Companionability Characterization for the Expansion of an O-minimal Theory by a Dense Subgroup
aa r X i v : . [ m a t h . L O ] J a n COMPANIONABILITY CHARACTERIZATION FOR THE EXPANSION OF ANO-MINIMAL THEORY BY A DENSE SUBGROUP
ALEXI BLOCK GORMAN
Abstract.
This paper provides a full characterization for when the expansion of a completeo-minimal theory by a unary predicate that picks out a divisible dense and codense subgrouphas a model companion. This result is motivated by criteria and questions introduced in therecent works [10] and [7] concerning the existence of model companions, as well as preservationresults for some neostability properties when passing to the model companion. The focus of thispaper is establishing the companionability dividing line in the o-minimal setting because thisallows us to provide a full and geometric characterization. Examples are included both in whichthe predicate is an additive subgroup, and where it is a multiplicative subgroup. The paperconcludes with a brief discussion of neostability properties and examples that illustrate the lackof preservation (from the “base” o-minimal theory to the model companion of the expansion wedefine) for properties such as strong, NIP, and NTP , though there are also examples for whichsome or all three of those properties hold. Introduction
The results in this paper follow in the spirit of “Generic structures and simple theories,” a seminalpaper of Chatzidakis and Pillay. In [5], they notably show, among other things, that the expansionof a theory T with uniform finiteness in the language L by a generic unary predicate (“generic”here meaning the predicate has no induced L structure) has a model companion T P , and that if T is simple in the sense of Shelah [13] then so too is T P . Many works in this vein have recentlyexplored questions related to generic predicates and properties preserved when passing to a modelcompanion, including [7], [8], [9], and [10].Among other results, we establish a neat dividing line for when the expansion of an o-minimaltheory T (which extends the theory RCF ) by a predicate for a dense and codense divisible subgroupof the positive elements has a model companion. Let T G be the expansion of an o-minimal theory T as described above, i.e. T extends the theory of a real closed field along with axioms statingthat the predicate G picks out a dense and proper (which implies codense by o-minimality) divisiblesubgroup of the positive elements. Then our characterization is the following: Theorem A.
Let M be any model of T . Then the theory T ×G as described above has a modelcompanion if and only if the graph of Ce x : M → M is not definable for any C ∈ M on anyinterval of M . However, the main result of this paper is a far more general theorem; we relax the assumptionthat G picks out a multiplicative subgroup to allow the predicate G to pick out a subgroup with Date : January 7, 2021.2010
Mathematics Subject Classification.
Primary 03C10 Secondary 03C64.
Key words and phrases. o-minimality, model companion, , dense pairs, lovely pairs, H -structures, Mann groups,NIP. respect to any binary relation ∗ in the language L of T of any L ( ∅ )-definable open subset D of theunderlying set, provided that T already dictates that ∗ is a group operation on D .The following theorem was conjectured by Erik Walsberg, whose work with Tran and Kruckmanin [10] motivated the author to investigate the question of when an o-minimal expansion of a groupwith a predicate for a suitable subgroup is companionable. Theorem B.
For T an o-minimal expansion of the theory of ordered abelian groups with language L , let T G be the L ∪ {G} -theory extending T that states also that G is a dense and codense subsetof L ( ∅ ) -definable unary set D , and G is an ordered divisible abelian group with respect to ∗ , an L ( ∅ ) -definable binary operation. Then T G has a model companion if and only if the only families ofadditive endomorphisms whose graphs are uniformly definable in a neighborhood of (which denotesthe ∗ identity) have finitely many germs at . The primary focus of this paper is to establish the above companionability dichotomy and under-stand its consequences.The structure of this paper is as follows. Notation and preliminaries occupy the remainder of thissection. In section 2 we establish Theorem B by carefully defining the sufficient condition for thenegative result (non-existence of a model companion) then prove both the negative result (sufficientcondition for non-existence of a model companion) and the positive result (sufficient condition forexistence of a model companion) as independent lemmas. In section 3, we prove Theorem A andoffer examples of o-minimal theories for which expanding by one type of group (say, additive)results in no model companion, whereas expanding by another type of group (say, multiplicative)does yield a model companion, along with other kinds of examples and non-examples that illustratethe companion dichotomy. Finally, in section 4 we show that for T an o-minimal theory thatexpands a real closed field, even if the expansion by a “generic” subgroup has a model companion,that model companion has T P . Conversely, if T is an o-minimal theory with only a vector spacestructure (definable subsets in the natural language are affine) then the model companion of thesame kind of expansion has NIP, but depending on the base field it may or may not be strong, i.e.have finite burden.1.1. Background.
Inspired by the work of Chatzidakis and Pillay in [5], Christian d’Elb´ee setabout describing a natural sufficient condition for when a geometric theory T that is model-completeaugmented by a predicate that picks out a “well-behaved” reduct of the theory has a model com-panion. He provides this criterion in [7], in which he also shows preservation of NSOP undercertain assumptions. In [10], Kruckman, Tran, and Walsberg generalize the work of d’Elb´ee byintroducing a new set up known as “interpolative fusions” which they use to produce even broaderconditions for when certain kinds of first-order theories have a model companion. In particular,they define the “fusion” of two theories under the assumption that the original theories exhibitcertain compatibility properties.Both d’Elb´ee and the authors of [10] found completely characterizing when the model companionexists to be elusive or quite complicated depending on which base theories one is concerned with.In the case that one considers a pair of disjoint theories with uniform finiteness, the existence of amodel companion is due to Peter Winkler in his thesis [15]. For a base theory that is not the emptytheory, however, characterizing companionability purely in terms of the properties of the theoryone is expanding can be quite nuanced and subtle. The results of this paper, Theorems A and B inparticular, illustrate exactly that subtlety, since we provide an entirely geometric characterizationof companionability for the expansion of an o-minimal theory by a predicate with extra structurein the base language, that of a dense and codense subgroup. OMPANIONABILITY CHARACTERIZATION 3
This dichotomy we establish with Theorem B bears some resemblance to the dividing line coinedas “polynomial boundedness” by Miller in [11], for which he establishes the “growth dichotomy.”The growth dichotomy states that for any o-minimal expansion of the real closed field, eitherthe exponential function is definable, or every definable function that is not eventually zero isasymptotically equivalent to a power function. Miller and Starchenko establish a similar growthdichotomy for “linearly bounded” expansions of the reals as an additive group in [12]. However, anotable difference from Theorem B, and the reason that neither growth dichotomy can be utilizeddirectly to obtain the results in this paper, is that the companionability dividing line coincides withlocal definability of exponentiation or multiplication, rather than global definability or boundedness.
Acknowledgements.
The author would like to thank Erik Walsberg for offering the statementof the main theorem of this paper as a remarkably apt conjecture, and for many conversationsabout the results which yield the proof of said conjecture. Deepest thanks go to the author’s thesisadvisor, Philipp Hieronymi, who not only helped with a number of arguments in this paper but alsogave diligent attention to correcting many rough drafts. Many thanks also to Christian d’Elb´ee fora detailed and helpful discussion of the main theorem, and for suggesting a nice proof of Theorem4.1. Finally, we would like to thank Minh Tran and Elliot Kaplan for some very helpful discussionsof various results in this paper. This material is based upon work supported by the National ScienceFoundation Graduate Research Fellowship Program under Grant No. DGE – 1746047.1.2.
Preliminaries.
Throughout this paper, let T be an o-minimal theory that expands the the-ory of ordered divisible abelian groups, and let L be a language in which T is model complete.Henceforth we also let M | = T and for a ∈ M , we let B ǫ ( a ) denote a ball of radius epsilon centeredat point a . Throughout, when we write that an interval ( a, b ) or [ a, b ] is “nontrivial,” we mean thatthe interval has nonempty interior. For the remainder of this paper, we use “ L ( ∅ )-definable” and“definable” to mean the same thing, where the language L is that of T . If a set X is definable withparameters A ⊆ M we write “ L ( A )-definable” and “ A -definable” interchangeably. Whenever a set X is L G ( A )-definable, we will refer to it as such and never simply call such a set “definable” or“ A -definable.” Definition 1.1.
Let G be a unary predicate, and let L G := L ∪ {G} . We define the theory T G asfollows:(1) T ⊆ T G (2) G picks out a divisible subgroup that is dense and codense as a subset of D , an L ( ∅ ) -definableopen unary set. We use ( M , G ) to denote models of T G . We will refer to the < -interval topology on M merelyas its topology, and similarly we call the topology induced by < on G simply the topology on G .For any function f : M n → M m we denote the graph of f by gr ( f ) ⊆ M n + m . For functions f, g : M → M we mean f ◦ g when we talk about precomposing g with f . Definition 1.2.
Let a ∈ M . Let A = ( f i ) i ∈ I be a definable family of unary functions such that a ∈ T i ∈ I dom( f i ) . Then for each f ∗ ∈ A the set G f ∗ := { ( ǫ, f ) : f ∈ A ∧ ∃ < δ < ǫ ∀ x ∈ B δ ( a ))( f ( x ) = f ∗ ( x )) } is L ( ∅ ) -definable.Call G := { G f : f ∈ A} the definable family of germs of A at a . BLOCK GORMAN
We say that f and f ∗ have equivalent germs if G f = G f ∗ for some a ∈ M . We will denote theequivalence class of the germs in G ∈ G by the partial function ˆ f G : B ǫ ( a ) → M , where ǫ > G is defined, and f ∈ A is such that G f = G forsome a ∈ M . Since we can generally assume a = 0 in our setting, we will often make no referenceto that parameter. Alternatively, we simply denote the equivalence class by the definable function f G when it is unambiguous from context that we mean the set of all partial functions in definablefamily A which coincide with ˆ f G ( x ) on some neighborhood of a . Notation.
For brevity, we define [ n ] = { , . . . , n } . Given a tuple ~x = ( x , . . . , x n ), we write ˆ x i =( x , . . . , x i − , x i +1 , . . . , x n ) to denote the vector of length | ~x | − ~x with the i th componentremoved. Additionally, if M | = T and D ⊆ M m is a definable set and I ⊆ [ m ], define π I ( D ) to bethe projection of elements in D onto the coordinates that are in the subset I . Below we let B nǫ ( x )denote Q ni =1 ( x − ǫ , x + ǫ ), i.e. the product of n intervals of width ǫ centered at x in M n . As isstandard, we let U denote the topological closure of the set U ⊆ M n . For a definable function f wedefine the “delta function,” written ∆ t f ( x ), as f ( x + t ) − f ( x ) for t in a neighborhood of 0 for which x + t is within the domain of f . Recall that we say f is definable or L ( ∅ )-definable interchangeably,and this means the graph of f is a L ( ∅ )-definable set in M . Definition 1.3.
