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ON WEAKLY O-MINIMAL NON-VALUATIONAL

EXPANSIONS OF ORDERED GROUPS

PANTELIS E. ELEFTHERIOU, ASSAF HASSON, AND GIL KEREN

Abstract. Let M = 〈M,<,+, . . .〉 be a weakly o-minimal expansionof an ordered group. This paper has two parts. In the �rst part, weintroduce the notion of M having no external limits and use it to provethat a large collection of non-valuational structures M do not admitde�nable Skolem functions. In the second part, we provide an alternativecharacterization of the canonical o-minimal completion from [11] andemploy it to produce new examples of non-valuational structures.

1. Introduction

A structure M = 〈M,<, . . .〉 is weakly o-minimal if < is a dense linearorder and every de�nable subset of M is a �nite union of convex sets [2, 5].Examples of weakly o-minimal structures are:

(a) M = 〈R,F in(R)〉, a non-archimedean real closed �eld R expandedby its natural valuation ring Fin(R).

(b) M = 〈Ralg, (0, π)〉, the �eld of real algebraic numbers expanded bythe convex set (0, π).

These are the archetypical examples of two categories of weakly o-minimalstructures that can be distinguished by their `de�nable cuts'. A pair (C,D)of non-empty subsets ofM is called a cut inM if C < D and C∪D = M . Itis called a de�nable cut if C (and D) are de�nable. IfM = 〈M,<,+, . . .〉 ex-pands an ordered group, then a de�nable cut (C,D) is called non-valuationalif the in�mum inf{y − x : x ∈ C, y ∈ D} exists in M (and must equal 0).Otherwise, it is called valuational. M is called non-valuational if every de-�nable cut is non-valuational; otherwise, valuational. Example (a) above isvaluational and (b) is non-valuational. We denote by M the set of all de�n-able cuts (C,D) inM such that C has no maximum element. Then M has anatural order, where (C,D) < (C ′, D′) if and only if C < C ′, which extendsthe order < ofM , where a ∈M is identi�ed with

((−∞, a), [a,+∞)

). In [5],

a weak cell decomposition theorem was proved for every weakly o-minimal

Date: September 16, 2015.1991 Mathematics Subject Classi�cation. 03C64.Key words and phrases. Weakly o-minimal structures, o-minimal traces.The �rst author was supported by an Independent Research Grant from the German

Research Foundation (DFG) and a Zukunftskolleg Research Fellowship.The second author was partially supported by an Israel Science Foundation grant num-

ber 1156/10.

1

2 P. ELEFTHERIOU, A. HASSON, AND G. KEREN

structure M. In [5, 11], a strong monotonicity theorem and a strong celldecomposition theorem were shown in caseM is non-valuational, exhibitingits resemblance to o-minimal structures. Moreover, it was shown that theproperty of being non-valuational is elementary [13].

Canonical examples of weakly o-minimal non-valuational structures arethe `o-minimal traces', obtained by considering dense pairs of o-minimalstructures, as introduced in [3]. Let N be an o-minimal structure andM1 ≺N a dense elementary substructure. The theory of dense pairs is the theoryof the structure 〈N ,M1〉 obtained by expanding N with a unary predicatefor the universe M ofM1. By a result of [1] the structureM induced from〈N ,M1〉 on M is weakly o-minimal. Moreover, any de�nable set inM is ofthe form Mn ∩ S where S ⊆ Nn is N -de�nable, [3, Theorem 2]. Thus, anyde�nable cut inM is of the form

((−∞, a) ∩M, (a,+∞)

)for some a ∈ N .

