aa r X i v : . [ m a t h . L O ] M a y DISTAL AND NON-DISTAL PAIRS
PHILIPP HIERONYMI AND TRAVIS NELL
Abstract.
The aim of this note is to determine whether certain non-o-minimalexpansions of o-minimal theories which are known to be NIP, are also distal.We observe that while tame pairs of o-minimal structures and the real fieldwith a discrete multiplicative subgroup have distal theories, dense pairs ofo-minimal structures and related examples do not. Introduction
Over the last two decades,
NIP (or dependent ) theories, first introduced by She-lah in [17], have attracted substantial interest. Properties of these theories havebeen studied in detail, and many examples of such theories have been constructed(see [19] for a modern overview of the subject). Recently, Simon [18] identified animportant subclass of NIP theories called distal theories . The motivation behindthis new notion is to single out NIP theories that can be considered purely unsta-ble. O-minimal theories, the classical examples of unstable NIP theories, are distal.The aim of this note is to determine whether certain non-o-minimal expansions ofo-minimal theories which are known to be NIP, are also distal.Let A = ( A, <, . . . ) be an o-minimal structure expanding an ordered group andlet B ⊆ A . We consider theories of structures of the form ( A , B ) that satisfy oneof the following conditions:1. A is the real field and B is a cyclic multiplicative subgroup of R > ( discretesubgroup ),2. A expands a real closed field, B is a proper elementary substructure suchthat there is a unique way to define a standard part map from A into B ( tame pairs ),3. B is a proper elementary substructure of A dense in A ( dense pairs ),4. A is the real field and B is a dense subgroup of the multiplicative group of R > with the Mann property ( dense subgroup ),5. B is a dense, definably independent set ( independent set ).Here and throughout this paper, dense means dense in the usual order topologyon A . All the above examples are NIP. For dense pairs this is due independentlyto Berenstein, Dolich, Onshuus [2], Boxall [3], and G¨unaydın and Hieronymi [12];for dense groups this was shown in [3] and [12]; for tame pairs and for the discretesubgroups NIP was first proven in [12]. For a later, but more general result implying Date : May 4, 2016.2010
Mathematics Subject Classification.
Primary 03C64 Secondary 03C45.
Key words and phrases.
Distal, NIP, expansions of o-minimal structures.A version of this paper is to appear in the
Journal of Symbolic Logic . The first author waspartially supported by NSF grant DMS-1300402.
NIP for all these theories, see Chernikov and Simon [4]. Our main results here areas follows.
Main result.
The theories of structures satisfying 1. or 2. are distal. The theoriesof structures satisfying 3., 4., or 5. are not distal.We observe the following interesting phenomenon: All examples of the above NIPtheories that do not define a dense and codense set, are distal. However, all theexamples that define a dense and codense set, are not distal. This not true ingeneral. The expansion ( R , <, Q ) of the real line by a predicate for the set ofrationals is dp-minimal and hence distal by [18, Lemma 2.10]. Definitions and notations.
Here are precise definitions of the properties underinvestigation. Let T be a complete theory in a language L and let M be a monstermodel of T . When a sequence ( a i ) i ∈ I from M p is indiscernible over a parameterset A , we say the sequence is A -indiscernible. We assume that such a parameterset A ⊆ M always has cardinality smaller than the cardinality of saturation of M .If we say a sequence is indiscernible, we mean the sequence is ∅ -indiscernible. Definition 1.1.
We call an L -formula ϕ ( x, y ) dependent (in T ) if for every indis-cernible sequence ( a i ) i ∈ ω from M p and every b ∈ M q , there is i ∈ ω such thateither M | = ϕ ( a i , b ) for every i > i or M | = ¬ ϕ ( a i , b ) for every i > i . The theory T is NIP (or is dependent ) if every L -formula is dependent in T .Here and in what follows, I, I , I will always be linearly ordered sets. When wewrite I + I , we mean the concatenation of I followed by I . By ( c ) we denotethe linearly ordered set consisting of a single element c . Definition 1.2.