As before, fix
M | = T .(1) Suppose that f : D ⊆ M → M is a definable function and U ⊆ D is a definable neighborhoodof ∈ M , and that for all x, y ∈ U such that x + y ∈ U we have f ( x + y ) = f ( x ) + f ( y ) .Then we say that f is a definable endomorphism on U , or local endomorphism if U is notspecified.(2) Let m, n ∈ N be such that m > . We call a function h : D ⊆ M m → M n a definabletransformation endomorphic in coordinates, or definable EIC transformation for brevity, iffor each i ∈ { , . . . , m } : ∀ t > ∀ ˆ a i , ˆ b i ∈ π [ m ] \ i ( D ) ∀ x ∈ π i ( D ) h ∆ t h ( a , . . . , x, . . . a m − ) = ∆ t h ( b , . . . , x, . . . b m − ) i and also ∀ ˆ a i ∈ π [ m ] \ i ( D ) ∀ x, y ∈ π i ( D ) h h ( a , . . . , x − y, . . . a m − ) = h ( a , . . . , x, . . . a m − ) − h (0 , . . . , y, . . . , i (we call this latter property endomorphic in coordinates).(3) Call a definable set X ⊆ M n with n > a definable hyperplane if there is some definableEIC transformation h : M n − → M such that X = gr( h ) .(4) We define a matrix as follows: A~x := m X i =1 g i, ( x i ) , . . . , m X i =1 g i,n ( x i ) ! where g i,j ( x ) is a definable endomorphism on π i ( D ) . Given a definable EIC transformation h : D ⊆ M m → M n , we say that A is the matrix representation of h if the coordinatefunctions of A are given by g i,j ( t ) = π j (∆ t h ( a , . . . , a i − , y, a i , . . . , a m − )) , where ~a =( a , . . . , a m − ) ∈ π [ m ] \ i ( D ) can be chosen arbitrarily. We see that A~x = h ( ~x ) in this case. What part (2) of the above definition says is that h : M m → M n is a “definable EIC transforma-tion” if when you view it as a function only of the i th coordinate (with i ≤ m ) then even if we varythe values of the other coordinates, the result is an endomorphism and has the same “behavior”(as measured by the delta function). OMPANIONABILITY CHARACTERIZATION 5
We will formulate many of the results in terms of definable EIC transformations because quan-tifier elimination for ordered divisible abelian groups tells us that these are precisely the definableunary functions in the language (+ , ,
1) up to affine shifts. Hence, for any other L -definable unaryfunction f , if we knew that f ( G ) = G and f ( G c ) = G c in every model of T G , then this would haveto follow from the axioms of T G itself. Yet T G is axiomatized in such a way that for every definablefunction f that is not definable in the group language, the property “ f sends some element of G to G c ” is realizable.To see why it makes sense that we define EIC transformations using the delta function, in thelemma below we illustrate the link between definable local endomorphisms and delta functions thatare constant. Lemma 1.4.
Suppose that f : D ⊆ M → M is definable and R > is such that ( − R, R ) ⊆ D and { x + t : x ∈ D, t ∈ [0 , R ) } ⊆ D . For all t ∈ [0 , R ) , the function ∆ t f ( x ) is constant (with respect to x ) on D and f (0) = 0 if and only if f is a definable endomorphism on ( − R, R ) .Proof. For the forward implication, suppose that f and R > g ( t ) := ∆ t f ( x ) = f ( x + t ) − f ( x ), for which x is chosen arbitrarily since ∆ t f does not dependon x . Suppose that y , y ∈ D are such that 0 < y < R and y + y ∈ D . Then f ( y + y ) − f ( y ) = g ( y ) = f (0 + y ) − f (0) = f ( y ). This yields f ( y + y ) = f ( y ) + f ( y ) as desired.For the backwards implication, suppose that f is a definable endomorphism on [0 , R ). Bydefinition of a local endomorphism f (0) = 0. Let t ∈ (0 , R ), and let x ∈ [0 , R ) be such that x + t ∈ D . Then ∆ t f ( x ) = f ( x + t ) − f ( x ) = f ( x + t − x ) since f is a local endomorphism, and f ( x + t − x ) = f ( t ), which is constant with respect to x , as desired. (cid:3) From the above lemma we can deduce that each definable EIC transformation can be representedby a matrix consisting of the coordinate-wise delta functions. Furthermore, we will observe thatthe definable EIC transformations that send G to itself in models ( M , G ) of T G are precisely thosewhose delta functions are ∅ -definable in ( G , , +). Lemma 1.5.
Let h : D ⊆ M m → M n be a L ( ~c ) -definable EIC transformation.(i) There is a unique matrix representation A for h (as defined in 1.3).(ii) The functions g i,j ( t ) used to define the matrix A are all definable in ( G , , +) , i.e. are Q -affine,precisely if T G ∪ tp( ~c ) ⊢ h ( G m ) ⊆ G n ∧ h ( M m \ G m ) ⊆ M n \ G n .Proof. For (i), given a definable EIC transformation h we define the corresponding matrix A by A~x = ( P i ≤ n g i,j ( x i )) j ∈ [ n ] where g i,j ( t ) = ∆ t π j ( h ( a , . . . , x, . . . a m − )) = π j ( h ( a , . . . , t, . . . a m − )).By definition of an EIC transformation, the coordinate functions of an EIC transformation arealso EIC. By Lemma 1.4, the functions g i,j ( t ) are well-defined functions of t alone, i.e. ˆ a i =( a , . . . , a m − ) ∈ π [ m ] \ i ( D ) can be arbitrarily chosen, and x is not a free variable in g i,j . FromLemma 1.4, we deduce that ∆ t applied to the coordinate functions of h are equivalent to thecoordinate functions as local endomorphisms. So we can write h as the sum of its coordinatefunctions in the following way: for ~x ∈ D , h ( ~x ) = m X i =1 g i, ( x i ) , . . . , m X i =1 g i,n ( x i ) ! . Yet this shows exactly the desired relationship of h ( ~x ) and A~x , and the uniqueness of the matrix A follows from A being determined, as a matrix, by A~x = h ( ~x ) for all ~x ∈ D . BLOCK GORMAN
For (ii), quantifier elimination for ordered divisible abelian groups tells us that definable unaryfunctions in ( G , , +) are all of the form f ( x ) = qx + g where q ∈ Q and g ∈ G . Hence if each g i,j isdefinable in ( G , , +), then because it is an endomorphism, it is of the form x qx with q ∈ Q . Theforward implication is immediate from this. For the other implication, suppose that h : M n → M is L ( ∅ )-definable, but not definable in ( G , , +), and that h ( ~g ) ∈ G n if and only if ~g ∈ G m . Thenfor each i ∈ [ m ] and j ∈ [ n ], it follows that g i,j ( x ) ∈ G if and only if x ∈ G for every ( M , G ) | = T G .Let us assume that for some i ∈ [ m ] and j ∈ [ n ] the function g i,j ( t ) is everywhere non-constant andeverywhere locally definable in L but not in ( G , , +). Then since the only axioms of T G concerning G are that it is a divisible subgroup dense and codense in D ⊇ G , the closure properties of additionand Q -affine functions on G need not hold for g i,j .For any ( M , G ) | = T G , we can pass to an |M| + -saturated elementary extension N < M . Let B ⊆ N \ M be a dense and codense subset of a maximal dcl L -independent set over D M , and define G ( N , G ) = ( G ( M , G ) ⊕ b ∈ B Q b ) ∩ D N . It follows from o-minimality and saturation that ( N , G ( N , G ) ) | = T G . By dcl L -independence of B , we know that g i,j ( b ) B \ { b } for any b ∈ B , and saturation plusthe non-definability of f in ( G , , +) ensures that for infinitely many b ∈ B we know f ( b ) is not inthe image of G ( M , G ) ∪ B under any Q -affine function. Hence f ( G ( N , G ) )
6⊆ G ( N , G ) , as desired. (cid:3) The following can be viewed as a corollary to Lemma 1.5, and will prove useful in section 2.2.
Corollary 1.6.
For any definable EIC transformation h : D ⊆ M n → M m with B nǫ ( ~ ⊆ D for some ǫ > and any ~x = ( x , . . . , x n ) ∈ D , the value of h ( ~x ) is uniquely determined by h (0 , x , . . . , x n ) , . . . , h ( x , . . . , x n − , , or, equivalently, by h ( x , , . . . , , . . . , h (0 , . . . , , x n ) .Proof. Since h is endomorphic in each coordinate, we can define ˆ h i ( ~x ) = h ( x , . . . , x n − , − h ( x , . . . , x i − , , x i +1 , . . . , x n ) = (0 , . . . , , x i , , . . . , , − x n ). Consider ˜ h ( ~x ) = P ni =1 ˆ h i = h ( x , . . . ,x n − , − nx n ). From this we can L ( ∅ )-define ˜ h ( x , . . . , x n − , − n x n ) on the same domain, and weobserve that ˜ h ( x , . . . , x n − , − n x n ) = h ( x , . . . , x n ) as desired. (cid:3) Model Companion in the General Setting
Recall that throughout this paper we assume T is an o-minimal theory that expands the theoryof ordered divisible abelian groups (which we call ODAG) and is both complete and model completein the language L . We require that T have an ∀∃ -axiomatization in L because we rely on the factthat for such theories, which are sometimes called “inductive,” the existence of a model companionis equivalent to the class of existentially closed models of T having a first-order axiomatization. Wewill often use without stating it explicitly that because the structure M is o-minimal, it has theuniform finiteness property, i.e. eliminates “ ∃ ∞ ”.In the theory T G , recall that we require that G is divisible, hence G | = ODAG as well. If G were not a divisible subgroup in some model ( M , G ) of T G , then it is immediate that ( M , G ) doesnot embed into any existentially closed model of T G , which is why we require divisibility. Notethat since T expands ODAG, stipulating that G is a dense subgroup and also a proper subset of D = int( G ) implies that G is codense as well. Remark 2.1.
Let
M | = T be a | T | + -saturated model, and let ϕ be an n + m -ary L -formula. Thenthe set of tuples ¯ b ∈ M m for which there is ¯ a ∈ M n with | = ϕ (¯ a, b ) and a i dcl L (ˆ a i ∪ ¯ b ) for each i ∈ [ n ] is definable.Proof. We know that A := { ~a : M | = ∃ ~bϕ ( ~a,~b ) } has a decomposition into finitely many cells, andthe cells on which there is a point ( ~a,~b ) such that M | = ϕ ( ~a,~b ) and a i dcl L (ˆ a i ∪ ¯ b ) correspond OMPANIONABILITY CHARACTERIZATION 7 precisely to the cells in A which have full dimension. By o-minimality, the set of full dimensioncells is finite and L -definable. (cid:3) The case that M defines an infinite family of distinct germs of endomorphisms atzero. We will first give a name to a property for a theory T which we then prove precludes thetheory T G from having a model companion. Below, when we say “endomorphism” we mean anendomorphism with respect to the binary operation (i.e. the corresponding symbol in the language L ) with respect to which G is a subgroup of models of T . We will use the symbol “+” for thisbinary operation and the language of additive groups throughout this section. Definition 2.2.
We say an o-minimal theory T has UEP (uniform endomorphisms property) ifthere is an L -formula ϕ ( x, ~y, z ) for which in every model M | = T , there is an infinite definable set J ⊆ M | ~y | such that for each ~c ∈ J there exists ǫ > such that the formula ϕ ( x, ~c, z ) defines thegraph of an endomorphism on a neighborhood of with radius at least ǫ , and for no other ~d ∈ J does ϕ ( x, ~d, z ) have the same germ at zero as ϕ ( x, ~c, z ) . Observe that in practice the property UEP reduces to the case that ~c is a singleton, since we canuse definable choice to define a path through the infinite set J that is parameterized by a singleinterval. Hence without loss of generality we shall work only with definable families of functionsthat vary with respect to a single parameter, but the formulas which define their graphs may requireadditional, fixed parameters. By taking an appropriate closed subset of J if necessary, we can alsoassume that J is topologically closed.Suppose T is as above, and that there is a ∅ -definable family of partial functions F Y := { f y : D ⊆ M → M : y ∈ Y } for which the definable family of germs G of F Y is infinite. Then by the uniformdefinability of F Y and by definable choice for T , we conclude that G is also uniformly definable. Infact, we can uniformly definably choose a representative partial function for each element of G . Thisgives us a definable sub-family F Y ′ of representative partial functions, i.e. each partial function f y does not have the same germ at zero as f y ′ for any y ′ = y ∈ Y ′ . Hence the definition of UEP isequivalent to the same statement with the uniqueness requirement for parameter ~c replaced by therequirement that the family of germs G contains infinitely many distinct equivalence classes. Lemma 2.3.
