t-height in weakly o-minimal structures

T-Height in Weakly O-Minimal StructuresAuthor(s): James TyneSource: The Journal of Symbolic Logic, Vol. 71, No. 3 (Sep., 2006), pp. 747-762Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/27588479 .

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The Journal of Symbolic Logic Volume 71, Number 3, Sept. 2006

r-HEIGHT IN WEAKLY O-MINIMAL STRUCTURES

JAMES TYNE

Abstract. Given a weakly o-minimal theory T, the T-height of an element of a model of T is defined as a means of classifying the order of magnitude of the element. If T satisfies some easily met technical

conditions, then this classification is coarse enough for a Wilkie-type inequality: given a set of elements

of a model of T, each of which has a different T7-height, the cardinality of this set is at most 1 plus the

minimum cardinality of a set that generates the structure.

?1. Introduction. When studying the models of a theory, it is often useful to have a bound on the size of a model, based upon the minimum cardinality of a set that generates the structure. One such bound?the "Wilkie inequality"?says that for certain power bounded o-minimal expansions of valued real closed fields, the dimension of the value group (as a Q-vector space) cannot exceed the rank of the field (see Wilkie [9] and Van den Dries [2]). This paper presents a bound that, while weaker than earlier bounds, is applicable in a wide variety of settings. An earlier version [8,3.11] of this bound has already found applications in [8] and [6].

Throughout, let T be an extension of the theory of dense linear orders with a

distinguished element, but without endpoints, and let & = (R, <, 0,... ) designate an arbitrary model of T unless otherwise indicated. All sets, unless otherwise

indicated, are understood to be subsets of R. As usual, elements of R larger than 0 are called positive, and those less than 0 are called negative. The set of positive elements of R will be denoted R+. Given a set A, the set of all ^4-definable elements of R?that is, the definable closure of A?will be denoted dcl(^4). The letters m and n will be used to represent nonnegative integers.

To simplify the results and arguments of this paper, assume that - : R ?> R is a

0-definable strictly decreasing bijection with ?0 = 0. This allows the absolute value function | | to be defined as usual: x \-^ x when x > 0 and x i?> ? x when x < 0. Call a set X symmetric if X = ?X. The symmetric hull of a set Y will mean the convex hull of Y U - Y.

This paper is concerned with two descriptions?J-class and T -height?of the

magnitude of an element of R, both of which are intended to capture the idea of one element being comparable in magnitude to another. These are presented in the

following paragraphs, in which r denotes an arbitrary element of R.

Received October 27, 2004.

This research was partially supported by a VIGRE postdoctoral fellowship. Thanks to Chris Miller and the referee for valuable comments.

? 2006, Association for Symbolic Logic 0022-4812/06/7103-0002/$2.60

747



748 JAMES TYNE

Let By denote the union of all r-definable bounded sets; in particular B0 denotes the union of all the bounded sets that are definable without parameters. Define x x y if x, y e Bo or if x, y G R \ Bo and Bx ? By. This relation is an equivalence, and the equivalence classes will be called T-classes. Let Cl (r) denote the T-class of r, and given a set X, let Cl (X) denote the union [jxeX Cl (x).

Let br denote the intersection of all the r -definable, but not 0-definable, symmetric convex sets that include B0. Iff* ? Bo, define the T-height of r to be B0. If r ̂ B0, define the T-height of r to be \Br \ br\. Equivalently, the T-height of r ? Bo is the union of all bounded r-definable sets S such that for each 0-definable bounded set

B, there is x G R with B < x < S. Let Ht (r) denote the T-height of r, and given a set X, let Ht (X) denote the union \Jxex Ht (x). Examples of ̂ -height in various structures can be found at the end of this introduction. For now, note that Ht (r) is a final segment of Br.

The goal of this paper is to show that two (mild) hypotheses on T imply the

following properties of 91.

Property I. For all r G R, the T-class ofr is Ht (r) U -Ht (r).

Property II. The cardinality of the set of T-classes is at most one plus the minimum

cardinality of a set of generators of 91.

The arguments in this paper generally follow the line of reasoning in [8, ?3], which dealt with the o-minimal setting (and in which 'T-height" was called "T-level"). The most notable additions are 6.1 and 6.4, which are trivial if the Exchange Principle is available (as is the case if T is o-minimal). We now proceed to the

hypotheses. The first hypothesis is that T is weakly o-minimal; that is, T is an extension of

the theory of dense linear orders without endpoints and, in any model of 7\ every

parametrically definable set of one variable is a finite union of convex sets. Thus, in any model of T, each convex component of a definable set of one variable is definable. Additional facts about weak o-minimality will be presented as needed. For further results, see the seminal papers [1] and [4].

The second hypothesis is that every model of T satisfies the following Key Lemma. This is the case for many theories of interest; the details are deferred to section 2.

Key Lemma. Suppose the set A is definably closed and ( Yr)reR is an A-definable family of definable bounded subsets of R. Then there is an A-definable family (Zr)reR of definable bounded subsets of R such that Yr = Zr whenever \r\ ? Ht (A), and

UreS Zr is bounded whenever S ? R is bounded.

It may be that these hypotheses are not necessary for the above properties. On the other hand, it is not hard to devise an example of an ordered structure in which

Property II fails.

Counterexample 1.1. Suppose 9t expands a field and Z is a discrete 0-definable set such that for each r G R the set {z e Z : r < z} has a smallest element (cf. the

expansions of the real field discussed in [5]). Note that this smallest element is r-definable. Now suppose zo G Z is larger than every 0-definable bounded set, and c G R is larger than every zo-definable bounded set. Thus, 0, zo, and c represent three distinct T-classes. If Property II were to hold, then a structure containing these three elements could not be generated by a single element. However, if Z is



r-HEIGHT IN WEAKLY O-MINIMAL STRUCTURES 749

suitably sparse?say no elements of Z lie between z0 and z0 - 1?then these three elements are in the structure generated by z0

? l/c.

We conclude this introduction with some examples of T-heights. For these

examples, let r denote a positive element of R that is larger than Ht (0).

Example 1.2. Suppose 9 is o-minimal; that is, in addition to T being weakly o-minimal, every definable set (of one variable) has a supremum and infinum in R U {?oo}. Then every definable bounded set has a supremum in R, so if X is an r-definable bounded set, then sup(Z) is an element of dcl(r). Thus, using the notation from the definition of T -height, Br is the convex hull of del (r) and br is the set of elements of R that are smaller in absolute value than every positive element of dcl(r) \ 2?o- That is, the T-height of r is the convex hull of the positive elements of dcl(r) \ Ht (0). Furthermore, the T-height of 0 is the convex hull of dcl(0).