We say T is distal if whenever A ⊆ M , and ( a i ) i ∈ I an indiscerniblesequence from M p such thata. I = I + ( c ) + I , and both I and I are infinite without endpoints,b. ( a i ) i ∈ I + I is A -indiscernible,then ( a i ) i ∈ I is A -indiscernible.It is an easy exercise to check that every distal theory as defined above is also NIP.When T is NIP, the definition of distality given above is one of several equivalentdefinitions. Here we will only use this characterization of distality, and we refer theinterested reader to [18, 19] for more information.For the purposes of this paper it is convenient to introduce the following notion ofdistality for a single L -formula. Definition 1.3.
Let ϕ ( x , . . . , x n ; y ) be a (partitioned) L -forumula, where x i =( x i, , . . . , x i,p ) for each i = 1 , . . . , n . We say ϕ ( x , . . . , x n ; y ) is distal (in T ) if for b ∈ M q and every indiscernible sequence ( a i ) i ∈ I from M p that satisfiesa. I = I + ( c ) + I , and both I and I are infinite without endpoints,b. ( a i ) i ∈ I + I is b -indiscernible,then M | = ϕ ( a i , . . . , a i n ; b ) ↔ ϕ ( a j , . . . , a j n ; b )for every i < · · · < i n and j < · · · < j n in I . ISTAL AND NON-DISTAL PAIRS 3
This definition of distality of a single formula depends on the indicated partition ofthe free variables. It is immediate that T is distal if and only if every L -formula isdistal in T . Using saturation of M , one can also see easily that in order to check thedistality of a formula, one may assume that I and I are countable dense linearorders without endpoints.We now fix some notation. We will use m, n for natural numbers. For X ⊆ M , weshall write dcl L ( X ) for the L -definable closure of X in M . When T is an o-minimaltheory, the closure operator dcl L is a pregeometry. We will use this fact freelythroughout this paper. For a tuple b = ( b , . . . , b n ) ∈ M n and X ⊆ M , by Xb wemean X ∪ { b , . . . b n } , and we say that b is dcl L -independent if the set { b , . . . , b n } is. For a function f , ar( f ) will denote the arity of f . Acknowledgements.
We thank Pierre Simon and the anonymous referee for veryhelpful comments. We also thank Danul Gunatilleka, Tim Mercure, Richard Rast,Douglas Ulrich, and in particular Allen Gehret for reading an earlier version of thispaper and providing us with excellent feedback.2.
The discrete case
In this section we give sufficient conditions for expansions of o-minimal theoriesby a function symbol to be distal. We prove in sections following this one thatboth tame pairs and the expansions by discrete groups mentioned above satisfythese conditions. This criterion for distality (and its proof) is closely related tothe criterion for NIP given in [12, Theorem 4.1]. Here we use the same set up.As in [12], let T be a complete o-minimal theory extending the theory of orderedabelian groups and let L be its language with distinguished positive element 1.Such a theory has definable Skolem functions. After extending it by constants andby definitions, we can assume the theory T admits quantifier elimination and has auniversal axiomatization. In this situation, any substructure of a model of T is anelementary submodel, and therefore dcl L ( X ) = h X i for any subset X of any model A of T ; here h X i denotes the L -substructure of A generated by X . For B (cid:22) A we write Bh X i for hB ∪ X i . Following the notation from [12] we extend L to L ( f )by adding a new unary function symbol f . We let T ( f ) be a complete L ( f )-theoryextending T . As usual, we take M to be a monster model of T ( f ). Theorem 2.1.
