ON ALMOST ORTHOGONALITY IN SIMPLE THEORIES

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ON ALMOST ORTHOGONALITY IN SIMPLE THEORIES ITAY BEN-YAACOV AND FRANK O. WAGNER Abstract. 1. We show that if p is a real type which is internal in a set ? of partial types in a simple theory, then there is a type p? interbounded with p, which is finitely generated over ?, and possesses a fundamental system of solutions relative to ?. 2. If p is a possibly hyperimaginary Lascar strong type, almost ?-internal, but al- most orthogonal to ??, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while fixing ? generically. In case p is ?-internal and T is stable, this is the binding group of p over ?. Introduction In this paper we shall study the interaction of a type p (over some set A in a simple theory) with a family ? of partial types over A. Recall that p is (1) (almost) ?-internal if for every realization a of p there are B |^ A a and real- izations c¯ of types in ? over B, such that a ? dcl(Bc¯) (resp. a ? bdd(Bc¯)). (2) (almost) generated over ? if there is B ? A such that for any realization a of a p there are realizations c¯ of types in ? over B with a ? dcl(Bc¯) (resp.

• let


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• uniformly algebraic over


• over ?


• ?c¯ c?


• has been shown


• then there


• almost ?-internal


• c¯ ?

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 31
• Langue: English

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ONALMOSTORTHOGONALITYINSIMPLETHEORIES ITAYBEN-YAACOVANDFRANKO.WAGNER Abstract. 1.Weshowthatif p isarealtypewhichisinternalinasetΣofpartial typesinasimpletheory,thenthereisatype p 0interboundedwith p ,whichisfinitely generatedoverΣ,andpossessesafundamentalsystemofsolutionsrelativetoΣ. 2.If p isapossiblyhyperimaginaryLascarstrongtype,almostΣ-internal,butal- mostorthogonaltoΣ ω ,thenthereisacanonicalnon-trivialalmosthyperdefinable polygroupwhichmulti-actson p whilefixingΣgenerically.Incase p isΣ-internal and T isstable,thisisthebindinggroupof p overΣ. Introduction Inthispaperweshallstudytheinteractionofatypep(oversomesetAinasimple theory)withafamilyΣofpartialtypesoverA.Recallthatpis (1)(almost)Σ-internalifforeveryrealizationaofpthereareB ^ | A aandreal- izationsc¯oftypesinΣoverB,suchthata ∈ dcl(Bc¯)(resp.a ∈ bdd(Bc¯)). (2)(almost)generatedoverΣifthereisB ⊇ Asuchthatforanyrealizationa ofaptherearerealizationsc¯oftypesinΣoverBwitha ∈ dcl(Bc¯)(resp. a ∈ bdd(Bc¯)). Inastabletheoryinternalityandfinitegenerationarethesame,andareanimportant toolintheanalysisofastructure(forinstanceinHrushovski’sproofthatunidimen- sionalstabletheoriesaresuperstable).Pillayhasgivenexamplesofsimpletheories (evenofSU-rank1)wheretheydiffer[SW02,Examples2and3].Thewayoutseemsto bealmostinternalityandalmostgeneration,astheyagreeinanysimpletheory.How- ever,definableasopposedtoalgebraicclosureplayedanimportantroˆleinthedefinition ofthebindinggroupofpoverΣ,namelythegroupAut(p/A ∪ Σ)ofallpermutationsof therealizationsofpinducedbyautomorphismsfixingAandallrealizationsofΣ.Ifp isΣ-internal,thisgroupanditsactiononparedefinableinthestablecase;moreover theactionistransitiveifpisastrongtypealmostorthogonaltoΣoverA.Formore details,thereadermayconsult[Bue96,Section4.4],[Pil96,Section7.4],and[Poi87, Section2.e]. OurTheorem1.2improvesTheorem6of[SW02].Recallthatforp∈S(A)andΣa familyofpartialtypesoverA,a(weak)fundamentalsystemofsolutionsforpoverΣ isatuplea¯ofrealizationsofpsuchthateveryrealizationofpisdefinable(bounded) overAa¯togetherwithsomerealizationsofΣ;notethattheexistenceofa(weak) fundamentalsystemofsolutionsimplies(almost)generation.ShamiandWagnershow Date :22January2002. 1991 MathematicsSubjectClassification. 03C46. Atthetimeofthewritingofthispaper,thefirstauthorwasagraduatestudentwiththeE ´ quipe deLogiqueMathe´matiqueofUniversite´ParisVII. ThesecondauthorwouldliketothankBraddHartforinterestingdiscussions. 1