Example 1.3. Suppose 9 is an o-minimal polynomially bounded expansion of an ordered field. Then for every 0-definable function f:R^R with f(r) > Ht (0) there is n > 0 such that rl/n < f(r) < rn. Thus, the convex hull of the

positive elements of dcl(r) \ Ht (0) is \J?=1[rx/n,rn]. Hence the T-height of r is the archimedean class of r with respect to (R+, <, -, 1).

Example 1.4. Suppose 91 is an o-minimal exponentially bounded expansion of an ordered field with a 0-definable exponential function E. Let En denote the nth iterate of E, and Ln the compositional inverse of En. For every 0-definable function f:R^R with f(r) > Ht (0) there is n > 0 such that Ln(r) <

f(r) < En(r). Thus, the convex hull of the positive elements of dcl(r) \ Ht (0) is U^Li [Ln(r),En(r)] Hence the T-height of r is the exponential comparability class of r.

Example 1.5. If 9t is not o-minimal, the connection between T-height and defin able closure weakens. Suppose 91 is an o-minimal expansion of a real closed field, further expanded by a T -convex subring V. That is, V is convex and f(V) ? F for each 0-definable continuous function / : R ?> 7?. The relative quantifier elimina tion presented in [3, 3.10] shows that definable closure is not affected by the choice of V?in particular, the set dcl(0) does not depend on V?and Ht (0)

= V. Hence Ht (0) could properly include the convex hull of dcl(0).

Example 1.6. Suppose 91' is an o-minimal expansion of a group with universe

R, and let H be the convex hull of dcW (0). Further suppose = is an equivalence

relation onR+\H such that x = y iffx is in the convex hull of the positive elements of dcl^/ (y)\H. Now suppose 9 is the expansion of 9' by =. Then Ht (r) includes the =-equivalence class of r and dcl^/(r) = dcl^(r). By compactness, 9 has an

elementary extension in which the =-equivalence class of r, hence also the T-height of r, properly includes the convex hull of the positive elements of dcl(r) \ Ht (0).

?2. The Key Lemma. This section explores some sufficient conditions for the Key Lemma.

Call a family ( Yr)reR of sets locally bounded if for each r G R there is an open

neighborhood U of r such that [jxeU Yx is bounded; call it regionally bounded if

[Jxes Y* is bounded whenever S ? R is bounded.



750 JAMES tyne

We will say that 9? is nice if either

every definable family of definable bounded sets is locally bounded, or

every r e Ris contained in some r-definable open subset of Ht (r) U -Ht (r).

Many common structures are nice. If 91 is an ordered group (without additional

structure), then every definable family of definable bounded sets is locally bounded. Hence every ordered group is nice. If 91 is a weakly o-minimal expansion of a

group, then each positive r G R is contained in the interval (r/2,2r), which is included in Ht (r), and each negative r G R is contained in the interval (2r, r/2),

which is included in ?Ht (r). If 91 contains a distinguished positive element e, then 0 G {?e, e) ? Ht (0). Hence every weakly o-minimal expansion of a group with a

distinguished positive element (e.g., a field) is nice.

Proposition 2.1. Suppose 91 is nice, and every locally bounded family of definable sets is regionally bounded. Then the Key Lemma holds.

Proof. Let i be a definably closed set and (Yr)reR be an ^-definable family of definable bounded sets; we need to find a locally bounded (hence regionally bounded) ^4-definable family (Zr)reR of definable sets such that Yr = Zr whenever

\r\ ? Ht (A). This is trivial if (Yr)reR is locally bounded, so assume (Yr)reR is not

locally bounded. By weak o-minimality (see [4, 3.3]) the set X of points at which

( Yr)rER is not locally bounded is finite. Since 91 is nice, there is an open ̂ -definable set U ? Ht (X) U -Ht (X) such that X CU. Now X, being ^-definable, is included in A, so Ht (X) ? Ht (^4). Thus we can define Zr := Yr if r ^ U and Zr := {r} otherwise. H

While every weakly o-minimal expansion of a group with a distinguished positive element is nice, the group structure does not ensure that every locally bounded

family of definable sets is regionally bounded. As an illustration, suppose V is a

proper convex subring of a field, and consider the structure induced on V by the field. Even though V expands a group, the graph of the function l/x, which is now

restricted to the units of V, is locally bounded but not regionally bounded. On the other hand, every locally bounded family of definable sets is regionally

bounded in many cases of interest.

Example 2.2. Suppose 9 is o-minimal. Then every definable bounded set has a

supremum and infinum in R, so a definable family ( Yr )reR of definable bounded sets

corresponds to the functions from R to R given byi^ sup Yx and x h-> inf Yx. By o-minimality, there is a decomposition of R into finitely many points and intervals such that these functions are (weakly) monotonie on each interval (see [7, 4.2]). Thus, if one of the functions is not bounded, then it is not bounded near an endpoint of at least one of the intervals. In other words, if ( Yr)reR is not regionally bounded, then it is not locally bounded. Consequently, if 9 is an o-minimal expansion of a

field, the Key Lemma holds.

Example 2.3. Suppose 9 is an o-minimal expansion of a field equipped with a

proper T-convex valuation ring. Consider a definable family (Yr)reR of definable bounded sets. Since 9 has definable Skolem functions after naming an infinite element (see [2, 2.7]), there are definable functions f,g: R ?> R such that Yr is

bounded above by f(r) and below by g(r), for all r G R. Now / and g are given




piecewise by finitely many functions /?, gn, each definable in the underlying o minimal structure (see [2,2.8]). So by replacing / with max? fn and g with min? gn, we may assume that / and g are definable in the underlying o-minimal structure.

Thus, we reduce to the o-minimal setting of the previous example. Therefore, the

Key Lemma holds.

?3. Property I.

Assumption. For the rest of this paper, assume T is weakly o-minimal and the Key Lemma holds in every model ofT.