Suppose that the following conditions hold: (i)
The theory T ( f ) has quantifier elimination. (ii) For every ( C , f ) | = T ( f ) , B (cid:22) C with f ( B ) ⊆ B and every c ∈ C k , there are l ∈ N and d ∈ f (cid:0) Bh c i (cid:1) l such that f (cid:0) Bh c i (cid:1) ⊆ h f ( B ) , d i . (iii) Let m ≥ n and let g, h be L -terms of arities m + k and n + l respectively, b ∈ M k , b ∈ f ( M ) l , ( a i ) i ∈ I be an indiscernible sequence from f ( M ) n × M m − n such that a. I = I + ( c ) + I , where both I and I are infinite and without end-points, and ( a i ) i ∈ I + I is b b -indiscernible, b. a i = ( a i, , . . . , a i,m ) for each i ∈ I , and c. f ( g ( a i , b )) = h ( a i, , . . . , a i,n , b ) for every i ∈ I + I .Then f ( g ( a c , b )) = h ( a c, , . . . , a c,n , b ) . P. HIERONYMI AND T. NELL
Then T ( f ) is distal.Proof. By (i), it is enough to show that every (partitioned) quantifier-free L ( f )-formula ψ ( x , . . . , x p ; y ) is distal. We will prove this by induction on the number e ( ψ ) of times f occurs in ψ . If e ( ψ ) = 0, this follows just from the fact thato-minimal theories are distal. Let e ∈ N > be such that every quantifier-free L ( f )-formula ψ ′ with e ( ψ ′ ) < e (with any partition) is distal. Let ψ ( x , . . . , x p ; y ) be aquantifier-free L ( f )-formula with e ( ψ ) = e . We will establish that ψ is distal. Takean indiscernible sequence ( a i ) i ∈ I from M s and b ∈ M k such that I = I + ( c ) + I ,where both I and I are countable dense linear orders without endpoints, and( a i ) i ∈ I + I is b -indiscernible. By b -indiscernibility we may assume that(A) M | = ψ ( a i , . . . , a i p ; b ) for all i < · · · < i p ∈ I + I .Let j ∈ { , . . . , p } , u < · · · < u j − ∈ I and v < · · · < v p − j ∈ I . It suffices toshow that(2.1) M | = ψ ( a u , . . . , a u j − , a c , a v , . . . , a v p − j ; b ) . Since e >
0, there is an L -term g such that the term f ( g ( x , . . . , x p , y )) occurs in ψ .Now let A be the L ( f )-substructure of M generated by { a i : i ∈ I + I } . By (ii),there is d ∈ f (cid:0) Ah b i (cid:1) l such that f ( Ah b i ) ⊆ h f ( A ) , d i (use M ↾ L as C and A ↾ L as B in the statement of (ii)). Take q, r ∈ N , u j < · · · u q + ( c ) + ( I ) ISTAL AND NON-DISTAL PAIRS 5 Discrete groups Let ˜ R be an o-minimal expansion of ( R , <, + , · , , 1) which is polynomially-bounded with field of exponents Q . We will establish distality for the theory ofthe expansion of ˜ R by a predicate for the cyclic multiplicative subgroup 2 Z of R > .Towards this goal, let T be the theory of ˜ R and L be its language. Let λ : R → R be the function that maps x to max( −∞ , x ] ∩ Z when x > 0, and to 0 otherwise.It is immediate that the structures ( ˜ R , Z ) and ( ˜ R , λ ) define the same sets. Vanden Dries [7] showed quantifier elimination for the latter structure when ˜ R is thereal field. This result was generalized by Miller [16] to expansions of the real fieldwith field of exponents Q . It is worth pointing out that by [14, Theorem 1.5] theassumption on the field of exponents can not be dropped.Let T disc be the theory of ( ˜ R , λ ) in the language L ( λ ), the extension of L by a unaryfunction symbol for λ . In order to show distality of T disc , we can assume that ˜ R has quantifier elimination and has a universal axiomatization. Theorem 3.1. T disc is distal.Proof. We need to verify that T disc satisfies the assumptions of Theorem 2.1. As-sumptions (i) and (ii) were already established in [12, Theorem 6.5]. It is left toprove (iii). Let M be a monster model of T disc . We denote λ ( M ) \ { } by G .Note that G is a multiplicative subgroup of M > . For p ∈ N , the set of p -powers G [ p ] := { g p g ∈ G } has finitely many cosets in G , since | Z : (2 Z ) [ p ] | = p . Indeed,1 , , . . . , p − are representatives of the cosets of G [ p ] .Take an indiscernible sequence ( a i ) i ∈ I from M m , where I = I + ( c ) + I and I and I are infinite without endpoints, such that a i, , . . . , a i,n ∈ λ ( M ) for every i ∈ I and a i = ( a i, , . . . , a i,m ). Let ( b , b ) ∈ M k × λ ( M ) l such that ( a i ) i ∈ I + I is b b -indiscernible. Suppose that there are L -terms g, h such that for i ∈ I + I λ ( g ( a i , b )) = h ( a i, , . . . , a i,n , b ) . It is left to conclude that λ ( g ( a c , b )) = h ( a c, , . . . , a c,n , b ). By definition of λ , wehave for every i ∈ I + I M | = 1 ≤ g ( a i , b ) h ( a i, , . . . , a i,n , b ) < . Since T is distal, the previous statement holds for all i ∈ I . It is left to show that h ( a c, , . . . , a c,n , b ) ∈ G . By [12, Corollary 6.4] and b -indiscernibility of ( a i ) i ∈ I + I ,there are t, q , . . . , q n ∈ Q , r = ( r , . . . , r l ) ∈ Q l such that for every i ∈ I + I h ( a i, , . . . , a i,n , b ) = 2 t · a q i, · · · a q n i,n · b r , where b r stands for b r , · · · b r l ,l . By distality of T , this equation holds for all i ∈ I . Itis left to show that 2 t · a q c, · · · a q n c,n · b r ∈ G . Let p ∈ N be such that p · t, p · q , . . . , p · q n ∈ Z and p · r ∈ Z l . It is enough to prove 2 p · t · a p · q c, · · · a p · q n c,n ∈ b p · r · G [ p ] . Let s ∈ { , . . . , p − } be such that b p · r is in 2 s · G [ p ] . Then for every i ∈ I ,(3.1) 2 p · t · a p · q i, · · · a p · q n i,n ∈ b p · r · G [ p ] iff 2 p · t · a p · q i, · · · a p · q n i,n ∈ s · G [ p ] . Since the second statement in (3.1) holds for i ∈ I + I and ( a i ) i ∈ I is indiscernible,it holds for all i ∈ I and in particular for i = c . (cid:3) P. HIERONYMI AND T. NELL Tame pairs For this section, let T be a complete o-minimal theory expanding the theory ofreal closed fields in a language L . In [10] van den Dries and Lewenberg introducedthe following notion of tame pairs of o-minimal structures. Definition 4.1. A pair ( A , B ) of models of T is called a tame pair if B (cid:22) A , A 6 = B and for every a ∈ A which is in the convex hull of B , there is a unique st( a ) ∈ B such that | a − st( a ) | < b for all b ∈ B > .The standard part map st can be extended to all of A by setting st( a ) = 0 for all a not in the convex hull of B . Instead of considering ( A , B ) we will consider ( A , st).It is easy to check that these two structures are interdefinable. Let T t be the L (st)-theory of all structures of the form ( A , st) . After extending T by definitions, we canassume that T has quantifier elimination and is universally axiomatizable. By [10,Theorem 5.9] and [10, Corollary 5.10], T t is complete and has quantifier elimination.We will also need to consider the theory of convex pairs. A T - convex subring of amodel A of T is a convex subring that is closed under all continuous unary L - ∅ -definable functions. A convex pair is a pair ( A , V ), where A | = T , V is a T -convexsubring of A , and V = A . We denote the theory of all such pairs by T c . By [10,Corollary 3.14], this theory is weakly o-minimal. By [5, Theorem 4.1], every weaklyo-minimal theory is dp-minimal and hence distal by [18, Lemma 2.10]. Therefore T c is distal.For every model ( A , st) of T t , the pair ( A , V ) is a model of T c , where V is theconvex closure of st( A ). It follows immediately that for every b ∈ st( A ) and a ∈ A st( a ) = b ⇐⇒ a = b or (( a − b ) − / ∈ V ) or ( b = 0 and a / ∈ V ) . We will not use the explicit description on the right, but we will use the fact thatthis gives us an L ( U )-formula ψ such that for all a ∈ A and b ∈ st( A )(4.1) ( A , st) | = st( a ) = b iff ( A , V ) | = ψ ( a, b ) . Theorem 4.2. T t is distal.Proof. We will show that T t satisfies the assumptions of Theorem 2.1. Assumptions(i) and (ii) were already established for [12, Theorem 5.2]. We only need to prove(iii). Let M be a monster model of T t , and V the convex closure of st( M ). Let( a i ) i ∈ I be an indiscernible sequence from M m , where I = I + ( c ) + I and I and I are infinite with no endpoints, such that a i, , . . . , a i,n ∈ st( M ) for i ∈ I and a i = ( a i, , . . . , a i,m ). Let ( b , b ) ∈ M k × st( M ) l such that ( a i ) i ∈ I + I is b b -indiscernible. Suppose that there are L -terms g, h such that for i ∈ I + I st( g ( a i , b )) = h ( a i, , . . . , a i,n , b ) . We need to show that st( g ( a c , b )) = h ( a c, , . . . , a c,n , b ). Since st( M ) is a modelof T , we have h ( a i, , . . . , a i,n , b ) ∈ st( M ) for every i ∈ I . By (4.1), there is an L ( U )-formula ψ such that for i ∈ I ( M , st) | = st( g ( a i , b )) = h ( a i, , . . . , a i,n , b ) ⇐⇒ ( M , V ) | = ψ ( a i , b ) . Since T c is distal, M | = ψ ( a c , b ). (cid:3) ISTAL AND NON-DISTAL PAIRS 7 Dense Pairs In this section we present sufficient conditions for non-distality of expansions ofo-minimal theories by a single unary predicate, and give several examples of NIPtheories satisfying these conditions. Let T be an o-minimal theory in a language L expanding that of ordered abelian groups, U a unary relation symbol not appearingin L , and T U an L ( U ) = L ∪ { U } -theory expanding T . Let M be a monster modelof T U . We denote the interpretation of U in M by U ( M ). We say that an L ( U )-definable subset X of M is small if there is no L -definable (possibly with parameters)function f : M m → M such that f ( X m ) contains an open interval in M . When wesay a set is dense in M , we mean dense with respect to the usual order topology on M . Theorem 5.1. Suppose the following conditions hold:(1) U ( M ) is small and dense in M .(2) For n ∈ N , C ⊆ M , and a, b ∈ M n both dcl L -independent over C ∪ U ( M ) , tp L ( a | C ) = tp L ( b | C ) ⇒ tp L ( U ) ( a | C ) = tp L ( U ) ( b | C ) . Then T U is not distal.Proof. Let b ∈ M be dcl L -independent over U ( M ). The existence of such a b follows immediately from smallness of U ( M ) and saturation of M . Let I , I betwo countable linear orders without endpoints. Consider a set Φ containing L ( U )- b -formulas in the variables ( x i ) i ∈ I +( c )+ I expressing the following statements:(i) { x i : i ∈ I + I } is dcl L -independent over U ( M ) b ,(ii) f ( x i , . . . , x i n , b ) < x i n +1 , for each i < · · · < i n +1 ∈ I + ( c ) + I and L - ∅ -definable function f ,(iii) there is u ∈ U ( M ) such that x c = u + b .We will show that Φ is realized in M . By saturation of M it is enough to showthat every finite subset Φ of Φ is realized. Let F = { f , . . . , f m } be the L -definable functions appearing in formulas of the form (ii) in Φ . Let i < · · · < i n ∈ I + ( c ) + I be the indices of variables occurring in Φ . We may assume c is amongthese, and by adding dummy variables that each f j is of the form f ( x i , . . . x i k , b )for some k < n . We now recursively choose ( a i , . . . , a i n ) realizing the type Φ .Suppose we have defined a i , . . . , a i k − . If k = 1, we will have defined no previous a i , and the functions below will be of arity 1 only mentioning b . If i k = c , then bydenseness of U ( M ) we may choose a c in (cid:0) b + U (cid:1) ∩ (cid:16) max f ∈F ,ar ( f )= k f ( a i , . . . , a i k − , b ) , ∞ (cid:17) . If i k = c , then by smallness of U ( M ) we may choose a i k in (cid:16) max f ∈F ,ar ( f )= k f ( a i , . . . , a i k − , b ) , ∞ (cid:17) \ dcl L ( U ( M ) ba i · · · a i k − ) . As ( a i , . . . , a i n ) realizes Φ , Φ is finitely satisfiable.By saturation, we can pick a realization ( a i ) i ∈ I +( c )+ I of Φ in M . This sequencecan be thought of as a very rapidly growing sequence; each element will realizethe type at + ∞ over the L -definable closure of everything before it. Therefore thesequence is a Morley sequence for the L -type of + ∞ over dcl L ( b ), and hence is L - b -indiscernible. As dcl L is a pregeometry and b is dcl L -independent over U ( M ),(i) and (iii) together imply that