Suppose that T has UEP. Then there is a ∅ -definable family H of definable partialfunctions such that for some interval I ∋ and infinitely many q ∈ Q ∩ (0 , we have x qx | I ∈ H .Proof. Suppose that UEP is witnessed in
M | = T by the infinite, definable family of local endo-morphisms F Y := { f y : D ⊆ M → M : y ∈ Y } where Y ⊆ M is the parameter space for the familyof partial functions. We remark that o-minimality and the definition of UEP guarantee that forsome x ∈ M the set { f y ( x ) : y ∈ Y } contains an interval. We claim there exists some a > Y ′ ⊆ Y such that [ − a, a ] ⊆ T y ∈ Y ′ dom f y . Suppose not, i.e. that for every ǫ > y ∈ Y such that [ − ǫ, ǫ ] ⊆ dom f y . Then by uniform finiteness,there is some N ∈ N such that for every ǫ >
0, at most N elements of F Y have the interval [ − ǫ, ǫ ]contained in their domains. Yet this implies there are at most N functions in F Y , since the domainof every function f y ∈ F Y contains an interval about zero in its domain. Hence there must existsuch an a > Y ′ ⊆ Y , and without loss of generality we may take Y := Y ′ .We now prove that o-minimality allows us to definably choose i, s ∈ Y for which on the interval[ − a, a ] the functions f s : [ − a, a ] → M and f i : [ − a, a ] → M are such that every point { ( x, z ) : x ∈ [ − a, a ] } between their graphs is in the graph of one of the other endomorphisms in F Y . Since UEPdictates that F Y ′ has an infinite family of germs at 0, by making a smaller if necessary, we ensurethat the fiber over each x ∈ [ − a, a ] contains an interval. We (definably) choose f s so that f s ( a ) is BLOCK GORMAN in the upper half of the right-most interval in F Y ( a ) := { f y ( a ) : y ∈ Y } , and f i so that f i ( a ) is inthe lower half of the right-most interval of F Y ( a ). We observe that if i = s and there is x suchthat f i ( x ) = f s ( x ) = y , then for every q ∈ Q ∩ [ − ,
1] we have f i ( qx ) = qy = f s ( qx ), so byo-minimality they agree on an entire interval.Using the fact that F Y ′ contains infinitely many functions with distinct germs as 0, and using theabove fact that if two functions in F Y ′ coincide at a point then they do so on an interval, we can findan f s and f i whose graphs do not coincide on a neighborhood of 0. By making a smaller if necessary,we can ensure the graphs of f i and f s on [ − a, a ] only intersect at 0. By our choice of i = s ∈ Y and a , we ensure that for all x ∈ (0 , a ] \ { } we know f i ( x ) < f s ( x ). Moreover we may assume { ( x, z ) : x ∈ (0 , a ] ∧ ( f i ( x ) < z < f s ( x )) } is contained in { ( x, z ) : x ∈ (0 , a ] ∧ ∃ y ∈ Y ( f y ( x ) = z ) } since each fiber is infinite, and cell decomposition allows us to choose f s and f i for which f s ( x ) and f i ( x ) are in the interior of an interval of the fiber over each x ∈ [ − a, a ], for a sufficiently small.Without loss of generality, we restrict to the case that the family of functions F Y are only thosewhose graphs lie between that of f i and f s on all of the interval [ − a, a ].We now define a new family of functions which will contain the endomorphism x qx foreach q ∈ Q ∩ ( − , δ = f s ( a ) − f i ( a ). As in the above argument that by restricting Y tosome Y ′ we can ensure that the domains of all f y contain some interval [ − a, a ], we can also finda subinterval ˜ Y ⊆ Y and some positive elements δ ′ ≤ δ such that for all f ˜ y with ˜ y ∈ ˜ Y , we knowthat [ − δ ′ , δ ′ ] ⊆ im f ˜ y . Without loss of generality we assume that Y = ˜ Y and change δ so that now δ < max { f s ( x ) − f i ( x ) : x ∈ [ − a, a ] } ≤ δ ′ .For each y ∈ Y , let g y ( x ) = f y ( x ) − f i ( x ) and let ˜ g y ( x ) = − g y ( x ). It is clear that { g y : y ∈ Y } ∪ { ˜ g y : y ∈ Y } is also a definable family of endomorphisms on [ − a, a ] with distinct germs at 0.Observe that the partial inverse g − s : [ − δ, δ ] → [ − a, a ] is a continuous endomorphism since g s isone. Hence the family { h y : [ − δ, δ ] → [ − δ, δ ] : y ∈ Y } ∪ { ˜ h y : [ − δ, δ ] → [ − δ, δ ] : y ∈ Y } given by h y ( x ) = g y ( g − s ( x )) , ˜ h y ( x ) = − h y ( x )is again a family of partial endomorphisms with distinct germs at 0. To see this, we observe that forany x, z ∈ [ − δ, δ ] such that | x − z | < δ , we have ˜ h y ( x − z ) = − g y ( g − s ( x − z )) = − g y ( g − s ( x ) − g − s ( z ))because x, − z ∈ dom g − s . Since g − s ( x ) , − g − s ( z ) ∈ [ − a, a ] ⊆ dom g y , we know − g y ( g − s ( x ) − g − s ( z )) = − g y ( g − s ( x )) + g y ( g − s ( z )) = ˜ h y ( x ) − ˜ h y ( z ), and similarly for h y ( x − z ). We will denotethis family H = { h y : y ∈ Y } ∪ { ˜ h y : y ∈ Y } .Finally, we now show that H contains the map x qx on [ − δ, δ ] for all q ∈ Q ∩ ( − , q ∈ Q ∩ ( − ,
1) and let y q ∈ Y be such that h y q ( δ ) = qδ . We note that such a y q must exist becausewe chose a > i, s ∈ Y , such that for all x ∈ [ − a, a ] and for all z ∈ [ f i ( x ) , f s ( x )] there exists y ∈ Y such that f y ( x ) = z . This ensures that same holds for all x ∈ [ − a, a ] and all z ∈ [0 , g s ( x )],and similarly for all x ∈ [ − δ, δ ] and all z ∈ [˜ h s ( x ) , h s ( x )] = [ − δ, δ ]. Letting x = δ and z = qδ , wededuce the existence of such a y q ∈ Y .We observe that for any r ∈ Q ∩ (0 ,
1) we have h y q ( rδ ) = rh y q ( δ ) = qrδ , hence the definablemaps h y q and x qx agree on infinitely many points between 0 and δ . Thus they must agreeon an interval containing infinitely many points of the form rqδ with r ∈ Q ∩ (0 , δ , δ ] ⊆ [0 , δ ] by o-minimality. By shifting the elements h ∈ H to the left by δ and down by h ( δ ), and by defining h y ( − x ) = − h y ( x ), we ensure that H contains the germ of x qx at zero, as desired. We note also the ∅ -definability of H follows fromthe ∅ -definability of a and δ , which is immediate by o-minimality and definable choice for T . (cid:3) OMPANIONABILITY CHARACTERIZATION 9
With this lemma we can now prove the following negative result:
Theorem 2.4.
Suppose that T has UEP. Then there is no model companion for the theory T G .Proof. By Lemma 2.3, there is a definable family of partial functions H = { h y : M → M : y ∈ Y } (where Y is the parameter space for the family of functions) as described in the above lemma. Wenow suppose for contradiction that the class of existentially closed models of T G are axiomatizable,say by theory T EC G . Let ( M , G ) | = T EC G be an ℵ -saturated model. We let I ⊆ T { dom( h y ) : y ∈ Y } be nonempty and ∅ -definable. Note that such an interval I ∋ I is contained in D = int( G ).First, let us reindex the family H using an element γ ∈ G ∩ T y ∈ Y dom h y in the following way.We let Y γ = { h y ( γ ) : y ∈ Y } , and by Lemma 2.3 we note that for each q ∈ Q ∩ (0 ,
1) the element qγ is in Y γ . Since we are simply reindexing, for each element y ∗ := h y ( γ ) ∈ Y γ where y ∈ Y , westill define h y ∗ ( x ) = h y ( x ) for all x ∈ dom h y . We let X := { y ∈ Y γ : ∀ g ∈ ( G ∩ I )( h y ( g ) ∈ G ) } . Wedefine a superstructure ( M ′ , G ′ ) ⊇ ( M , G ) by taking M M ′ to be an | M | + -saturated elementarysuperstructure, and let G ′ = G L b ∈B Q b where B ⊆ I ′ is dcl L -independent over M and, usingchoice, we may ensure that B is both dense and codense in I ′ . By construction it is clear that( M ′ , G ′ ) | = T G , and ( M , G ) is existentially closed in ( M ′ , G ′ ) since we assume ( M , G ) is in the classof existentially closed models of T G .Let X ′ be the interpretation in ( M ′ , G ′ ) of the formula that defines X in ( M , G ). Since forevery y ∈ X ′ we have ( M ′ , G ′ ) | = h y ( b ) ∈ G ′ , we know b ∈ B implies h y ( b ) = P i ∈ [ m ] q i b i + a forsome b , . . . , b m ∈ B and q , . . . , q m ∈ Q and a ∈ M . We suppose that y ∈ X ′ ∩ G ′ , and note thatthere are infinitely many elements in X ′ ∩ G ′ since γ ∈ X ′ ∩ G ′ and so too is every positive rationalmultiple smaller than γ . Without loss of generality, we may assume that h y ( x ) is not identicallyzero on any nontrivial interval containing zero.We know that y = P i ∈ [ m ′ ] q ′ i b ′ i + a ′ for some q ′ , . . . , q ′ m ′ ∈ Q and b ′ , . . . , b ′ m ′ ∈ B and a ′ ∈ G ,hence for each b ∈ B we conclude h P i ∈ [ m ′ ] q ′ i b ′ i + a ′ ( b ) = X i ∈ [ m ] q i b i + a. This in turn implies that b ∈ dcl L ( { b , . . . , b m , b ′ , . . . , b ′ m ′ } ∪ G ). This would contradict that B isdcl L -independent over G unless h y ( b ) is also L ( { b } ∪ G )-definable. Hence we conclude that q i = 0precisely if b i = b . Since h y ( b ) = q i b + a we also know for each n ∈ N that h y ( b/n ) = q i b/n + a/n .By o-minimality the definable function h y ( x ) − q i x is either non-constant with respect to x or takeson finitely many values on any interval to the right of zero. Since a ∈ G , it cannot be a non-constantfunction of b ∈ B , which means a = 0. Moreover, since h y agrees with x q i x on all rationalmultiples of b , it agrees with this function on an interval by o-minimality. Since this is true foreach b ∈ B , we conclude that on a sufficiently small interval of zero (depending on y ) we must have h y ( x ) = qx for some q ∈ Q . Since y, y ′ ∈ Y and y = y ′ implies h y = h y ′ , we conclude that y = qγ for some q ∈ Q .Finally, we will observe that X ∩ G ⊆ X ′ ∩ G as subsets of M ′ , and we conclude that X ∩ G is a countable set. Since M M ′ , we have Y M γ = M ∩ Y M ′ γ . It only remains to see that( M , G ) | = ∀ g ∈ G ∩ I ( h y ( g ) ∈ G ) implies that ( M ′ , G ′ ) | = ∀ g ∈ ( G ∩ I )( h y ( g ) ∈ G ) for each y ∈ Y ∩ M . This is immediate from the existential closedness of ( M , G ) in ( M ′ , G ′ ), so y ∈ X ∩ G implies y ∈ X ′ ∩ G ′ . This contradicts the axiomatizability of the existentially closed models, sincethen ( M , G ) must be an uncountably-saturated model which defines a countable infinite set. (cid:3) The case that definable families of endomorphisms in M have finitely many germs. Throughout this section, let T be a theory that does not have UEP. Let M | = T be a saturatedmodel. To establish our criterion, we will examine the interaction of definable curves in M n withdefinable endomorphisms in multiple variables in the context of linearly bounded structures. Here,we use “curve” to mean a function f : D → M n where D ⊆ M is an open interval.We are now ready to state and prove the demarcation lemma for companionability of ( M , G ).This lemma, and the theorem which follows, will prove Theorem B in conjunction with Theorem2.4 from the previous subsection. In essence, the following lemma shows that we can recover anyhyperplane that intersects an arbitrary definable curve in M n on an open subset of its domain. Lemma 2.5.