In this section we will show that 9 has Property I, i.e., Cl (r) = Ht (r) U -Ht (r)

for all r G R. For each r e R, again let Br denote the union of all r-definable bounded subsets of R, and let br denote Br \ (Ht (r) U ?Ht (r)). Since Br = B-r we may (and shall) restrict our attention to nonnegative elements of R. Since

C1(0) =

Ht(0) =

-Ht(0), we may further restrict our attention to the positive elements of R \ Ht (0). We proceed in three steps.

Lemma 3.1. Suppose r G R+ \ Ht (0) and s G Ht (r). Then r G Bs.

Proof. Let X be an r-definable, but not 0-definable, symmetric convex set that includes Bo, but does not contain s. (So s is larger than every element of X.) Let (Xx)xeR be a 0-definable family of definable sets such that Xr = X. Since r ? Ht (0), the Key Lemma allows us to assume this family is regionally bounded. Let Y := {x G R+ : ?s < Xx < s}. Then Y is ^-definable and contains r, so it suffices to show that Y is bounded.

For a contradiction, assume Y contains arbitrarily large elements of R. By weak

o-minimality, there is some x0 G R+ such that (x0, oo) ? Y. So the union Ux>x0 Xx is bounded above by s and below by ?s. By regional boundedness, the union

Uo<x<jco ^x *s a'so bounded. Therefore, the set X' := Ux>o ^ is bounded and

0-definable, hence X1 ? Bo. However, this means X' ? X, since Bo ? X ? X'',

contradicting the assertion that X is not 0-definable. H

Lemma 3.2. Suppose r G R+ and s G Br \ Bo. Then Bs ? Br.

Proof. Consider an arbitrary s-definable bounded set Y; it suffices to find an r-definable bounded set X such that Y ? X. Let (Yy) eR be a 0-definable family of definable sets such that Ys = Y. Since s ? Ht (0), the Key Lemma allows us to assume this family is regionally bounded. Let Z be an r-definable bounded set such that s G Z. Define X to be \JzeZ Yz. -\

Lemma 3.3. Suppose r G R+ \ Ht (0) and s G br \ Ht (0). Then Bs ^ Br.

Proof. Noter G Br. For a contradiction, assumer G Bs. Let Y be an ^-definable bounded set containing r. Let ( Yy) R be a 0-definable family of definable sets such

that Ys = Y. Since s ? Ht (0), the Key Lemma allows us to assume this family is

regionally bounded. So for each x G R, the set Zx := Uiv|<M Yy is bounded. Let

X := {x G R : r ? Zx}. Note that X is symmetric, convex, and does not contain s. We claim Bo ? X. Suppose b G Bo. Let B' be a 0-definable bounded set

containing b, and let B be the symmetric hull of B'. (Hence B is a 0-definable bounded set containing all y G R with \y\ < \b\.) Since

[jyeB Yy is bounded and

0-definable, it is included in Bo, hence it does not contain r. Thus, r is not contained in any subset of this union, specifically r ̂ Z?, so b G X.



752 JAMES tyne

We claim X is not 0-definable. Consider the set Z := C\X?XZX. This set is bounded since Z ? Zs and Zs is bounded. This set contains r since r G Zx for each x ? X. If Z was 0-definable, then Z would be a subset of Bo, contradicting r ̂ Bo. Thus, Z is not 0-definable, hence X is not 0-definable.

Therefore, X is an r-definable, but not 0-definable, symmetric convex set that includes Bq. Hence br ? X, contradicting s G br \ X. H

Proposition 3.4. Ifr g R+ \ Ht (0), then Cl (r) n i?+ - Ht (r). Proof. Fix r e R+\Ht{0). If s e br\ Ht (0) then r ̂ s by 3.3. If s G Ht (r)

then r e Bs by 3A, hence Br = Bs by two applications of 3.2. If s > Ht (r) then s G Bs\Br, hence r ̂ s. H

This establishes Property I. Consequently, for each r > Ht (0), the set br is the union of all those Bs properly included in Br. Furthermore, the sets Ht(r) are

pairwise disjoint, hence naturally ordered by the order of 9 (i.e., Ht (x) < Ht (y) if?V </for all*' G Ht (x) and/ eUt(y)).

?4. Elementary extensions. This section will lay the groundwork for looking at an elementary extension of 9. For this section, let 9' =

(Rf,... ) be an elementary extension of 9, and let Ht' (x) and Cl' (x) denote the T-height and T-class of x,

respectively, in the context of 9'.

Lemma 4.1. For every r G R, Ht (r) = Ht7 (r) n R.

Proof. Fix r G R. If r is contained in a bounded 0-definable subset ofR, then r is contained in the corresponding (bounded, 0-definable) subset of Rf. The converse also holds. That is, r G Ht (0) if and only ifr G Ht' (0). So suppose r (? Ht (0).

Consider s G Ht (r). By Property I, every r-definable bounded subset of R is included in an s-definable bounded subset of R, and vice versa. By elementary equivalence, the same holds in 9': every r-definable bounded subset of R1 is included in an s-definable bounded subset of Rf, and vice versa. So s G Ht7 (r) by Property I. Similarly, if s G Ht' (r) H R, then s G Ht (r). H

We will say that a set S ca/w a set X if S is nonempty and bounded and for all

nonnegative r G R, if r < S then r < |x| for some x G X, and if S < r then \x\ < r for all x G X. (This is analogous to saying sup \X\ G S.) Call a set X comparable to r G R if either r G Ht (0) and X is included in a 0-definable bounded set, or r ̂ Ht (0) and there is an r-definable subset of Ht (r) that caps X. Note that a set is comparable to r if and only if its symmetric hull is comparable to r. Call a set

comparable to a set A if it is comparable to some element of A. The preceding definitions are elementary, except for the condition that a set be

included in a T-height. To help deal with that aspect of these definitions, the

following lemma will be useful. (This lemma is expanded upon in 5.3.)

Lemma 4.2. Suppose c > Ht (0) and (Zr)reR is a ^-definable family of definable sets such that Zc is included in Ht (c). Then Zr ? Ht {r) for all r G Ht (c).

Proof. By weak o-minimality, either Zr is unbounded for all r > Ht (0) or Zr is bounded for all r > Ht (0). In the former case, Ht (c) is cofinal in R, so Zr cannot contain an element larger than Ht (r) for all r G Ht (c). In the latter case, Zr does not contain an element larger than Ht (r) for all r > Ht (0) by the definition of




T-height. Since T-heights are convex, it remains to show that Zr does not contain an element smaller than Ht (r) for all r G Ht (c).