the full sequence is dcl L -independent over U ( M ). P. HIERONYMI AND T. NELL Thus by (2), the L -indiscernibility of these sequences lifts to L ( U )-indiscernibility;that is, ( a i ) i ∈ I + I is L ( U )-indiscernible over b , and the full sequence is L ( U )-indiscernible. However, since a c = b + u for some u ∈ U ( M ), the full sequence isnot L ( U )-indiscernible over b . Hence T U is not distal. (cid:3) Optimality. Note that the assumption that T expands the theory of orderedabelian groups can not be dropped. As pointed out in the introduction the theoryof the structure ( R , <, Q ) is distal. However, it is not hard to check that the theoryof ( R , <, Q ) satisfies the other assumptions of Theorem 5.1. Dense pairs. Let A , B be two models of an o-minimal theory T expanding thetheory of ordered abelian groups such that B (cid:22) A , B 6 = A , and B is dense in A .We call ( A , B ) a dense pair of models of T . Let T d be the theory of dense pairsin the language L ( U ). By van den Dries [8] T d is complete. Moreover, for everydense pair ( A , B ), the underlying set of B is small by [8, Lemma 4.1]. While notstated explicitly, it follows almost immediately from [8, Claim on p.67] that T d alsosatisfies (2) of Theorem 5.1 (see [11, Proposition 2.3] for detailed proof). Therefore T d is not distal. Dense groups. Let R be the real field ( R , <, + , · , , R > that has the Mann property , that is for every a , . . . , a n ∈ Q × , there arefinitely many ( γ , . . . , γ n ) ∈ Γ n such that a γ + · · · + a n γ n = 1 and P i ∈ I a i γ i = 0for every proper nonempty subset I of { , . . . , n } . Every multiplicative subgroup offinite rank in R > has the Mann property. Let L be the language of R expanded bya constant symbol for each γ ∈ Γ. Let T Γ be the L ( U )-theory of ( R , ( γ ) γ ∈ Γ , Γ) inthis language. This structure was studied in detail by van den Dries and G¨unaydın[9]. A proof that every model satisfies (1) of Theorem 5.1 is in [13, Proposition3.5]. Similarly to dense pairs, it is not mentioned in [9] that these theories satisfycondition (2) of Theorem 5.1. However, it can easily be deduced from the proof of[8, Theorem 7.1] (see also [11, p. 6]).The argument can easily be extended to related structures (see [1, 13, 15]). Independent sets. We finish with another class of structures that were studiedrecently by Dolich, Miller and Steinhorn [6]. Let T be an o-minimal theory in alanguage L expanding that of ordered abelian groups. Let T indep be an L ( U )-theoryextending T by axioms stating that U is dense and dcl L -independent. By [6], T indep is complete. Every model of T indep satisfies (1) of Theorem 5.1 by [6, 2.1]. By [6,2.12], condition (2) of that theorem also holds for T indep . References [1] Oleg Belegradek and Boris Zilber. The model theory of the field of reals with a subgroup ofthe unit circle. J. Lond. Math. Soc. (2) , 78(3):563–579, 2008.[2] Alexander Berenstein, Alf Dolich, and Alf Onshuus. The independence property in generalizeddense pairs of structures. J. Symbolic Logic , 76(2):391–404, 2011.[3] Gareth Boxall. NIP for some pair-like theories. Arch. Math. Logic , 50(3-4):353–359, 2011.[4] Artem Chernikov and Pierre Simon. Externally definable sets and dependent pairs. Israel J.Math. , 194(1):409–425, 2013.[5] Alfred Dolich, John Goodrick, and David Lippel. Dp-minimality: basic facts and examples. Notre Dame J. Form. Log. , 52(3):267–288, 2011.[6] Alfred Dolich, Chris Miller, and Charles Steinhorn. Expansions of o-minimal structures bydense independent sets. Modnet Preprint No. 691 , 2014. ISTAL AND NON-DISTAL PAIRS 9 [7] Lou van den Dries. 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Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801 E-mail address : [email protected] URL : Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 WestGreen Street, Urbana, IL 61801 E-mail address : [email protected] URL ::