Suppose m, n ∈ N and let n ≥ . For every definable family of germs of curves atthe origin: { F ( x, ~a ) = ( f ( x, ~a ) , . . . , f n ( x, ~a )) : ~a ∈ A ⊆ M m } there are only finitely many definable hyperplanes H ⊆ M n for which there exists an n -box B nǫ ( ~ and a tuple ~a ∈ A such that M | = ∀ ~y ∈ B nǫ (0)( ∃ xF ( x, ~a ) = ~y → ~y ∈ H ) .Proof. First, remark that we can expand the family of functions { F ( x, ~a ) : a ∈ A } to the higher-dimensional family { F ′ ( x, ~a ) = ( x, F ( x, ~a )) : a ∈ A } and in doing so regard the graphs of the originalfamily of functions as the image of the new family of functions. In light of this, we may addressthe case that some coordinate function f i ( x, ~a ) is itself a local endomorphism of x by applying thestatement of the lemma to the expansion of F ( x, ~a ) by the coordinate function x x . By themonotonicity theorem, we can define the finite set of intervals D ~a ⊆ dom F ( x, ~a ) on which each f i is injective and continuous. We will proceed by induction on n , the dimension of the image of F .We exclude the intervals on which some f i is constant since on those intervals the result will followby induction hypothesis.We perform a series of manipulations on F ( x, ~a ), in which we iteratively precompose the inverseof the coordinate function of the i th coordinate and then take the ∆-function of the resultingcurve. We will do this for the first coordinate function, conclude that we can do the same for eachcoordinate, and the result will follow by induction. We may further expand the family F ( ~x, ~a )and the parameter tuple ~a so that F ( ~x, ~a~b ) := F ( ~x − ~b, ~a ) − F ( ~b, ~a ) is in the family of functionsfor each ~b ∈ dom F ( ~x, ~a ) and each ~a ∈ A . Below we will suppress the parameter tuple ~a , using F ( x ) := F ( x, ~a ), and f i ( x ) := f i ( ~x, ~a ). For each H a definable hyperplane that intersects the imageof F as specified in the hypotheses, let h : M n − → M be a definable EIC transformation witnessingthat H is such a definable hyperplane.If n = 2, we can precompose f − with F ( x ) and conclude that on some interval f ( B ǫ a (0)) wecan define the function ˜ F ( y ) = F ( f − ( y )) = ( y, f ( f − ( y ))). If we assume that for some ~a theimage of F intersects the graph of a definable endomorphism h : M → M on some interval, then weconclude that F ( f − ( y )) = ( y, f ( f − ( y ))) = ( y, h ( y )) on that interval. By the definition of T nothaving UEP, this h can only be one of finitely many local definable endomorphisms for all ~a ∈ M m .The case n = 2 thereby reduces to the n = 1 case, so we now take our base case to be n = 3.Consider F ( x ) = ( f ( x ) , f ( x ) , h ( f ( x ) , f ( x )))defined on B ǫ (0), where h : M → M is a definable EIC transformation. As indicated by ourchoice of domain decomposition, we assume that F is continuous and injective in each coordinateon B ǫ (0). We make the assumption that f i (0) = 0 for i ∈ [2] since if some curve in the family F nontrivially intersects a shift of a definable hyperplane, then the curve shifted to pass through OMPANIONABILITY CHARACTERIZATION 11 the origin (which we have included in the family F ) intersects the unshifted definable hyperplane.We define ˜ F ( y ) = ( y, f ◦ f − ( y ) , h ( y, ( f ◦ f − )( y ))) on B ǫ (0) ⊆ f ( B ǫ (0)) for a suitably chosen ǫ >
0. We now consider the function ∆ t ˜ F ( y ) = ˜ F ( y + t ) − ˜ F ( y ) and observe˜ F ( y + t ) − ˜ F ( y ) = ( t , f ◦ f − ( y + t ) − f ◦ f − ( y ) , h ( t , ( f ◦ f − )( y + t ) − ( f ◦ f − )( y ))) . Let f , − ( x ) := f ( f − ( x )). There are now two cases, by the o-minimality of M . Either there is no˜ ǫ > t > t f , − ( y ) is nowhere constant on (0 , ˜ ǫ ) with respect to y , or thereexists a t > , ˜ ǫ ) on which ∆ t f , − ( y ) is monotone increasing or monotonedecreasing with respect to y .Suppose we are in the first case, i.e. there is some ˜ ǫ > t the function ∆ t f , − ( y ) is constant on (0 , ˆ ǫ ) for some ˆ ǫ >
0. We appeal to Lemma 1.4 toconclude that ∆ t f , − ( y ) being constant on a neighborhood (0 , ˆ ǫ ) means that f , − ( y ) is itself alocal endomorphism on (0 , ǫ ). Consequently, we observe that h ( y, f , − ( y )) intersects a definablehyperplane on (0 , ǫ ). If there were infinitely many distinct choices (as a function of parameters ~a )for h , then projecting ˜ F ( x ) onto its first and third coordinates would yield a definable family ofendomorphisms, contradicting the assumption that T does not have UEP. So we conclude that theclaim holds for case 1.Assume now we are in case 2. Without loss of generality, let ˜ ǫ > t ∈ (0 , ǫ ) for which the function ∆ t ˜ F ( y ) = ˜ F ( y + t ) − ˜ F ( y ) is continuous and injective as afunction of y in each coordinate on interval I := (0 , ˜ ǫ ). Fix one such t , and we define this functionas F (1) := ∆ t ˜ F ( y ), which equals˜ F ( y + t ) − ˜ F ( y ) = ( y + t , f , − ( y + t ) , h ( y + t , f , − ( y + t ))) − ( y, f , − ( y ) , h ( y, f , − ( y )))= ( t , ∆ t f , − ( y ) , h ( t , ∆ t f , − ( y ))) . We remark that by our assumption, ∆ t f , − ( y ) is invertible as function of y on a neighborhood of0, so we now define ˜ F ′ : ∆ t f , − ( I ) → M given by˜ F ′ ( x ) := F (1) ((∆ t f , − ) − ( x )) = ( t , x, h ( t , x )) . Since we definably chose t >
0, the tuple ( t , , h ( t )) is uniformly definable in the same parameters ~a as ˜ F ′ , hence the function ˆ F ( x ) = ˜ F ′ ( x ) − ( t , , h ( t , ~a .Shifting ˆ F to have 0 in its domain if necessary, we see that ˆ F projects onto the definablehyperplane given by ˜ h ( x ) = ( x, h (0 , x )), which is uniformly definable in terms of parameters ~a .Exchanging the roles of f and f in the above proof, we similarly see that ˜ h = ( x, h ( x, ~a as well. By Lemma 1.6, together ˜ h and ˜ h uniquelydetermine the definable EIC transformation h . By Lemma 2.3, this means there are finitely manydefinable EIC transformations that h can possibly be as ~a ranges over the parameter space.The induction step proceeds analogously to the base case. Let n > M n . Let F ( x ) = ( f ( x ) , . . . , f n ( x ) , h ( f ( x ) , . . . , f n ( x )))be as in the statement of the lemma, with h : M n → M a definable transformation. We makethe same assumption as in the base case, that on B ǫ (0) each coordinate function is injective, andsurjects onto a neighborhood of the origin of at least diameter ǫ . We define:˜ F ( y ) = (cid:0) y, f ◦ f − ( y ) , . . . , f n ◦ f − ( y ) , h ( y, f ◦ f − ( y ) , . . . , f n ◦ f − ( y ) (cid:1) on B ǫ (0) := f ( B ǫ (0)). Note that since f is continuous it must map open balls to open balls, andsince we can again replace f ( x ) with f ( x ) − f (0), and let ǫ be half the diameter of f ( B ǫ (0)). We now consider F ′ ( y ) = ∆ t ˜ F ( y ) = ˜ F ( y + t ) − ˜ F ( y ) = ( t , f ◦ f − ( y + t ) − f ◦ f − ( y ) , . . . , f n ◦ f − ( y + t ) − f n ◦ f − ( y ) , h ( t , f ◦ f − ( y + t ) − f ◦ f − ( y ) , . . . , f n ◦ f − ( y + t ) − f n ◦ f − ( y )).We definably choose t > t f i, − ( y ) is constant with respect to y for the fewestpossible number of indices i ∈ { , . . . , n } . We define the components of a new tuple ~g ( t ) as follows: g i ( t ) = t , i = 1∆ t f i, − ( ǫ ) , if ∆ t f i, − ( y ) is constant on an interval (0 , δ ) , i ∈ { , . . . , n } , if ∆ t f i, − ( y ) non-constant on each interval (0 , δ ) , i ∈ { , . . . , n } h ( y, g ( t ) , . . . , g n ( t )) i = n + 1 . We observe that if ∆ t f i, − is not constant on any interval with left endpoint 0 for any 2 ≤ i ≤ n ,then ~g = ( t , , . . . , , h ( t , , . . . , ~g ∈ M n +1 is uniformly definable inthe same parameters ~a as F . Hence the function ˆ F ( x ) = F ′ ( x ) − ~g ( t ) is uniformly definable over ~a as well.In the case that ~g ( t ) = ( t , , . . . , , h ( t , , . . . , F ( x ) = (cid:0) , ∆ t f , − ( y ) , . . . , ∆ t f n, − ( y ) , h (0 , ∆ t f , − ( y ) , . . . , ∆ t f n, − ( y )) (cid:1) is equal to { } × ˆ F where ˆ F is an n -dimensional curve. We apply the induction hypothesis toconclude that there are finitely many possible n − F can locally coincide. Hence the n th coordinate of ˆ F can only coincide with one of finitely manyEIC transformations with respect to the change of variables ˆ z := (∆ t f , − ( y ) , . . . , ∆ t f n, − ( y )).Let ˜ h (ˆ z ) := h (0 , z , . . . , z n − ) denote this coordinate function with the change of variables.For each f k with k ∈ { , . . . , n } , instead of precomposing f i with f − , we precompose f i with f − k and repeat the argument to conclude that there are similarly only finitely many possible n − F k , which is defined analogously to ˆ F , can locallycoincide. Then we perform a similar change of variables to that in the paragraph above, and define˜ h k (ˆ z k ) = h ( z , . . . , z k − , , z k +1 , . . . , z n ) where z i := ∆ t k f i, − k ( y ) for each i ∈ [ n ] \ { k } . By Lemma1.6, an EIC transformation is uniquely determined by how it acts on each coordinate, so we canrecover the EIC transformation h uniquely from the coordinate functions ˜ h , . . . , ˜ h n . We concludethat there are only finitely many n -dimensional EIC transformations that h can be.Now consider the case that g k ( t ) = 0 for the indices k ∈ { , . . . , n } such that k ∈ J , where ∅ 6 = J ⊆ { , . . . n } . We observe that by Lemma 1.4, the vector-valued function ~f ∗ , ( y ) where f ∗ , i ( y ) = f i, − ( y ) ⇐⇒ i = 1 or i ∈ J and f ∗ , i ( y ) = 0 if i ∈ [ n ] \ J , and f ∗ , n +1 ( y ) = h ( y, f ∗ , ( y ) , . . . , f ∗ , n ( y )),coincides with the graph of an endomorphism in each pair of coordinates ( f , f i ) where i ∈ J or i = n + 1. We iterate this process with f i, − k instead of f i, − for each k ∈ [ n ] to obtain a function ~f ∗ ,k ( y ) analogous to the function ~f ∗ , ( y ) described above.For each iteration, we apply the induction hypothesis to the projection of ˆ F ( y ) onto the coordi-nates which are not identically zero on some interval with left endpoint 0. Define h k ( ~x ) to be the n th coordinate of ˆ F ( y ) for the k th iteration of this process. We know that each h k ( ~x ) is a definableEIC transformation that is obtained from h ( ~x ) by setting x i = 0 for all i in some subset J k ⊆ [ n ],with | J k | ≥
2. By induction, there are only finitely many definable EIC transformations that each h k can be. Using a generalized version of 1.6, we know we can recover h uniquely from the h k ’s,hence there also are only finitely many possible EIC transformations that h can be, finishing theinduction step. (cid:3) OMPANIONABILITY CHARACTERIZATION 13
Corollary 2.6.