Suppose for each bounded 0-definable set B there is some x G R with B < x < Zc. Then by weak o-minimality, for each bounded 0-definable set B and each r > Ht (0) there is some x G R with B < x < Zr. Thus, Zr contains no element smaller than

Ht(r) for every r > Ht(0), by the definition of T-height. Now suppose B is a bounded 0-definable set for which there is no x G R with B < x < Zc. Then B is cofinal in Ht (0), so Ht (0) is the symmetric hull of B, hence is 0-definable. Furthermore, Ht (c) is the least T-height larger than Ht (0). By weak o-minimality, Ht (0) < Zr for all r > Ht (0). In particular, Zr does not contain an element smaller than Ht (r) for all r G Ht (c). H

One useful characteristic of comparability is that if a set is comparable to some r > Ht (0), then this r is unique up to T-height. Before demonstrating this, we need the following lemma about uniform definability.

Lemma 4.3. Suppose s > Ht (0) is such that some bounded s-definable set is cofinal in Ht(s). Then there is a 0-definable family (Zr)reR of definable sets such that

Zr = Ht (r)for all r > Ht (0) (and Zr ? Ht {Q) for all r G Ht (0)). Proof. First, (considering symmetric hulls) note that Ht (s) is cofinal in some

convex, symmetric, bounded s-definable set. Let (Xr)reR be a 0-definable family of convex, symmetric, bounded definable sets such that Ht(s) is cofinal in Xs. Then every bounded ?--definable set is included in Xs. By weak o-minimality, for all r > Ht(0), every bounded r-definable set is included in Xr. (If (Yr)reR is a 0-definable family of bounded definable sets, then the set of r such that Yr ? Xr contains s, hence contains every r > Ht (0).) That is, if r > Ht (0), then Xr is the union of all r-definable bounded sets.

By the Key Lemma, we may assume (Xr)reR is regionally bounded. So if B

is a 0-definable bounded set, then \JxeB Xx is bounded, hence included in Ht (0). Taking the union of these unions as B varies, we see that

UxeHt(o) Xx is included in Ht (0). Thus, given r > Ht (0), the T-class of r is the set ofxG^ with X\x\

= Xr. Recall Property I, and let Zr := {x > 0 : Xx =

Xr}. H

Lemma 4.4. Suppose s G R \ Cl (0) and r G Cl (s). For each s-definable bounded set Z ? Ht (?*), there is an r-definable bounded subset ofY?X (s) that includes Z.

Proof. Let Z be an s-definable bounded subset of Ht (s). If Z is not bounded above by an element of Ht (s), then Ht (s) is r-definable by 4.3, and we are done. So

suppose Z is bounded above by c2 G Ht (s). Then there is an r-definable bounded set Z2 ? Ht (s) that contains c2. If Z is bounded below by c\ G Ht (s), then there is an r-definable bounded set Z\ ? Ht (s) that contains c\, and Z is included in the convex hull of Z\\JZ2. So suppose Z is not bounded below by an element of Ht (s ). Then x < Z is equivalent to x < Ht (s), so the set X := {x G R : \x\ < Ht (s)} is s -definable. Note that X is the intersection of all s -definable, but not 0-definable, symmetric convex sets that include Ht (0). If X is not 0-definable, then an argument similar to that in the proof of 4.3 shows that Ht (r) is r-definable, and we are done. If X is 0-definable, then the set of x > X which are less than some y G Z2 is r-definable and includes Z. H



754 JAMES tyne

Corollary 4.5. If a set is comparable to s G R, then it is comparable to every element ofC$s).

Proof. For s G Cl (0), this result is trivial. Suppose s ? Cl (0) and r G Cl (s). If a set is comparable to s, then this fact is witnessed by a capping set Z, where Z is an ?--definable bounded subset of Ht (s). Note that every bounded set including Z

caps the original set, so it suffices to find an r-definable bounded subset of Ht (s) that includes Z. Thus, applying 4.4 finishes this proof. H

Proposition 4.6. Suppose n,r2 > Ht(0) and X ? R is comparable to both r\

andri. Then?Lt(r$ =

Ht(r2). Proof. For a contradiction, assume Ht (n) < Ht (r2). For i = 1,2, let Z? be an

r,-definable subset of Ht (r?) that caps X. Then Z\ < Z2, and from the definition of "caps", there is no element of R between Z\ and Z2. Hence Z\ is cofinal in Ht (ri). Using 4.3, let (Yx)xeR be a 0-definable family of definable sets such that

Yx = Ht(x)forallx>Ht(0). Let Y' be the set of x > r\ for which Z\ < Yx and there is no element of R

between Z\ and Yx. Then Y' is an r\-definable set that contains r2 (an element

larger than Ht (n)) so Y' is not bounded. On the other hand, Y' does not contain an element larger than Ht (r2). Therefore, Ht (r2) is not bounded.

Let Y" be the set of x > 0 for which Yx is bounded. This set is 0-definable and included in the interval (0,r2). Thus, Y" is included in Ht(0). This is a

contradiction since r\ E Y". H

Another useful characteristic of comparability is that it is preserved under ele

mentary extensions, with one exception.

Lemma 4.7. Let X be a definable set, and X' the corresponding subset ofRf. (That is, X1 is the interpretation in 9' of the formula that defines X.) If X' is comparable to some a G R, then X is comparable to a. If X is comparable to some a G R, then

X' is comparable to either aorO.

Proof. Suppose X' is comparable to a G R. If a G Ht' (0), then a G Ht(0) by 4.1, and Xf is included in some bounded 0-definable subset of R''. In this case, X is included in the corresponding bounded 0-definable subset of R, hence X is

comparable to a. So suppose a ? Ht' (0), hence a fi Ht(0). Let Z' be an in

definable subset of Ht' (a) that caps X', and let Z be the corresponding a-definable subset of R. By elementary equivalence, Z caps X, so it remains to observe that

Z = Z' n R ? Ht' (a) H R = Ht (a) is a consequence of 4.1.

Now suppose X is comparable to a G R, and assume X' is not comparable to 0.

