Minimal types in simple theories

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

Minimaltypesinsimpletheories

AnandPillay

UniversityofLeeds

September4,2006

Abstract

WeprovethatifM0isamodelofasimpletheory,andp(x)isacompletetypeofCantor-Bendixonrank1overM0,thenpisstation-aryandregular.AsaconsequenceweobtainanotherproofthatanycountablemodelM0ofacountablecompletesimpletheoryThasin- nitelymanycountableelementaryextensionsuptoM0-isomorphism.Thelatterextendsearlierresultsoftheauthorinthestablecase,andisaspecialcaseofarecentresultofTanovic[4].

1Introduction

Thispaper,whichextendsearlywork[1]oftheauthor,iscloselyrelatedtoandmotivatedbycurrentworkofPredragTanovicontheauthor’soldcon-jecturethatanycountablemodelM0(inacountablelanguage)hasin nitelymanycountablemodelsuptoisomorphismoverM0.In[4]Tanovicprovestheconjecturefortheorieswithoutthestrictorderproperty,andinprivatee-maildiscussionshehasdescribedaroutetothefullconjecture.

Theexpression“minimaltype”inthetitlereferstoatypeofCB-rank1overamodelM0,ratherthantoatypeofSU-rank1.Forp(x)∈S(M0)tobeofCantor-Bendixonrank1meansthatp(x)lythereissomeformulaφ(x)overM0suchthatp(x)isaxiomatizedby{φ(x)}∪{x=a:a∈ SupportedbyaMarieCurieChair

1

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

M0,M0|=φ(a)}.ThesetXde nedinM0bysuchaformulaφ(x)iswhatissometimescalledminimal:namelyitisin nite,buteveryde nable(withparametersinM0)subsetis niteorco nite.IfM0isanin nitestructuresuchthatS(M0)iscountable,thentherewillexist1-typesoverM0ofCB-rank1.In[1]itwasobservedthatifTh(M0)isstableandp(x)∈S(M0)hasCB-rank1thenpisaregulartype:forkingonrealizationsofpinabig¯isapregeometry.HenceifMnisprimeoverM0togetherwithanmodelM

independentset{a1,..,an}ofrealizationsofp,then(bytheopenmappingtheoremandregularity),n=mimplesMnandMmarenotisomorphic(overM0).TheresultsinthispapershowthatthesameconstructionworksifTh(M0)isjustassumedtobesimple.

Ontheotherhand,themaintechnicalresultofthispaper(stationarityofCB-rank1typesovermodelsinsimpletheories)suggeststhatifTisacountablesimpletheorysuchthatforsomecountablemodelM0ofT,S(M0)iscountable,thenTisclosetobeingstable.OfcourseifthemodelM0happenstobeω-saturated(soforexamplewhenTisω-categorical),countabilityofS(M0)directlyyieldsω-stabilityofT.Buthereisanunstableexample(alsosuggestedbyB.Kim):Eisanequivalencerelation,andinM0thereisexactlyoneequivalenceclassCnofeach nitecardinalityn(andnoin niteclasses).AlsoRisabinaryrelation,andthe nitestructures(Cn,R|Cn)approximatetherandomgraph,sothatifCisanin niteE-classinanelementaryextensionMofM0then(C,R|C)istherandomgraph.ThiscanbesetupsothatS(M0)iscountable(andinfactsuchthatthereisauniquenon-algebraic1-typeoverM0).

2Proofofmainresult

AssumeTtobeacompletesimpletheoryinalanguageLofarbitrarycar-dinality.Weassumefamiliaritywiththebasicmachineryofstabilityandsimplicity(see[2]and[5]).Wewillmakeheavyuse(amongotherthings)ofthefactthatpisatypeoveramodelthenanyheirorcoheirofpisanonforkingextension.¯ofT.AllothermodelsofTweWeworkinabigsaturatedmodelM

considerareassumedtobesmallelementarysubmodelsofT.