Suppose that F ( ~x, ~y ) : D × A ⊆ M n + k → M m is L ( ∅ ) -definable, and for every ~a ∈ A ⊆ M k (the parameter space) F ( ~x, ~a ) is a continuous function with domain D ⊆ M n . Then { ˆ F ( ~x ) = [ F ( ~x + ~b, ~a ) − F ( ~b, ~a )] | B nǫ (0) : ~a ∈ A,~b ∈ D, ǫ > , gr ( ˆ F ) ⊆ H ⊆ M m a definable hyperplane } collapses to a finite definable family of germs of functions (as in definition 1.2) through the origin.Proof. We proceed by induction on n = | ~x | . For the base case, suppose that ~x = x is a singlevariable. We consider the following family of functions with restricted domain: F := { ˆ F ( x + b, ~a ) := F ( x + b, ~a ) − F ( b, ~a ) | B ǫ (0) : b ∈ D, ~a ∈ A, ǫ > , gr ( ˆ F ) ⊆ H a def. hyperplane } and consider the set of ~a ∈ A and b ∈ M such that ∃ δ > ∀ x, y ∈ B δ (0)( ˆ F ( x + y + b, ~a ) =ˆ F ( x + b, ~a )+ ˆ F ( y + b, ~a )). This carves out a ∅ -definable subfamily of F that collapses (as in definition1.2) to a collection of germs of endomorphisms in each coordinate. We apply Lemma 2.5 to concludethat for all b ∈ M and ~a ∈ A there are only finitely many definable hyperplanes which coincidewith ˆ F ( x + b, ~a ) on some neighborhood.Now let n > n −
1. Write ~x = ( x , . . . , x n ), and let H = { H ~b,~a ⊆ M m : ~a ∈ A,~b ∈ D } enumerate the definable hyperplanes H ~b,~a that coincide with ˆ F ( ~x + ~b, ~a ) forsome parameters ~a ∈ A and ~b ∈ D . Since Lemma 2.5 holds for every arity of parameter tuple, wesimply “move” the last variable x n from the domain of the function to the parameter space. Bythis, we mean that we can think of the n -dimensional hypersurface ˆ F ( ~x + ~b, ~a ) as a definable familyof n − { ˆ F ( x + b , . . . , x n − + b n − , c, ~a ) : M n − → M m : ~b ∈ π [ n − ( D ) , c ∈ π n ( D ) , ~a ∈ A } , in which ( c, ~a ) is now the parameter tuple ranging over π n ( D ) × A ,with domain π nn − ( D ). We will write ˆ x n = ( x , . . . , x n − ). We now use the induction hypothesisto conclude there are only finitely many definable hyperplanes, say H , . . . , H ℓ ⊆ M m , for whichthere is some ǫ -neighborhood of zero and some ˆ F (ˆ x n + ˆ b n , c, ~a ) in this family of n − H i on said ǫ -neighborhood.Finally, we appeal to the fact that the set of all m -dimensional hyperplanes that contain somelevel set of a hypersurface (intersected with a neighborhood of zero) is by definition a superset of the m -dimensional hyperplanes that contain the entire surface (intersected with the same neighborhoodof zero). This can easily be seen by writing out the set H for the family F , and the set H definedanalogously for the family F (ˆ x n , z, ~y ) (in which the n th coordinate is regarded as a parameter) asfollows: H F = { ˆ F ( ~x + ~b, ~a ) | B nǫ (0) : ~a ∈ A,~b ∈ D, ǫ > , gr ( ˆ F ) ⊆ H ⊆ M m a definable hyperplane } ⊆{ ˆ F (ˆ x n + ˆ b n , c, ~a ) | B n − ǫ (0) : c ∈ π n ( D ) , ~a ∈ A,~b ∈ D, ǫ > , gr ( ˆ F ) ⊆ H ⊆ M m a def. hyperplane } = ˆ H F . So we conclude that the set H F is finite as well. It is now easy to see that H F is a subset becauseevery hyperplane H ∈ H F that coincides with ˆ F ( ~x, ~a ) on a neighborhood U also coincides with thelevel sub-hypersurface ˆ F (ˆ x n , c, ~a ) for any c ∈ π n ( U ). This concludes the induction argument. (cid:3) By restricting our attention to o-minimal theories T which do not have UEP, we can ensure byLemma 2.5 that we can axiomatize the following property: given a generic input for a definablefunction f : M n → M m that is not a definable EIC transformation, we can obtain points in theimage of f both inside and outside of the predicate group. Definition 2.7.
Let ( M , G ) | = T G and n ∈ N > . Let ~f = ( f , . . . f k ) : D ⊆ M n → M k be an n -aryvector function that is L ( ∅ ) -definable. For I ⊆ [ k ] , let M I denote Q ki =1 P i where P i = M if i ∈ I and P i = { } if i I . Given ~d ∈ D and I ⊆ [ k ] , let ˜ f d,Ii ( ~x ) = ( f i ( ~x ) − f i ( ~d ) , i ∈ I , i I .
We let Q denote the (non-definable) set of definable hyperplanes H that have a matrix represen-tation with exclusively Q -affine entries. Let ˜ f d,I = ( ˜ f , . . . , ˜ f k ) . We define the following: C f = n ~d = ( d , . . . , d n ) ∈ D : for some ∅ 6 = I ⊆ [ k ] there is H ⊆ M n + | I | with H ∈ Q and ∃ ǫ > ∀ x ∈ B ǫ ( ~d ) (cid:16) ( ~x, ˜ f d,I ( ~x )) ∈ H (cid:17) o Furthermore, let ˜ C f denote int( D \ C f ) , the topological interior of the complement of C f in D . Intuitively, we think of C f as being the open subset of the domain of ~f on which ~f or someprojection of ~f onto a subspace of M n is locally Q -affine. We think of ˜ C f as the obstacle-free partof the domain of a definable function. The following remark is really a corollary to Lemma 2.5. Remark 2.8.
For all ~f as described in the above definition, the set C f is L -definable. By the L -definability of C f and the fact that T is a complete theory, we conclude that for L ( ∅ )-definable sets and functions, the set C f should have the same first-order properties with respect L in every model of T G . By the L -definability of a conducive configuration we conclude that for L ( ∅ )-definable sets and functions, configurations which are conducive in one model are conducivein every model. Below, for ~x = ( x , . . . , x n ) ∈ M n we will use ~x ι to denote ( x i , . . . , x i k ) where ι = ( i , . . . , i k ) ∈ [ n ] k . Let [ n ] k denote the set of all k -element subsets of n .We now show in Lemma 2.9 that we may apply Proposition 1.12 from [7] to the theory T G when T does not have UEP. Below T is a reduct of T with sub-language L ⊆ L such that the algebraicclosure operation (call it acl ) is a pregeometry, and | ⌣ is the independence relation associated toacl . D’Elb´ee uses the following definition in his companionability criterion, in which T S is thetheory of pairs ( M , M ) where M | = T and M | = T . Definition ([7], 1.10) . We say that a triple ( T, T , L ) is suitable if it satisfies the following: ( H ) T is model complete; ( H ) T is model complete, and for all infinite A , acl ( A ) | = T ; ( H +3 ) acl defines a modular pregeometry; ( H ) for all L -formula ϕ ( x, y ) there is some L -formula θ ϕ ( y ) such that for b ∈ M | = T , M | = θ ϕ ( b ) ⇐⇒ there exists N < M and a ∈ N such that ϕ ( a, b ) and a is | ⌣ -independent over M . Below, the theory
T S mentioned in the criterion D’Elb´ee gives for companionability is axioma-tized in [7] as follows. For x = x x , and for each L -formula ϕ ( x, y ) and each ( τ i ( t, x, y )) i T S , and the set of all such sentences along with those in T axiomatize T S . Proposition ([7], 1.12) . Let ( T, T , L ) be a suitable triple. Then T S exists and is the model-companion for the theory T S . Lemma 2.9. If T does not have UEP, then the triple ( T, ODAG , (0 , + , ( q · ( − )) q ∈ Q )) is suitable.Proof. We already require ( H ) of T , and in the language L = (0 , + , q · ( − )) q ∈ Q ) the theory ODAGis model complete and has quantifier elimination. Given a set A in a model of T , the L -algebraicclosure acl ( A ) is the Q -linear span of A , and as such is also a model of ODAG, so condition ( H ) issatisfied. Moreover, since acl is the Q -linear closure of a set, the pregeometry it defines is modularand hence ( H +3 ) is satisfied.To see that condition ( H ) holds, suppose that ϕ ( ~x, ~z ) is a L -formula with | ~x | = d and | ~z | = m .Let Z := { ~z : ∃ ∞ ~xϕ ( ~x, ~z ) } and let Y := { ( ~x, ~z ) : ~z ∈ Z ∧ ϕ ( ~x, ~z ) } . We partition Y into definablesets D , . . . , D d and E such that the following holds. For each n ∈ [ d ] and all ( ~x, ~z ) ∈ D n , n isthe unique element of [ d ] for which there is a sub-tuple ~x [ n ] of ~x and there are definable functions ~f , . . . , ~f k n such that ~f i ( ~x [ n ] , ~z ) = ( f n +1 ( ~x [ n ] , ~z ) , . . . , f d ( ~x [ n ] , ~z )) for each i ∈ [ k n ] and a definablepartition of D n into D n , . . . , D k n n such that T | = ∀ ( ~x, ~z ) ∈ D in (cid:16) ϕ ( ~x, ~z ) ↔ ϕ ( ~x [ n ] , ~f i ( ~x [ n ] , ~z ) , ~z ) (cid:17) ∧ ~x [ n ] C ~f i ! . Recall the L ( ∅ )-definability of the set C f as defined in Definition 2.7. Additionally, for each n ∈ [ d ]and each i ∈ [ k n ], we require that for each ~z ∈ D n the set { ~x : ( ~x, ~z ) ∈ D in } either is empty orhas dimension exactly n , and we also require that for each such ~z the projection of D n onto thesub-tuple ~x [ n ] is open. We may take each D n to be the maximal subset of Y on which the aboveholds, and we also may assume each k n could not be made smaller without violating one of theabove properties. Let E contain all the tuples ( ~x, ~z ) in Y \ D ∪ . . . ∪ D d .We think of D n as being the intersection of C c~f with the subset of ϕ that has natural dimension n . We assume without loss of generality that on each cell D n , the tuple ~x is ordered in such a waythat the set of components of ~x that can be chosen acl-independent of each other come before therest. Consider the following formula: θ ϕ ( ~z ) := d _ n =1 k n _ i =1 ∃ ~xϕ ( ~x, ~z ) ∧ ( ~x, ~z ) ∈ D in . We now show that θ ϕ ( ~z ) holds precisely if there is N < M and a ∈ N such that N | = ϕ ( a, b ) and a is | ⌣ -independent over M .First, we suppose that M | = T , ~b ∈ M | ~z | and M | = θ ϕ ( ~b ). Let N < M be a M + -saturatedelementary superstructure (in particular, also max {|L| + , ℵ } -saturated). Let n ≤ d and i ≤ k n besuch that ~b ∈ D in . Since the projection onto the first n coordinates of D in is open, we know thatthe partial type over M given byΘ ~b = ( ϕ ( ~x [ n ] , ~f i ( ~x [ n ] ,~b ) ,~b ) ∧ (cid:16) ~x [ n ] C ~f i (cid:17)) [ ( ~q · ( ~x [ n ] , ~f i ( ~x [ n ] ,~b )) = a : ~q ∈ Q d , a ∈ M ) is consistent with the theory of M because it is finitely satisfiable, by the definition of C c~f . Henceit has a realization ~a ∈ N d . It is clear that if ~a satisfies ϕ ( ~x,~b ) and the sub-tuple ~a [ n ] satisfies the partial type Θ ~b , then it does not lie in any set that is acl -dependent over M , since those areprecisely the Q -affine