Then, by an argument similar to the preceding paragraph, X is not comparable to 0, hence a ? Ht (0). Let Z be an ?-definable subset of Ht (a) that caps X, and let Z' be the corresponding (bounded) a -definable subset of Rf. By elementary equivalence, Z' caps X', so it remains to show that Z' ? Ht' (a). Consider the sets Y := {x g R : \x\ < Z} and Y' := {x G R' : \x\ < Z'}. Note that Y includes

every 0-definable bounded set, so Y' does also, hence Ht' (0) ? Y'. Note also that Y' is symmetric and convex. If Y is not 0-definable, then neither is Y', so

Z' ? Ht' (a) by the definition of ̂ -height. Thus, it suffices to show that Z can

be chosen so that Y is not 0-definable. Suppose Y is 0-definable. Since X is not




comparable to 0, we can find c G \X\ \ Y. Since Z caps X, this c is contained in the convex hull of Z, hence in Ht (a). So there is some a -definable, but not 0-definable,

symmetric convex subset of R that includes Ht (0), but does not contain c. If Yo is such a set, then we replace Z with Z \ 70. H

A set will be called rational if it is comparable to R. Every rational set is bounded. At present, there is no proof that every definable bounded set is rational (except in the trivial case where T is o-minimal). On the other hand, there are no known

counterexamples. We will say a set B is T-bounded by a set A if every ̂ -definable rational set is

comparable to A. In particular, if B is T-bounded by A, then dcl(i?) ? Cl (A). This property is preserved under elementary extensions to some extent.

Proposition 4.8. Suppose A and B are subsets of R and A contains 0. Then B is T-bounded in 9 by A if and only if B is T-bounded in 9' by some Ao ? R' such that

Aof\C\(R)=A. Proof. Suppose B is T-bounded in 9 by A. Let Ao contain (and only contain)

the elements of A together with a single element from each T-class of 91 that contains no elements of R. So Ao n Cl' (R)

= A. Consider a Indefinable set X' ? R' that is comparable (in 9') to c' G R'. Suppose Cl7 (c') n R = 0. Then an element of Cl7 (c') is contained in Ao, so Xf is comparable to Ao by 4.5. Suppose c G Cl7 (c1) D R. Then X' is comparable to c by 4.5. Let X be the interpretation in 9 of the formula that defines X'. By 4.7, X is comparable (in 9) to c. So X is ^-definable and rational, hence comparable to A. Thus, X' is comparable (in 9') to Ao by 4.7. Therefore, B is T-bounded in 9' by Ao.

Suppose B is T-bounded in 9' by some A0 Q Rf such that A0 n Cl7 (R) = A.

Consider a B -definable set X ? R that is comparable to c G R. Let X' be the

interpretation in 9' of the formula that defines X. By 4.7, X' is comparable (in 9') to either c or 0. In the latter case, X is comparable (in 9) to 0 by 4.7, so X is comparable to A. Suppose Xf is not comparable to 0. Now X' is comparable to some c' G Ao, and by 4.6, c' is an element of Cl7 (c). Thus, c' is an element of

Ao n Cl7 (c), which is included in A. Thus, c7 G i^, and using 4.7 once again, X is

comparable to A. Therefore, B is T-bounded in 9 by A. H

?5. Saturated models. This section will examine properties of saturated models of T. In this paper, "saturated" means "/^-saturated for some uncountable rc larger than the cardinality of the language of 7"'. A small set will be one whose cardinality is less than this k. Thus, given r G R, the set of r-definable sets is small. In

particular, each T-height is a small union of definable sets, hence the complement of finitely many T-heights is a small intersection of definable sets.

Assumption. For this section, assume 9 is saturated.

We begin by looking at the order type of the set of T-heights, and some conse

quences of this order type.

Theorem 5.1. The set of T-heights of elements of R is densely ordered without

supremum.

Proof. By saturation, given r G R, the union of all r-definable bounded sets does not include all of R, hence there is no largest T-height.



756 JAMES TYNE

Consider nonnegative r.seR such that Ht(r) < Ht(s) and either r = 0 or

Ht (0) < Ht (r); we need to find x e R such that Ht (r) < x < Ht (s). Note s is

larger than each r-definable bounded set. Let X be an r-definable bounded set, and Y an ?"-definable, but not 0-definable, symmetric convex set that includes Ht(0). By saturation, it suffices to show 7\I^8. (The intersection of all such Y \ X is the set of x G R satisfying Ht (r) < \x\ < Ht (s).)

For a contradiction, assume Y ? X. (Hence Y is bounded.) Let (Yv) R be a

0-definable family of definable bounded symmetric convex sets such that Ys ? Y.

By weak o-minimality, every 0-definable set either includes or excludes R+ \ Ht (0). In particular, for each 0-definable bounded B ? R, the set of y G R such that Y y ? B does not contain s, so this set does not contain any y > Ht (0). That is, for each y > Ht (0), Yv properly includes every 0-definable bounded set, hence contains all x G R with |x| < Ht(j) by the definition of T-height. Thus, Yt contains s

whenever t > Ht(s). Again using weak o-minimality, every r-definable set either includes or excludes

{x G R : x > Ht(r)}. In particular, the set of y G R such that Yv ? X does contain

s, so it contains every y > Ht (r). So if t > Ht (s) then Yt ? X, hence Yt < s.

Since there is some t > Ht (s), we have a contradiction. H

Corollary 5.2. Suppose A ? R is small. Then the set {Ht (a) : a G A} is not dense in the set of T-heights of elements of R.

Proof. By replacing A with \A\, we may assume no element of A is negative. Let a be an element of A for which the set A' := {x G A : Ht (a) < Ht (x)} is not

empty. If a G Ht (0), then we may assume 0 e A and replace a with 0. It suffices to find x G R with Ht (a) < x < Ht (A') (since then there would be x' G R with a < Ht (xf) < x). Proceed as in the proof of 5.1, with the addition that s is allowed to vary over A'. H

Corollary 5.3. Suppose c > Ht (0) and (Zr)reR is a (^-definable family of defin able sets such that Zc is included in Ht (c). Then Zr ? Ht (r) for all r > Ht (0).

Proof. The proof proceeds as in 4.2, with the additional observation that by 5.1, there is no largest T-height, nor is there a least T-height larger than Ht (0). H

Corollary 5.4. Suppose r\, r2 G R and X ? R is comparable to both r\ and r2.

ThenHt(ri) =Ht(r2). Proof. For a contradiction, assume Ht (n) < Ht (r2). For ? = 1,2, let Zz be an

rrdefinable subset of Ht (r,) that caps X. Then Z\ < Z2, and from the definition of "caps", there is no element of R between Z\ and Z2, hence no element between Ht (n) and Ht (r2), contradicting 5.1. H

The remainder of this section establishes a technical property of saturated models of T that will be used in the next section.