Proposition2.1LetM0beamodelofT.Letp(x)∈S(M0)beatypeof

2

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

CB-rank1.Thenpisstationary,namelyhasauniquenonforkingextensionoveranysetcontainingM0.

Proof.Wewillassumethatp(x)isnonstationaryandgetacontradiction.ClaimI.ThereisamodelMcontainingM0,anddistinctnonforkingextensionsp (x),p (x)∈S(M)ofp(x)suchthatp (x)isanheirofp.

Proof.Thisisclear,asanyheirofpisanonforkingextensionandweareassumingthatphasdistinctnonforkingextensionsoversomemodel.

ClaimII.ThereisamodelMofTandanonforkingextensionp (x)∈S(M)ofpwhichisnotacoheirofp(namelyisnot nitelysatis ableinM0).¯isProof.IfnotthenanynonforkingextensionofpoverthemonstermodelM¯ nitelysatis able nitelysatis ableinM0.ButthenumberoftypesoverM|M|inM0isboundedby220,sowehaveaboundednumberofnonforkingextensionsofp.Thisimpliesthatpisstationary.(BytheIndependencetheoremforexample:see[3].)

ByClaimIthereisa nitetupleaanddistinctnonforkingextensionsp1(x),q(x)∈S(M0a)ofp(x)suchthatp1isanheirofp.

ByClaimII,letcbea nitetuple(c0,)andr(x)anonforkingexten-sionofpoverM0cwhichisnot nitelysatis ableinM0.

Thereisnoharminextendingc,sowemayassumethatc0realizesp,namely,tp(c0/M0)=p.Also,asbyautomorphismwemayreplacecbyanyrealizationoftp(c/M0),wemayassumethatc0realizesthetypep1(overM0∪{a})mentionedabove.Notethatthentp(a/M0c0)is nitelysatis ableinM0sohasacompleteextensionoverM0cwhichwhichis nitelysatis ableinM0.Thus,byautomorphismagainwemayassumethattp(a/M0c)is nitelysatis ableinM0,namelythattp(c/M0a)isanheiroftp(c/M0).Letussummarisethesituationsofar.Wehavetuplesaandc=(c0,..,cn)¯andcompletetypesq(x)∈S(M0a)andr(x)∈S(M0c)suchthatinM

(i)tp(c0/M0)=p(x),

(ii)tp(c/M0a)isanheiroftp(c/M0),soinparticularcaswellasc0isinde-pendentfromaoverM0.

(iii)q(x)isanonforkingextensionofp(x),andq(x)=tp(c0/M0a).

3

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

(iv)r(x)∈S(M0c)isanonforkingextensionofp(x)butisnot nitelysat-is ableinM0.

BytheIndependenceTheoremoveramodel(forsimpletheories),wecan ndbrealizingbothq(x)andr(x)(suchthatmoreover{a,c,b}isM0-independent).

By(iii)letψ(x,y)beaformulaoverM0suchthat

(v)|=ψ(b,a)∧¬ψ(c0,a).

By(iv)letχ(x,z)beaformulaoverM0suchthat

(vi)|=χ(b,c)andχ(x,c)isnotrealizedinM0.

Letusnow xaformulaφ(x)overM0whichisolatespamongnonalgebraictypesoverM0(whichexistsaspisassumedtohaveCB-rank1).Soby(v)and(vi)weclearlyhave

|=¬ψ(c0,a)∧( x)(φ(x)∧χ(x,c)∧ψ(x,a))

By(ii)thereisa inM0suchthat

( )|=¬ψ(c0,a )∧( x)(φ(x)∧χ(x,c)∧ψ(x,a ))

Soletb ly

|=φ(b )∧χ(b ,c)∧ψ(b ,a )

By(vi)andourassumptiononφweseethattp(b /M0)=p(x).Soψ(x,a )∈p(x).Ontheotherhandby( )and(i),¬ψ(x,a )∈p(x).Thisisacontradictionandprovestheproposition.