hyperplanes defined over M . So the ( = ⇒ ) part of condition ( H ) holds.Now suppose that there is N < M and ~a ∈ N such that N | = ϕ ( ~a,~b ) and ~a is | ⌣ -independentover M . Note that if ( ~a,~b ) 6∈ Y , then ~a would have to be in the dcl of ~b in which case ~a ∈ M ,which would contradict the | ⌣ -independence of ~a over M . If ( ~a,~b ) ∈ E , due to o-minimal celldecomposition either one of two things must hold. The first possibility is that for some n ∈ [ d ] anddefinable vector function ~f we must have ~a [ n ] ∈ C ~f . Yet by definition of C ~f it would then followthat ~a is not | ⌣ -independent over M . The other possibility is that the projection of { ~x ∈ B ǫ ( ~a ) : ϕ ( ~x [ n ] , ~f ( ~x [ n ] ,~b ) ,~b ) ∧ ~x C ~f ) onto the first n coordinates is not open for each n ∈ [ d ] and for each ǫ > B ǫ ( ~a ) is a box and ~f is a definable (partial) function for which the following holds: N | = ∀ ~x ∈ B ǫ ( ~a ) (cid:16) ϕ ( ~x,~b ) ↔ ϕ ( ~x [ n ] , ~f ( ~x [ n ] ,~b ) ,~b ) ∧ ~x C ~f (cid:17) . If this is the case for all n ≤ d , then by our requirement that for each n ∈ [ d ] the set D n is maximalwith respect to containing all ǫ -balls satisfying the sentence above, subject to the condition thatthe projection onto ~x [ n ] is open, we conclude that ~a ∈ dcl( ~b ). Yet this would again contradict | ⌣ -independence over M . So for some 0 < n ≤ d and i < k n we must have ( ~a,~b ) ∈ D in . Hence M | = θ ϕ ( ~b ), and condition ( H ) holds, as desired. (cid:3) As a corollary to Lemma 2.9, and Proposition 1.12 in [7], we conclude that Theorem B holds.For the theory T G , let D be the open and unary L ( ∅ )-definable superset of G in which G is denseand codense. We define a slight variant of the formula θ ϕ from before:˜ θ ϕ ( ~z ) := d _ n =1 k n _ i =1 ∃ ~xϕ ( ~x, ~z ) ∧ ~x ∈ D ∧ ( ~x, ~z ) ∈ D in . Theorem (B) . For T an o-minimal expansion of the theory of ordered abelian groups with language L , let T G be the L ∪ {G} -theory extending T that states also that G is a divisible subgroup both denseand codense in D . If the only families of additive endomorphisms whose graphs are uniformlydefinable in a neighborhood of have finitely many germs at , then T G has a model companion T ∗G given as follows. For ~x = ~x ~x , and for each L -formula ϕ ( ~x, ~y ) and each ( τ i ( ~t, ~x, ~y )) i 6⊆ G (cid:17)(cid:17)! is in T ∗G , and T G ⊆ T ∗G . Otherwise, T G does not have a model companion.Proof. Under the given hypotheses, the axiomatization for T ∗G follows from Proposition 1.12 in [7].We can apply Proposition 1.12 because Lemma 2.9 shows the hypotheses of the proposition aresatisfied if T does not have UEP. The proof of the “otherwise” statement is given by Theorem2.4. (cid:3) We will see in the examples section below that this characterization translates to an even cleanerdichotomy when applying the characterization to an o-minimal expansion R of a real closed fieldwith an added predicate for a multiplicative subgroup dense in R > . OMPANIONABILITY CHARACTERIZATION 17 Examples We consider the connections that the results in this paper have to the framework of [4]. Let R K denote ( R , <, + , ( x kx ) k ∈ K ), where K ⊆ R is a subfield. We observe that the structure( R , Q ) = ( R , <, + , ( x kx ) k ∈ K , Q ) is a model of T G . The theory of this structure is not modelcomplete in general, though T G in this case does have a model companion. In the language L U =(0 , , <, + , ( k ( x )) k ∈ K , U ( x )), where k ( x ) is the symbol for scalar multiplication by k ∈ K and U isthe predicate that picks out Q , we know that T dK := T h ( R K , Q ) is what the authors of [4] call an“ML theory.” In particular, this implies it is near-model complete. Yet it is not model complete,and what fails is linear disjointness. Example 3.1. Consider the field K = R alg ( e ) . Let R = R alg ( e, ζ, η ) where ζ is an algebraicallyindependent transcendental number K , and η is an algebraically independent transcendental numberover K ( ζ ) . Let R = R alg ( e, ζ ) . Let Q = Q , and let Q = Q ( − eη + ζ, η ) . Then it is easy to check ( R , Q ) and ( R , Q ) are models of T dK , and by construction ( R , Q ) ⊆ ( R , Q ) . However it isnot the case that ( R , Q ) ( R , Q ) since ( R , Q ) | = ∃ q ∃ q U ( q ) ∧ U ( q ) ∧ ( e ( q ) + q = ζ ) , but ( R , Q ) | = ∀ q , q ( U ( q ) ∧ U ( q ) → e ( q ) + q = ζ . The case that + is multiplication in a field. We can apply the companionability char-acterization to pairs in which the underlying o-minimal structure M expands a real closed field,and the group G is a multiplicative subgroup of M > . Interpreting 0 , + in L from section 2 as 1 M and · M , we can clearly establish the same companionability dichotomy for o-minimal expansions ofRCF. We make this more concrete in the examples below. Example 3.2. An example of a non-companionable structure is ( R exp , G ) where G < R > is adense, divisible multiplicative subgroup. This follows since ln( x ) is definable in R exp , so y x = e x ln( y ) is definable. Hence the family of multiplicative endomorphisms { x r : 0 < r < } is uniformlydefinable in R exp , and witnesses that T h ( R exp ) has UEP. By Theorem 2.4, this implies no modelcompanion exists. Theorem (A) . Let T be an o-minimal theory expanding the theory of real closed fields (RCF), andlet T ×G be the expansion by a predicate for a divisible multiplicative subgroup of the positive elementsthat is dense and codense as a subset of an open unary set definable in L . Then T ×G has a modelcompanion if and only if for every model M | = T there is no constant C ∈ M such that the graphof Ce x is locally definable on some interval.Proof. ( = ⇒ ) If Ce x is definable on some interval I ⊆ M , then we can apply Theorem 2.4 toconclude that T ×G does not have a model companion, as illustrated in Example 3.2.( ⇐ = ) We suppose that M defines an infinite family of local endomorphisms of the multiplicativegroup M > , and we show this implies that a function defined on some interval I satisfies thedifferential equation that characterizes functions of the form Ce x . We let ˜ F ( x, y ) be any definablepartial function in M such that its domain is the box I × ˜ J ⊆ M > × M , where we assume 1 ∈ I ,and ˜ F ( x , y ) ˜ F ( x , y ) = ˜ F ( x x , y ) for all x , x ∈ I and y ∈ ˜ J . Such a function F can be defined in M using the existence of an infinite definably family of functions, the o-minimality of the structure M , and the definable choice functions o-minimality gives us.We write ˜ F ( x, y ) = ˜ f y ( x ) for a fixed y ∈ ˜ J . We now define J = { ˜ f ′ y (1) : y ∈ ˜ J } , where˜ f ′ y ( x ) = ddx ˜ f y ( x ), and let η ( y ) = ˜ f ′ y (1), which we can without loss of generality take to be continuousand injective, restricting ˜ J if needed. One can show that by possibly inverting and shifting someof the functions ˜ f y , we can make 0 the left endpoint of J . Let F ( x, y ) : I × J → M be given by F ( x, y ) = ˜ F ( x, η − ( y )). We now deduce that since f y ( x ) f y ( x ) = f y ( x x ) we also have ∂∂z f y ( xz ) = ∂∂z f y ( x ) f y ( z ) = f y ( x ) f ′ y ( z ) and ∂∂z f y ( xz ) = xf ′ y ( xz ), so f y ( x ) = f ′ y ( xz ) · xf ′ y ( z ) . Letting z = 1, we conclude that for each y ∈ J we know f y ( x ) = f ′ y ( x ) · xf ′ y (1) characterizes the function f y subject to the constraints f y (1) = 1 and ∀ x, z ∈ I ( f y ( xz ) = f y ( x ) f y ( z )). We hence conclude forany y , y ∈ J that f ′ y (1) = f ′ y (1) ⇐⇒ f y = f y .We will use this to show that F ( x, y + y ) = F ( x, y ) · F ( x, y ). As a product of definableendomorphisms, we observe F ( x, y ) · F ( x, y ) is a definable endomorphism as well. By our aboveremarks, f ′ y + y (1) uniquely determines the function f y + y subject to the constraint of being adefinable endomorphism. We know f ′ y + y (1) = y + y by how we defined F ( x, y ) and J withthe use of η ( y ). We also observe ∂∂x F ( x, y ) F ( x, y ) = F ( x, y ) F ′ ( x, y ) + F ( x, y ) F ′ ( x, y ). Since f y ( x ) is an endomorphism on I , we know that f y (1) = 1 for all y ∈ J . So ∂∂x F ( x, y ) F ( x, y ) at x = 1 is F (1 , y ) F ′ (1 , y ) + F (1 , y ) F ′ (1 , y ) = y + y , hence f y ( x ) f y ( x ) has y + y = f ′ y + y (1)as its derivative at x = 1, which makes them the same function on I .Finally, we show that ∂∂y F ( x, y ) = C ( x ) F ( x, y ) where C ( x ) is purely a function in x to concludethat F ( e, y ) is equivalent to an exponential function (or shift thereof) on some subinterval of J .Observe thatlim h → F ( x, y + h ) − F ( x, y ) h = lim h → F ( x, y ) F ( x, h ) − F ( x, y ) h = F ( x, y ) lim h → F ( x, h ) − h so letting C ( x ) = lim h → F ( x,h ) − h we obtain the desired result. Since the differential equation ∂∂y F ( e, y ) = C ( e ) F ( e, y ) characterizes all functions of the form c e c y + c where c , c , c ∈ R , weconclude that the function F ( e, y ) coincides with an exponential function on a subinterval of J . (cid:3) When T is an o-minimal expansion of RCF and T ×G is the expansion described in Theorem A,we use T ∗G to denote the model companion of T ×G , if it exists. Depending on the theory T , it willbe clear from context which kind of model companion T ∗G denotes for the remainder of this paper. Example 3.3. For the structure R ∗ := ( R , , , + , · , ( k ) k ∈ K , ( x k ) k ∈ K ) , where K ⊆ R is any subfield,the pair ( R ∗ , G ) , with G ⊆ R > dense and codense in R > , has a model companion.Proof. If r x is definable on some interval in R , then it is definable from the real field ¯ R augmentedwith finitely many functions of the form x k , say x k , . . . , x k n . By results of Bianconi [2], however,a function x β is definable in a structure only if β is in the field generated by k , . . . , k n . Since wecan find such a β , and r x can be used to define x β , we must conclude r x is not defined anywherein R . Hence by the above proposition, the model companion exists. (cid:3) Neostability and Tameness In [10], Kruckman, Tran, and Walsberg also give many general criteria under which neostabilityproperties, such as NIP and NSOP , are preserved by the fusion of theories with these properties.These preservation results are very much in the spirit of [5], [7], and [9]. However, they also provideexamples of interpolative fusions that have TP despite both theories being fused having NTP .In this section, we similarly show that depending on the o-minimal base theory T , the modelcompanion can have IP or NIP, can be strong or not strong, and can have NTP or TP .For a comprehensive list of definitions and equivalent formulations of the properties NIP, NTP2,and strong or finite burden, please refer to [14] and [1], respectively. We first observe that if T is just the theory of ordered divisible abelian groups, then the theory T G is a dense pair in thesense of [6]. In [6], van den Dries shows that the theory T d of a dense pair is complete if T is OMPANIONABILITY CHARACTERIZATION 19 complete. Since T G aligns with T d when T is the theory ODAG, we conclude that the theory T ∗G isaxiomatized by T d as defined in [6]. Moreover, by [3] this theory has NIP (it is not hard to showthat T ∗G has the property they call “innocuous” in [3]). On the opposite end of the spectrum, themodel companion T ∗G has TP if T expands the theory of real closed fields, G is a subgroup of themultiplicative group, and T G is companionable. Theorem 4.1. For T the theory of a real closed field, the theory T G , where G is a subgroup of themultiplicative group, has a model companion T ∗G that has TP .Proof. That for this T the theory T G has a model companion follows from Theorem A. Let M | = T ∗G be an ℵ -saturated model, and let ( b i ) i ∈ N and ( c ( i,j ) ) i,j ∈ N be countable sequences of elements in M , where every finite subset of ( c ( i,j ) ) j ∈ N is Q -linearly independent over G . We can choose thesequence ( b i ) i ∈ N to be Q -linearly independent from each other and the sequence ( c ( i,j ) ) ( i,j ) ∈ N over G , since as a divisible subgroup G must be infinite index in M , and since we take M to besuitably saturated. Consider the array generated by these indiscernible sequences and the formula ϕ ( x, b i , c ( i,j ) ) = b i · x + c ( i,j ) ∈ G . For any n ∈ N let A be the n × n + 1-matrix which representsa Q -linear homogenous system of equations generated by a path of length n through the array, i.e. A i,k = b i if k = 1, for 1 < k < n + 2 we let A i,k = − k = i + 1 and A i,k = 0 otherwise, and if n + 2 ≤ k ≤ n + 1 then A i,k = 1 if k = n + i + 1 and A i,k = 0 otherwise, where { j , . . . , j n } ⊆ N .Any two formulas in a row of the array are inconsistent, since each column requires that b i · x is in a different coset of G . By the Q -independence of the sequences ( b i ) i ∈ N and ( c ( i,j ) ) ( i,j ) ∈ N over G , for any n ∈ N and j < . . . < j n ∈ N , the corresponding matrix A has rank n , whichimplies n+1 free variables for the corresponding solution set. So there are infinitely many tuples( x, y , . . . , y n , c (1 ,j ) , . . . , c ( n,j n ) ) that satisfy the equation A · ( x, y , . . . , y n , c (1 ,j ) , . . . , c ( n,j n ) ) = ~ c (1 ,j ) , . . . , c ( n,j n ) ) is regarded as the part uniquelydetermined by the system of equations, we can then regard the sub-tuple ( x, y , . . . , y n ) as the freevariables. In particular, we conclude that the set of components ( x, y , . . . , y n ) that correspond toa fixed ( c (1 ,j ) , . . . , c ( n,j n ) ) as a solution set for the above matrix equation has interior in M n +1 .Hence the companion axioms tell us we can find a solution such that y , . . . , y m ∈ G . So T ∗G hasTP , as witnessed by this array. (cid:3) Let V S be the theory of a real ordered vector space with base field K in the language L = { <, , + , ( k ) k ∈ K , ( k ( x )) k ∈ K } where ( k ) k ∈ K enumerates constant symbols for each element of K , and( k ( x )) k ∈ K enumerates scalar multiplication functions for each element of K . In particular, V S contains the axioms for an ordered divisible abelian group as well as the axioms for an orderedvector space. In the language L we know V S has quantifier elimination and V S G does not haveUEP, yet V S G is not model complete by example 3.1. However, it does have a model companion V S ∗G . Using quantifier elimination we can show that V S ∗G has NIP and may or may not have finiteburden, depending on the base field. To show both NIP and that finite burden occurs in a specialcase, we first need the following lemma. Below, we use “ · ” to denote the usual dot product. Definition 4.2. Let V S be the theory of a real ordered vector space with base field K , and let ( M , G ) | = V S ∗G . Define the coset type of an element a ∈ M over C ⊆ M to be a maximalconsistent set of formulas of this form: ua − ~v · ~c ∈ G where u, ~v ∈ K , ~c ∈ C , and each of these formulas are modeled by ( M , G ) . Lemma 4.3. Let V S be the theory of a real ordered vector space with base field K . For every ( R , G ) | = V S ∗G and for each r ∈ R , the following hold:(1) For any acl -independent set C ⊆ R , the type tp( r/C ) is implied by the < -cut of r in acl( C ) := K h C i plus the coset type over C .(2) Suppose K = Q ( η , . . . , η n ) is a finite-dimensional extension of Q as a vector space. Thenfor every ( R , G ) | = V S ∗G , for each r ∈ R there is a finite set of elements { d , . . . , d n } ⊆ R such that for any C ⊆ R containing these elements, the type tp( r/C ) is implied by the < -cutof r in K h C i plus the coset type over { d , . . . , d n } .Proof. For (1), let ( R , G ) | = V S ∗G and { r } , C ⊆ R be as in the hypotheses, and suppose ϕ ( x, ~c ) ∈ tp( r/C ). By model completeness, this is equivalent to a disjunct of formulas of the form ∃ ~yψ ( x, ~y, ~c ) ∧ ^ i ∈ I k i x + ~u i · ~y + ~v i · ~c ∈ G ^ i I k i x + ~u i · ~y + ~v i · ~c 6∈ G where m ∈ N , I ⊆ [ m ], for all i ∈ [ m ] we have k i , ~u i , ~v i ∈ K , and ψ is a quantifier-free L -formulawithout disjuncts. By quantifier elimination for ordered real vector spaces, we know that everydefinable function used in ψ ( x, ~y, ~c ) is a K -linear function.Since V S eliminates ∃ ∞ , either there is ℓ ∈ N such that R | = ∃ ≤ ℓ ~yψ ( r, ~y, ~c ) or else R | = ∃ ∞ ~yψ ( x, ~y, ~c ). If the former holds, we can write each y i as a K -linear function of r and ~c , and thesubformula ^ i ∈ I k i r + ~u i · ~y + ~v i · ~c ∈ G ^ i I k i r + ~u i · ~y + ~v i · ~c 6∈ G (1)of ϕ ( r, ~c ) is implied by V i ∈ I k ′ i r + ~v ′ i · ~c ∈ G V i I k ′ i r + ~v ′ i · ~c 6∈ G for some other k ′ i ∈ K and ~v ′ i ∈ R | ~c | for each i ∈ { , . . . , m } . Subtracting ~v i · ~c on each side, we see that these conjuncts form a cosettype over C , which we see implies tp( r/C ) in conjunction with the < -cut of r in K h C i .Suppose now that R | = ∃ ∞ ~yψ ( r, ~y, ~c ). Without loss of generality, we assume there is a witnessfor ∃ ~yψ ( r, ~y, ~c ) such that ~y is acl L -independent over C ∪ { r } . Otherwise, we could write the s < | ~y | dependent coordinates as a { r } ∪ C -definable function of the s independent coordinates of ~y , whichcan be subsumed into ψ . Hence, by modifying the way we express ϕ ( x, ~c ) slightly to be in theappropriate form, we can find an axiom in the model companion axiom scheme of V S ∗G which tellsus that the formula ∃ ~yψ ( x, ~y, ~c ) implies ϕ ( x, ~c ). So ϕ ( x, ~c ) is implied by an L ( C )-formula, as desired.Therefore this formula is already implied by part of the cut of r over K h C i , so we are done.For (2), we show that in the special case that K is finite-dimensional over Q as a vector space,we get the further quantifier reduction from a similar analysis of the formulas in tp( r/C ). Above inequation 1, each k i , and each component of ~u i and ~v i is equal to q i, + q i, η + . . . + q i,n η n for some q i,j ∈ Q for each i ∈ { , . . . , m } and j ∈ { , . . . , n } , and for notational convenience define η = 1.Hence V i ∈ I k ′ i r + ~v ′ i · ~c ∈ G V i I k ′ i r + ~v ′ i · ~c 6∈ G is implied by ^ j ∈ I ′ η j r + ~v ′ j · ~c ′ ∈ G ^ j I ′ η j r + ~v ′ j · ~c ′ 6∈ G for some I ′ ⊆ { , . . . , n } . Moreover, by the hypotheses each of the negated subformulas η i x + ~v ′ i · ~c 6∈ G is implied by any formula of the form η i x − d i +1 ∈ G for i ∈ { , . . . , n } , one of which holds for r . (cid:3) That V S ∗G has NIP follows from an analysis of indiscernible sequences in light of this quantifierreduction. OMPANIONABILITY CHARACTERIZATION 21 Theorem 4.4. For V S the theory of a real ordered vector space over base field K , the theory V S ∗G has NIP.Proof. Let ( R , G ) | = V S ∗G be a monster model, though we will only use that it is | K | + -saturated.We suppose for contradiction that there is a formula ϕ ( x, ~y ) along with an element a ∈ R andindiscernible sequence ( ~b i ) i<ω that witnesses IP for V S ∗G , i.e. ( R , G ) | = ϕ ( a,~b i ) precisely if i is even.Let | ~y | = n . By model completeness of V S ∗G , the formula ϕ is equal to a disjunct of formulas of theform σ ( x, ~y ) := ∃ ~z (cid:16) ψ ( x, ~y, ~z ) ∧ ^ j ∈ I k j x + ~u j · ~y + ~v j · ~z + c j ∈ G ^ j I k j x + ~u j · ~y + ~v j · ~z + c j 6∈ G (cid:17) where I ⊆ [ m ] and | ~z | = d for some m, d ∈ N , each k j , ~u j , ~v j and c j is in K , and ψ is a quantifierfree L -formula without disjuncts. Since NIP is preserved under boolean combinations, one suchdisjunct must itself witness IP. For convenience of notation, we will change the conjunct ^ j ∈ I k j x + ~u j · ~y + ~v j · ~z + c j ∈ G to A ( x, ~y, ~z ) + ~c ∈ G | I | where A is the matrix representation for the EIC transformation thatcorresponds to the concatenation of the linear transformations appearing in the specified conjuncts.Similarly for V j I k j x + ~u j · ~y + ~v j · ~z + c j 6∈ G and A ′ ( x, ~y, ~z ) + ~c ′ ∈ ( G c ) | I c | .Since V S has NIP, either R | = ∃ ∞ ~zψ ( a,~b i , ~z ) for cofinitely many i < ω , or R | = ∃ ≤ N ~zψ ( a,~b i , ~z )for