Lemma 5.5. Suppose c G R and B,X ? R are such that X is B-definable and

comparable to c. Then Ht (c) is a union of B-definable sets.

Proof. By replacing c with -c, we may assume c > 0. This lemma is trivial if c G Ht (0), so assume c > Ht (0). We begin by finding a B-definable subset of

Ht (c). Let (Zr)reR be a 0-definable family of definable sets such that Zc caps X and is included in Ht (c). Note that Zr is included in Ht (r) for each r > Ht (0) by 5.3. Let D' be the set of positive r G R such that Zr caps X. This set is indefinable and,




by 5.4, is included in Ht (c) UHt (0). Let D denote the convex component of D' that contains c. Since 9 is saturated, there is some x G R with Ht (0) < x < Ht (<?). Thus, D is a indefinable subset of Ht (c).

Recall that Ht (c) is a union of c-definable sets, and let Y be one set of this union. It suffices to find a ^-definable set Y' such that Y ? Y' ? Ht (c). Let (Yr)reR be a 0-definable family of definable sets such that Yc = Y. By 5.3, Yr is included in Ht (c) for each r G Ht (c). In particular, Yd is included in Ht (c) for each d e D. Hence 7 ? U?GjD ?? ? Ht (c). H

Call a set T-distinct if xi ^ x2 for ail distinct x\,x2 in the set. (Recall x\ ^ x? means Ht (x\) ^ Ht (x2).) We will say that xi,..., xn G R are T-distinct if they are distinct and the set {x\,..., xn} is T-distinct.

Lemma 5.6. Suppose the set B is T-bounded by the small set A. Further suppose

(Xi)ieI is a small collection of B-definable sets (i.e., I is a small set). Let X denote the intersection f]ieI X?. If X \ Cl (A) ^ 0, then there are T-distinct r,s<ER such that X \ Cl (A) includes the interval [r,s].

Proof. Suppose r7 G X \ Cl(A). By replacing X with ?X if necessary, and

noting that Cl(A) contains 0, we reduce to the case r7 > 0. For each i G I, we may replace X? with the convex component of X? n R+ containing r7, hence we reduce to the case X is convex and each X? contains only positive elements.

Fix / G I. Since B is T-bounded by A, X? is not comparable to r7 by 5.4. Thus, for each /-definable bounded set Z, there is ?yz G X? with Z < ?yz- Let i (and Z) vary. By saturation, there is s' e X with Ht (r7) < sf. Since A is small, there is, by 5.2, a subinterval [r, s] of [r7, s*7] whose intersection with Ht (A) is empty, such that Ht (r) ̂ Ht (s). Since X is convex, it includes [r, s]. H

Proposition 5.7. Suppose B is a small set that is T-boundedby the set A. Further

suppose, for each a G A, there is a B-definable set comparable to a. Let (p(y, z) be a formula with parameters only from B, such that there are T-distinct elements c, d in R+ \ Ht (A) with <p(c,d). Then there are T-distinct elements c\1d[,c2,d2,... in

jR+\Ht(^) with^(ci,di)fori -1,2,.... Proof. Recall that each T-height is a union of definable sets. Using 5.3, we

can make this uniform: there is a collection Oo of 0-definable families of definable bounded sets such that, for every r > Ht (0), the T-height of r is the union, over families from ?o, of the sets indexed by r. Let Oo(x) denote the collection of the sets indexed by x in families from Oo. By 5.5, we can take O^ to be a collection of B-definable sets whose union is Ht (A). Let O(x) denote the collection of all finite unions of sets in Oo(x) U 0^. When convenient, we will regard O as a collection of ^-definable families of definable sets, each family indexed by R. Given r > Ht (0), the union of the sets in O(r) is Ht (r) U Ht (A). Since A contains some element of

Ht (0), the union of the sets in O(0) is Ht (A). Note that O(x) is a small collection of sets because B is small. Given a family 6 G O and x G R, let 6(x) denote the set of this family indexed by x.

Case 1: Suppose d' G R+\ Ht (A) is such that there are infinitely many T distinct y G R+ \ Ht (A) with y ? Ht (d1), for each of which there is z G Ht (d') with (p(y,z). We define Koo, which will be seen to be a set of z G R+ \ Ht (A) for which there are infinitely many T-distinct y G R+ \ Ht (^4) with y ? Ht(z), for each of which there is z7 G Ht (z) with (p(y, z7). For each n G N, choose 6n G Oo



758 JAMES tyne

so that there are at least n T-distinct y G R+ \ Ht (A) with y ? Ht (d'), for each of which there is z G 9n(d') with (p(y,z). For each 9 e ? and n G N, consider the set of z G R+ \ 9(0) for which there are z\,..., zn G 9n (z) and ji,..., yn G R+ \ 9(z) such that (p(yt,Zi) and ^z ^ O(yi') for i = I,... ,n and z' = 1,..., i - 1. Note that d' is an element of this set. Let K^ denote the intersection of these sets as 6 and n

vary.

By saturation, for each z G K^ and n G N, there are elements zi,... ,zn of

0?(z) and T-distinct elements y\,... ,yn of R+\Ht (A) such that yt ? Ht (z) and

(p(yi,Zi) for z = 1,..., n. Hence, for each z G K^ there are infinitely many T

distinct y G R+ \ Ht (^4) with y ? Ht (z), for each of which there is z' G Ht (z) with

<??(>>, z'). Note that ̂oo is not empty as it contains df, so 7^ contains elements from infinitely many T-heights by 5.6 and 5.1. Sequentially choose c\, d\, c2, ?/2,... as follows. Suppose Wn := {ci, d\,..., c?_i, ?Z?_i} is given. Since W? is finite, we

can choose d" in ?^ \Ht ( ^?), then find cn in i?+ \Ht (A U H^) with c? ? Ht (J"), for which there is dn G Ht (d") with tp(cn,dn).

Case 2: Suppose there is no such d'. In analogy to T^o of the previous case, we define J, which will be seen to be the set of y G R+ \ Ht (A) for which there is z G R+ \ Ht (^4) with z ^ Ht (y) and (p(y, z). For each 6 G O, consider the set of

j G i?+ \ 0(0) for which there is z e R+\ 9(y) such that <p(y, z). Note that c is an element of this set since d G R+\9(c) and (p(c,d). Let / denote the intersection of these sets as 9 varies over O.

