoforbits
on5-tuples,andin[Mp2]itwasindicatedhowtheproofmaybemodifiedso
as to work with a bound on 4-tuples. (Using only [KLM], we would also be
forcedtostateeverythingdoneherewith5inplaceof4.)
Inmodeltheory,techniquesforgoingfromagooddescriptionofprimitive
piecestomeaningfulstatementsaboutimprimitivestructuresgenerallyfallun-
dertheheadingof“geometricalstabilitytheory,”whoserootslieinearlywork
of Zilber on ℵ -categorical theories, much developed subsequently. Though
1
the present theory lies slightly outside stability theory (it can find a home in
themorerecentdevelopmentsrelatingtosimpletheories),geometricalstability
theoryprovidedaveryusefultemplate[Bu,PiGS].
Before entering into greater detail regarding the present work, we make
somecommentsontheGaloiscorrespondencebetweenstructuresandpermu-
tationgroupsimplicitintheabove,andonitslimitations.
LetX beafiniteset. ThereisthenaGaloiscorrespondencebetweensub-
groupsofthesymmetricgroupSym(X)onX,andmodeltheoreticstructures
with universe X, associating to a permutation group the invariant relations,
and to a structure its automorphism group. This correspondence extends to
ℵ -categoricalstructures([AZ1,Introduction],[CaO]).
0
Whenweconsiderinfinitefamiliesoffinitestructuresingeneral, orapas-
sagetoaninfinitelimit,thiscorrespondenceisnotwellbehaved. Forinstance,
theautomorphismgroupofalargefiniterandomgraphofordern(withcon-
stant and nontrivial edge probability) is trivial with probability approaching
1 as n goes to infinity, while the natural model theoretic limit is the random
countablegraph,whichhasmanyautomorphisms.
It was shown in [CHL], building on work of Zilber for totally categorical
structures, thatstructureswhicharebothℵ -categoricalandℵ -stablecanbe
0 0
approximated by finite structures simultaneously in both categories. Lachlan
emphasizedtheimportanceofthisproperty,whichwillbedefinedpreciselyin
§2.1,andproposedthattheclassofstructureswiththisproperty,thesmoothly
approximablestructures,shouldbeamenabletoastrongstructuretheory,ap-
propriately generalizing [CHL]. Moreover, Lachlan suggested that the direc-
tion of the analysis can be reversed, from the finite to the infinite: one could
classify the large finite structures that appear to be “smooth approximations”
toaninfinitelimit, orinotherwords, classifythefamiliesoffinitestructures
which appear to be Cauchy sequences both as structures and as permutation
groups.ThislineofthoughtwassuggestedbyLachlan’sworkonstablefinitely
homogeneousstructures[La],muchofwhichpredatestheworkin[CHL],and
providedanadditionalideologicalframeworkforthatpaper.
Inthecontextofstablefinitelyhomogeneousstructuresthisanalysisinterms
offamiliesparametrizedbydimensionswascarriedoutin[KL](cf. [CL,La]),
4 INTRODUCTION
butwasnotknowntogothrougheveninthetotallycategoricalcase. Harring-
tonpointedoutthatthisreversalwouldfollowimmediatelyfromcompactness
ifonewereabletoworksystematicallywithinanelementaryframework[Ha].
Thisideaisimplementedhere: wewillreplacetheoriginalclassof“smoothly
approximable structures” by an elementary class, a priori larger. Part of our
effortthengoesintodevelopingthestructuretheoryfortheostensiblybroader
class.
From the point of view of permutation group theory, it is natural to begin
the analysis with the case of finite primitive structures. This was carried out
usinggrouptheoreticmethodsin[KLM],andwerelyonthatanalysis. How-
ever, there are model theoretic issues which are not immediately resolved by
such a classification, even for primitive structures. For instance, if some fi-
nitegraphsG areassumedtobeprimitive,andtohaveauniformlybounded
n
numberof4-types,ourtheoryshowsthatanultraproductG∗ oftheG isbi-
n
interpretablewithaGrassmannianstructure,whichdoesnotappeartofollow
from[KLM]bydirectconsiderations. ThepointhereisthatifG is“thesame
n
as” a Grassmannian structure in the category of permutation groups, then it
is bi-interpretable with such a structure on the model theoretic side. To deal
with families, one must deal (at least implicitly) with the uniformity of such
interpretations;see§8.3,andthesectionsonreducts. Itisnoteworthythatour
proofinthiscaseactuallypassesthroughthetheoryforimprimitivestructures:
anynonuniforminterpretationofaGrassmannianstructureonG givesriseto
n
a certain structure on G∗, a reduct of the structure which would be obtained
from a uniform interpretation, and one argues that finite approximations (on
themodeltheoreticside)toG∗wouldhavetoomanyautomorphisms. Inother
words, we can obtain results on uniformity (and hence effectivity) by ensur-
ingthattheclassforwhichwehaveastructuretheoryisclosedunderreducts.
This turns out to be a very delicate point, and perhaps the connection with
effectivityexplainswhyitshouldbedelicate.
1.2 RESULTS
Arapidbutthoroughsummaryofthistheorywassketchedin[HrBa],withoc-
casionalinaccuracies. Foreaseofreferencewenowrepeatthemainresultsof
thetheoryaspresentedthere,makinguseofaconsiderableamountofspecial-
izedterminologywhichwillbereintroducedinthepresentwork. Thevarious
finitenessconditionsreferredtoareallgiveninDefinition2.1.1.
Theorem1(StructureTheory)
Let M be a Lie coordinatizable structure. Then M can be presented in
afinitelanguage. AssumingMissopresented,therearefinitelymanyde-
finabledimensioninvariantsforMwhichareinfinite,uptoequivalenceof