cofinitely many i < ω . We consider the first case. Let a ∈ M and ( b i ) i<ω be such that forcofinitely many i < ω there are infinitely many ~r in R | ~z | that witness ∃ ~zψ ( a,~b i , ~z ). Without loss ofgenerality, we suppose this holds for all i < ω . By cell decomposition, we can assume that the setof ~z that satisfy ψ ( a,~b i , ~z ) has interior in R d for cofinitely many i < ω .We now adjust the form that the formula σ ( x, ~y ) takes in order to apply the model companionaxioms as listed in section 2. We can replace the tuple ~z with the tuple ˜ z = ( ~z, ~z ′ ) where ℓ := | ~z ′ | = | I | . We consider the following formula that is equivalent to σ ( x, ~y ):˜ σ ( x, ~y ) := ∃ ˜ zψ ( x, ~y, ˜ z [ d ] ) ∧ (cid:16) A ˜ z [ d ] + ~c = (˜ z d +1 , . . . , ˜ z d + ℓ ) (cid:17) ∧ d + ℓ ^ j = d +1 ˜ z j ∈ G ∧ (cid:16) A ′ ˜ z [ d ] + ~c ′ 6∈ G | I c | (cid:17) where ˜ z [ d ] = (˜ z , . . . , ˜ z d ). Now we can apply the model companion axiom in which the L -formula is ψ ( x, ~y, ˜ z [ d ] ) ∧ A ˜ z [ d ] + ~c = (˜ z d +1 , . . . , ˜ z d + ℓ ) and ~x = (˜ z d +1 , . . . , ˜ z d + ℓ ), and the functions ( τ i ) i<ℓ aregiven by A ′ ˜ z [ d ] + ~c ′ . The corresponding model companion axiom tells us that ˜ σ holds precisely if θ ψ does, which then therefore holds precisely if σ does. Since θ ψ is purely an L -formula, NIP for V S tells us that ∃ ∞ ~zψ ( a,~b i , ~z ) holds for cofinitely many i < ω , or does not hold for cofinitely many i < ω , but this contradicts ϕ having IP.We now consider case two, that for some N ∈ N we have R | = ∃ ≤ N ~zψ ( a,~b i , ~z ) for cofinitely many i < ω . Without loss of generality, we suppose this is true for all i < ω . If there are at most N elements of R | ~z | that witness R | = ∃ ~zψ ( a,~b i , ~z ), then each such witness is in dcl L ( { a } ∪ ~b i ). So byo-minimality we can enumerate them as L -definable functions of a and ~b i and whichever parametersappear in ψ , say as ~f ( a,~b i ) , . . . , ~f N ( a,~b i ). We conclude that for x = a and ~y = ~b i for any i < ω the following sentence: N _ ℓ =1 (cid:16) ψ ( a,~b i , ~f ℓ ( a,~b i )) ∧ ^ j ∈ I k j a + ~u j · ~b i + ~v j · ~f ℓ ( a,~b i ) ∈ G ^ j I k j a + ~u j · ~b + ~v j · ~f ℓ ( a,~b i ) 6∈ G (cid:17) holds precisely if ϕ ( a,~b i ) does.By pigeonhole principle and by restricting (if necessary) to some cofinal subset of ω , we mayassume that the only part of ϕ ( a, ~y ) that alternates in truth value on a cofinal subset of ω is somesubformula corresponding to ℓ ∈ [ N ] of the form ^ j ∈ I k j a + ~u j · ~y + ~v j · ~f ℓ ( a, ~y ) ∈ G ^ j I k j a + ~u j · ~y + ~v j · ~f ℓ ( a, ~y ) 6∈ G . By pigeonhole principle, at least one of the conjuncts in this subformula alternates in truth valueon a cofinal subset of ω , and we may make the further assumption that one such conjunct holdsprecisely if i < ω is odd. Without loss of generality, suppose that k x + ~u · ~y + ~v · ~f ( x, ~y ) ∈ G issuch a subformula. Since we know that f ( x, ~y ) is a definable function in the vector space language,by quantifier elimination for ordered vector spaces we know that f ( x, ~y ) = u ′ x + ~v ′ · ~y + ~c for some u ′ , ~v ′ ∈ K .Hence the subformula is equal to k x + ~u · ~y + ~v · ( u ′ x + ~v ′ · ~y + ~c ) ∈ G , which is furthermoreequivalent to k ∗ x + ~v ∗ · ~y + c ∗ ∈ G for the requisite k ∗ , ~v ∗ , c ∗ in K . Hence if i < ω is odd then ~v ∗ · ~b i ∈ G − ( k ∗ a + c ∗ ). However, if i < ω is even then there exists c i ∈ ( G + c ∗ ) c such that ~v ∗ · ~b i ∈ G − ( k ∗ a + c i ). So if i < ω is even then ~v ∗ ( ~b i − ~b i +1 ) ∈ G − ( c i − c ∗ ), where c i − c ∗ 6∈ G .However, ~v ∗ ( ~b i +1 − ~b i +3 ) ∈ G since ~v ∗ b i +1 and ~v ∗ b i +3 are in the same coset by the formula thatthe odd index ~b j ’s satisfy. By the indiscernibility of the sequence ( b i ) i<ω , for any j > i we mustthereby conclude that ~v ∗ ( ~b i − ~b j ) ∈ G , a contradiction. So no such subformula can alternate intruth value on such an indiscernible sequence. By preservation of NIP under boolean combinationsof formulas, we conclude that σ and hence ϕ has NIP, as desired. (cid:3) Recall that the notion of “finite burden” as defined in [1] is equivalent to finite inp-rank. Corollary 4.5. For V S n , the theory of a real ordered vector space with base field K = Q h η , . . . , η n i ,i.e. K is an algebraic extension with linear degree n over Q , the model companion V S ∗ n, G of V S n, G has finite burden.Proof. By the Lemma 4.3, for any M | = V S ∗ n, G and for any x ∈ M and C ⊆ M countable, the typeof x over C is determined by the L -type of x over C in conjunction with the formulas x ∈ G + y and η x ∈ G + y , . . . , η n x ∈ G + y n for some y , . . . , y n in K h C i . We suppose for contradiction thatthere is an inp-pattern of depth n + 3, and that the array of formulas h ψ i ( x, y ) : i < n + 3 i and thearray of indiscernible sequences ( c i,j ) i ∈ [ n +3] ,j ∈ N witnesses this. In particular, assume that the first n + 2 rows form an inp-pattern of depth n + 2 on their own.By dp-minimality of o-minimal structures, we know that the purely L -definable part of ψ i ( x, y )is trivial (or is exactly the same for all i ∈ [ n + 2] and is independent of the parameter tuple y ) forall but one i ∈ [ n + 2]. Without loss of generality let ψ ( x, y ) be the lone formula in the array with anontrivial L -definable component that varies as y does. The rest of the formulas must be definablein L G but not in L , and without loss of generality we may assume they do not define intervals ofany kind.By the quantifier reduction result Lemma 4.3, we know that each ψ i is equivalent to an L -formulaor the disjunct of conjuncts of L -formulas with some conjuncts of the form η ℓ x + ~v · y + k ∈ G orthe form η ℓ x + ~v · y + k 6∈ G , where ~v and k are in K . Since for i = 0 the purely L -definable partof ψ is trivial, we conclude that ψ i ( x, y ) defines a finite union of a finite intersection of cosets of G for one of x, η x, . . . , η n x to lie in, and coset-complements for x or one of those scalar multiples OMPANIONABILITY CHARACTERIZATION 23 not to lie in. Since G is an infinite-degree subgroup of M , if ψ i were purely a finite union of finiteintersections of coset-complements for x and its scalar images, then for any j , . . . , j ℓ < ω with ℓ ∈ N and j < . . . < j ℓ we could find an x that satisfies all of ψ i ( x, c i,j ) , . . . , ψ i ( x, c i,j ℓ ). So tohave finite inconsistency across each row of the array, for all but finitely many i < ω each disjunctof ψ i ( x, y ) must include a conjunct that for any given y dictates the coset of G for x or a scalarmultiple of X , and that coset varies as y does.We know from part (2) of Lemma 4.3 that for any element r ∈ M there are at most n + 1disjoint formulas, each defining a particular coset of G for one of r, η r, . . . , η n r to lie in, which, inconjunction with the < -cut and formulas excluding r or some η i r from being in any C -definablecoset of G , isolates the type of r over M . Hence if for each i > ψ i defines a disjunctof cosets of G for some subset of { x, η x, . . . , η n x } to lie in, then for paths to be consistent each ψ i can define the coset of at most one element of { x, η x, . . . , η n x } . Yet this yields at most n + 1distinct formulas to which one each of the ψ i ’s may be equivalent. So if ψ n +1 ( x, y ) is the formulafor the n + 2 th row, where y = c ( n +2) ,j is the parameter used for the j th column, then by Lemma4.3 the formula ψ ( x, c n +2 ,j ) is implied by formulas from the previous rows. Hence the values of theparameters c n +2 ,j are dictated by formulas and parameters of the previous n + 1 rows, contradictingindiscernibility of ( c i,j ) i,j ∈ N . (cid:3) Below we see that this V S ∗G is not strong in the sense of [1] if the base field K has infinite lineardegree over Q . The proof that V S ∗G is it not strong is directly analogous to the above proof thatthe model companion for an expansion of a real closed field has TP . Remark 4.6. For V S ∞ the theory of a real ordered vector space with base field K | = RCF , thetheory T ∞ , G has a model companion T ∗∞ , G that is not strong.Proof. Consider the array of formulas where the formula with coordinates ( i, j ) is λ k i ( x )+ c ( i,j ) ∈ G ,where k i ∈ K and c ( i,j ) 6∈ G . For each i ∈ N , let ( c ( i,j ) ) j ∈ N be a sequence of constants such thatfor all k > j it is not the case that c ( i,k ) − c ( i,j ) ∈ G . This can be arranged by the fact that G hasinfinite Q -linear degree over K , by divisibility and saturation. Thus any two formulas in a row ofthe array are inconsistent. Since K has infinite linear degree over Q , we can arrange that the set A of elements from K that appear in these formulas are Q -linearly independent. By saturation, wecan arrange that the array ( c ( i,j ) ) ( i,j ) ∈ N is such that the tuples { ( k i , c ( i,j ) ) : i, j ∈ N } are K -linearlyindependent as well. Hence it follows that for any m ∈ N and any j , . . . , j m ∈ N it is true that theset of equalities { k ( x ) + c (1 ,j ) = y , . . . , k m ( x ) + c ( m,j m ) = y m } has infinitely many solutions for( x, y , . . . , y m ), and the solution space as we vary the elements c ( i,j i ) has linear degree 2 m + 1 over Q . Hence we can rewrite the conjunct of these formulas to apply the companion axioms, whichtell us for each ( c (1 ,j ) , . . . , c ( m,j m ) ) we can find a solution such that y , . . . , y m ∈ G . So T ∗G is notstrong, as witnessed by this array. (cid:3) Since we require that T be o-minimal and the theory T ∗G is a model-complete expansion of thistheory, one might expect it to have more model-theoretic tameness. Therefore the lack of correlationof the model companion T ∗G with any of the widely employed neostability properties may suggesta need for a more robust notion of tameness that captures the kind that the theory T ∗G exhibits asan expansion of T G . 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