By saturation, for each y G / there is z G i?+ \ Ht (A) with z ^ Ht (y) and

<?>(;>, z). Note that / is not empty as it contains c, so J contains elements from

infinitely many r-heights by 5.6 and 5.1. Sequentially choose c\,d\, c2, ?/2,... as follows. Suppose Wn := {c\,d\,..., c?_i, dn-\} is given. By the case distinction,

there are at most finitely many T-distinct y G / for which there is z' G Ht ( Ww) with cp(y, z'). That is, we can choose cn e J \ Ht (Wn) so that -?p(cn, z') for all z' G Ht(PF?). Since cn G /, we can find dn in jR+ \ Ht(^) with dn ? Ht(c?) and (f(cn,dn). H

?6. Property II. In this section, we will show that 9 has Property II. Our

approach involves showing that if a set X is T-bounded by a set Y, then for each x e R there is y G R such that X U {x} is T-bounded by F U {j}. This will first be done in the case where X is T-distinct, and then the general case will follow. We

begin with a preliminary lemma.

Lemma 6.1. Suppose A is a T-distinct set that contains 0, and there is an A

definable bounded set that is not comparable to A. Then there is a T-distinct set B that contains 0, and a B-definable bounded set Y that is not comparable to B, such

thatcard(B) < card (^4) and B ? Y.

Proof. Without loss of generality, we may assume A is finite. We may further assume A is minimal: if A' ? A contains 0, then every ̂ '-definable bounded set is comparable to A'. By replacing A with \A\ (i.e., {\a\ : a e A}), we reduce to the case A contains only nonnegative elements of R. Let a be the largest element of A. Note a > 0 since every 0-definable bounded set is trivially comparable to 0. Let X be an .4-definable bounded set that is not comparable to A. We may (by replacing X with the symmetric hull of X) assume X is symmetric and convex. We




may also assume X < a since otherwise we could take B = A and Y = X. In fact X <Ht(a) since otherwise there would be an a-definable bounded subset of Ht (a) that contains both a and an element of X, hence that caps X.

Let ^47 := A \ {a} and note X is not A'-definable. Let (Xr)reR be an ̂-definable

family of definable bounded convex symmetric sets such that Xa = X. We can find c > a such that Xc ^ X since otherwise X would equal f]r \Js>r Xs, which is ^-definable.

Claim. It suffices to find a B-definable set Y' that contains a but not c, where B is either A' or A' U {b} for some b G [0, a] \ Ht (A).

It suffices to find an A' -definable set Y" that contains a but not all of Ht (a). Proof of claim. Since Y' has finitely many convex components, the component

C containing a is indefinable. Let Y be the (B-definable) symmetric hull of R+ n C. Since Y is included in [?c, c], it is bounded. Since Y includes [0, a], it includes B and is not comparable to B (because Ht (B) < a).

The second part of the claim is proven similarly, once it is observed that either Y" or its complement has a bounded-above convex component that contains an

element of Ht (a). H

Case 1: Suppose^ c X. Choose some positive b G X\XC, hence b < Ht (a). If b ? Ht (A1) then define B := ^47 U {b}, and note the set {x G R : b G Xx} contains a but not c. Suppose b G Ht(Af). Then there is an ̂ -definable bounded set Z ? Ht (,47) that contains b. Since Z does not cap X, but does contain an element of X, there is x G X with Z < x. Define B := Af, and note that a, but not c,

belongs to the set of r G R for which there is x G Xr with Z < x.

Case 2: Suppose X c Xc. If we can find some positive b G Xc \ X with b < Ht (a), then the argument proceeds as in the previous case. Assume there is no such b, hence X = {r G R : \r\ < Ht (a)} and Xc n Ht (a) ̂ 0.

Case 2.1: Suppose a' G Ht (a) is such that Xa> n Ht (a) ^ 0. Let (Zr)reR be a 0-definable family of definable sets such that Za> ? Ht (a1) and Xa> n Za/ ^ 0.

By 4.2, Za ? Ht (a), so the set {r G R : Xr n Zr = 0} is ̂-definable and contains

a but not a' (i.e., contains a but not all of Ht (a)). Case 2.2: Suppose Xa> n Ht (a)

= 0 for all a' G Ht (a). Let Z ? Ht (a) be a-definable such that Z 0 Xc ^ 0, and let Y' be the convex component of

{r>0:Znlr =

U} that contains a. Note F7 is bounded, since 0 < Y' < c. We claim F7 is not comparable to A, hence we can take B to be A and 7 to be the

symmetric hull of Y'. For a contradiction, assume Y' is comparable to A. Then F7 cannot contain an element larger than Ht (a). On the other hand, Y' does contain

every a' G Ht (a) because Z n Xa> ? Ht (a) n Xa> = 0. Thus, an a-definable set that caps Y' is cofinal in Ht (a). By assumption, there is such a set. Thus, by 4.3, Ht (a) is a-definable. But we are in a case where Ht (a) caps X, so this would mean that X is comparable to a. -\

Proposition 6.2. Suppose A is a T-distinct set that contains 0. Then all A

definable bounded sets are comparable to A.

Note: This is stronger than saying such a set is T-bounded by itself.

Proof. For a contradiction, assume this proposition is false. Then there is a

counterexample in which A is finite. Let m + 1 be the minimum cardinality of



760 JAMES TYNE

a counterexample. Note m > 0 since every 0-definable bounded set is trivially comparable to 0. Furthermore, m > 1 by the definition of T-height. (Using the notation from the definition of T-height, if r ̂ 0 and X is an r-definable bounded

symmetric convex set, then either X is included in a 0-definable bounded set or

br ? X ? Br. Thus, every r-definable bounded set is comparable to either 0 or r.) Let A be a T-distinct set of cardinality m + 1 that contains 0, and X an ^4-definable bounded set that is not comparable to A. By 6.1, we may assume X includes A, hence no set including X is comparable to A. Enumerate A =

{a0,... ,am} with

\ao\ < < \am\, hence ao = 0. Let A' := {0, a2,..., am}, so every v4'-definable

bounded set is comparable to A!. Let (Xr)reR be an ̂ '-definable family of definable bounded sets such that Xax

=

X. By the Key Lemma, we may assume this family is regionally bounded. (Note that in using the Key Lemma here, we look at the definable closure of A', which is included in Cl (A') because A! is T-bounded by itself.) Hence the set

(J|r|<ia2| ̂r *s

^'-definable and bounded, so it is comparable to A'. However, this set includes X, so it is not comparable to A'. H

Proposition 6.3. Suppose A is a T-distinct set that contains 0, and B is a set that is T-bounded by A.IfceR\C\(A) then B U {c} is T-bounded by AU {c}.