Corollary2.2LetM0andp(x)∈S(M0)beasinProposition2.1.Thenp(x)isde nableandregular.Moreoverthenonforkingextensionsofp(x)arepreciselythecoheirsofp(x).

Proof.Themoreoverclauseisclearfromstationarityofp:foranysetA M0phasanonforkingextensionswhichisacoheir,souseuniqueness.

De nabilityisalsoclear:p(x)hasauniqueheiroveranyset(itsnonfork-ingextension)sobyBethde nability,pisde nable(alternativelysee[3]).MoreoverclearlythenonforkingextensionofpoveranyA M0isgivenbyapplyingthede ningschemaofptoA.

4

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

Regularityofp(x)isstandard.Butwegothroughtheproofforcom-pleteness.Wehavetoprove:

(*)IfMisamodelcontainingM0,brealizesaforkingextensionofpoverMandcrealizesa(orratherthe)nonforkingextensionofp(x)overMthencisindependentfromboverM.

Letdbethede ningschemaforp.Asbeforeletφ(x)isolatepamongnonalgebraictypesoverM0.AssumethatcforkswithMboverM0.Soforsomeψ(x,y,z)overM0,anda∈Mwehave|=ψ(c,b,a)∧¬dψ(b,a).AsbforkswithMoverM0wehavesomeχ(x,z )overM0anda ∈Msuchthat|=χ(b,a )∧¬dχ(a ).Sothefollowingholds:

y(φ(y)∧ψ(c,y,a)∧¬dψ(y,a)∧χ(y,a )∧¬dχ(a ))

Thereisnoharminassuminga=a .Astp(c/M0a)istheheiroftp(c/M0)=p,we nda0∈M0andb0suchthat

|=φ(b0)∧ψ(c,b0,a0)∧¬dψ(b0,a0)∧χ(b0,a0)∧¬dχ(a0)

Socforkswithb0overM0wherebyb0∈/M0,soasb0realizesφ,tp(b0/M0)=p(x).Butthenthefactthat|=χ(b0,a0)∧¬dχ(a0)givesacontradiction.Corollary2.3([4])IfTisacountablecompletesimpletheory,andM0acountablemodelofTthenM0hasin nitelymanycountableelementaryextensionsuptoisomorphismoverM0.

Proof.WemayassumeS(M0)tobecountablesocontainsaCB-rank1typep(x).Proposition2.1andCorollary2.2applytop.Leta1,..,anbeindependentrealizationsofp(overM0)andletMnbetheprimemodeloverM0∪{a1,..,an}.Ifb∈Mnrealizespthentp(b/M0,a1,..,an)isisolatedhencebythemoreoverclauseinCorollary2.2forkswitha1,..,anoverM0.Hence{a0,..,an}isamaximalindependent(overM0)setofrealizationsofpinMn.So(byregularity)thedimensionofpinMnisn.Thisconcludestheproof.Question1.SupposeM0isamodelofasimpletheory,andthateverytypeinS(M0)hasCB-rank.IseverycompletetypeoverM0stationary?Theresultsin[1]wereactuallyprovedunderaweakerassumptionthanstabilityofTh(M0).TheassumptionwasthatM0hasnoorder:thereisnoin nitesetoftuplesfromM0totallyorderedbysomeformula.

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We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

Question2.UnderthesameasumptionsasinQuestion1,isitthecasethatM0hasnoorder?

References

[1]A.Pillay,Dimensiontheoryandhomogeneityforelementaryextensions

ofamodel,JournalofSymbolicLogic,47(1982),147-160.

[2]A.Pillay,Anintroductiontostabilitytheory,OxfordUniversityPress,

1983.

[3]A.Pillay,De nabilityandde nablegroupsinsimpletheories,Journal

ofSymbolicLogic,63(1998),788-796.

[4]P.Tanovic,Onconstantsandthestrictorderproperty,Archivefor

Math.Logic,45(2006),423-430.

[5]F.O.Wagner,Simpletheories,Kluwer,2000.

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