Proof. Use 4.8 to pass to an elementary extension, and reduce to the case where 9 is saturated. For a contradiction, assume this proposition is false. Then there is a counterexample in which B is finite. Let A, B, and c be a counterexample

with minimum cardinality of B \ A. Let n denote this cardinality, and note that n is nonzero by 6.2. We may assume A is minimal with respect to inclusion: for each a e A there is a B-definable set comparable to a. By replacing c with ? c if

necessary, we reduce to the case c > 0.

Let F be a (B, c)-definable rational set that is not comparable to A U {c}. Then Y is comparable to some positive element of R \ Ht (A U { c } ) ; call one such element d. Let (Yr)reR be a indefinable family of definable sets such that Yc = Y, and let

(Zr)reR be a 0-definable family of definable sets such that Z? is a subset of Ht (d) that caps Yc. Using 5.7, let c\,d\,c2, d2, c^, ?/3 be T-distinct elements of R+ \Ht (^4) such that Zdi caps YCj for i

= 1,2, 3. Fix b G B \ A and let B' denote B \ {b}. Let

(Wqj)qr^R be a ^'-definable

family of definable sets such that (Wbj)r^R =

(Yr)reR. Thus, for z = 1,2,3, the set

{q G R : Zdt caps WqXi} contains b\ let Qi be the convex component of this set that contains b, and note that Qt is (Bf, c?, dt)-definable. Now the Qi are convex subsets of R that contain a common point, so one of them includes the intersection of the other two. By re-indexing, we may assume Q\ D Q2 D Qi, hence Z?x caps WqM for all q G 02 H 03. Consequently, Z?x caps the intersection, over q G ?2 n ?3, of the symmetric hulls of the Wq,Cl. By 5.3, Z^ is included in Ht(d\), so this intersection is comparable to d\. Thus, by 5.4, this intersection is not comparable to AU C, where C denotes {c\,c2, c^,d2,d^}. On the other hand, this intersection is (Bf, C)-definable, so it is comparable to A U C by the minimality of n. H

Theorem 6.4. Suppose A and B are sets such that B is T-bounded by A. Then for every c G R there is a G R such that B U {c} is T-boundedby A U {a}.

Proof. Assume not, and let A, B, and c be a counterexample. Without loss of

generality, A is T-distinct and contains 0. Let Yl and Y2 be (B, c)-definable sets




that are not comparable to A, but are comparable to d\ and d2 respectively, where

d\ and d2 are T-distinct. By replacing Yl and Y2 with their symmetric hulls, we

may assume they are symmetric and convex.

Let i = 1,2. Let (Y*) be a ̂ -definable family of definable symmetric convex

sets such that Ylc ? Y1, and let Zl be a ?/?-definable subset of Ht (d?) that caps

Yl. We may choose Z1 so that {x G R : \x\ < Z1} is not 0-definable. (If Zl can be chosen so that this is not the case, then, since it is not comparable to 0, Yl contains an element of Ht (di). This element is larger than some d?-definable set that properly includes Ht(0). Choose Zl so that it does not intersect this set.) Consider the set of r G R such that Zl caps Ylr. Note c is contained in this set. Let

Qi denote the convex component of this set that contains c. By weak o-minimality, Qi is (B,di)-definable.

By re-indexing if needed, we may assume Q\ does not contain an element that is

larger than every element of Q2. (One might say sup Q\ < sup Q2.) Thus, if r G Q\ and r > c, then r G Q2, so Z2 caps Y}. In particular for such r, the set Y} contains all x G R with |x| < Z2, and contains no element larger than Z2. Therefore, Z2

caps the (B, d\)-definable set

*==n U yl

hence X is comparable to d2. By the choice of Z2, X is not comparable to 0, so X is not comparable to A U {d\} by 4.6. However, since it is rational, X is comparable to AU {di} by 6.3. H

Property II now follows easily. We state here a more refined version.

Corollary 6.5. Every set X is T-boundedby a set Y with card ( Y) < 1 +card (X). Proof. If there were a counterexample, then there would be a finite counterex

ample. However, the empty set is T-bounded by {0}, so repeated applications of 6.4 show that there is no finite counterexample. H

Corollary 6.6. Given a set X,

card ({Cl (x) : x G dcl(X)}) < 1 + card (X).

REFERENCES

[1] M. A. Dickmann, Elimination of quantifiers for ordered valuation rings, this Journal, vol. 52

(1987), no. Lpp. 116-128.

[2] Lou van den Dries, T-convexity and tame extensions II, this Journal, vol. 62 (1997), no. 1,

pp. 14-34.

[3] Lou van den Dries and Adam H. Lewenberg, T-convexity and tame extensions, this Journal, vol. 60 (1995), no. 1, pp. 7^102.

[4] Dugald Macpherson, David Marker, and Charles Steinhorn, Weakly o-minimal structures and real closed fields, Transactions of the American Mathematical Society, vol. 352 (2000), no. 12,

pp. 5435-5483 (electronic).

[5] Chris Miller, Tameness in expansions of the real field, Logic colloquium '01 (M. Baaz, S.-D.

Friedman, and J. Kraj?cek, editors), Lecture Notes in Logic, vol. 20, ASL and A K Peters, 2005,

pp. 281-316.

[6] Chris Miller and James Tyne, Expansions of o-minimal structures by iteration sequences, Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 1, pp. 93-99.



762 JAMES TYNE

[7] Anand Pillay and Charles Steinhorn, Definable sets in ordered structures I, Transactions of the American Mathematical Society, vol. 295 (1986), no. 2, pp. 565-592.

[8] James Tyne, T-levels and T-convexity, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2003.

[9] A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfajfian functions and the exponential function, Journal of the American Mathematical Society, vol. 9 (1996), no. 4, pp. 1051-1094.

DEPARTMENT OF MATHEMATICS

THE OHIO STATE UNIVERSITY

231 WEST 18TH AVENUE

COLUMBUS, OH 43210, USA

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t-height in weakly o-minimal structures

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