Since M is dense in N , we obtain thatM is non-valuational.In this paper, we study model theoretic and topological properties in the

setting of weakly o-minimal non-valuational expansionsM = 〈M,<,+, . . .〉of ordered groups. In Section 3, we introduce the notion of having no ex-ternal limits (De�nition 3.1) and use it to prove that a large collection ofsuchM do not admit de�nable Skolem functions. We then explore di�erentnotions of topological compactness and illustrate their inadequacy in the cur-rent context, due to the main feature that the topology is de�nably totallydisconnected. In Section 4, we turn to the canonical o-minimal completionM of M, introduced by Wencel [11], and propose that a model theoreticanalysis of the pair (M,M) may be fruitful for the the geometric study ofde�nable sets inM. To illustrate this proposal, we prove that the dimensionof a de�nable set equals to the o-minimal dimension of its closure in M, and(re-)establish the basic properties of dimension, such as its invariance underde�nable injective functions. In Section 5, we give yet another applicationof appealing to the canonical o-minimal completion; namely, we employ itto produce new examples of structures M, which, in particular, cannot beobtained as reducts o-minimal traces.

We now list the main results of the paper. In Section 2 we prove thefollowing theorem.

Theorem 1. LetM be an o-minimal expansion of an ordered group. Thenthere existsM≺ N such that M is dense in N and N realizes all irrationalnon-valuational cuts inM.

In Section 3, we introduce the notion of having no external limits andcombine it with Theorem 1 to derive the next theorem, which generalizes aresult of Shaw [9].

Theorem 2. Let M be an o-minimal expansion of an ordered group, M∗any proper expansion ofM by non-valuational cuts. Then any reduct ofM∗preserving the order and the group structure is weakly o-minimal and doesnot admit de�nable Skolem functions.

WEAKLY O-MINIMAL NON-VALUATIONAL STRUCTURES 3

In Section 4 we give a somewhat less technical description of the canonicalo-minimal completion M introduced in [11].

Proposition 1. Let M be a weakly o-minimal non-valuational expansionof an ordered group. Consider the structure M∗ whose universe is M andwhose atomic sets are

{ClM (S) : S ⊆Mn (some n)M-de�nable}.

Then M∗ and M have the same de�nable sets.

Section 5 is dedicated to a collection of results that could be summarizedas follows.

Theorem 3. The class of o-minimal traces is not closed under taking reducts.There exists a weakly o-minimal non-valuational expansion of an orderedgroup which is not a reduct of an o-minimal trace.

Our conclusion from the results of the present paper is that the realm ofweakly o-minimal non-valuational expansions of ordered groups is wider thanwe �rst expected. It seems that a topological and geometric study of suchstructures cannot directly borrow ideas from the o-minimal context. Rather,one has to study the theory of the pairs 〈M∗,M〉. The right abstract contextfor studying such pairs is yet to be found, as the context of dense pairs isnot general enough.

Acknowledgements. We wish to thank Salma Kuhlmann for hosting GilKeren in Konstanz in the Winter term 2013/2014, during which time thiscollaboration began. We would also like to thank Moshe Kamensky for hiscomments and ideas.

2. Non-valuational cuts

If (C,D) is a cut in an ordered structure (M, <, · · · ) we identify it withthe (partial) type {x > C}∪ {x ≤ D}. We recall that a cut inM is rationalif it is realized in M (equivalently, if D has a minimum). As mentionedin the introduction, a cut (C,D) in an ordered group is de�ned to be non-valuational if

(*) inf{d− c : c ∈ C, d ∈ D} = 0.

As was pointed out by Kamensky, (∗) is, in fact, equivalent to the existenceof the in�mum, provided that the group is p-divisible for some p. Indeed,assume that (C,D) is such that

inf{d− c : c ∈ C, d ∈ D} = γ > 0.

Take d ∈ D, c ∈ C such that d−d′ < γp for all d

′ ∈ D with d′ < d and similarly

for c′ ∈ C. Then d−cp ≥

γp , implying that c + d−c

p /∈ C so c + d−cp ∈ D. But

d− c− d−cp = (p−1)(d−c)

p ≥ γp , a contradiction.


Since weakly o-minimal groups are divisible ([5, Section 5]) in our settingthe two notions are indeed equivalent.

In [6, Lemma 2.2] it is shown that if M is o-minimal, and a realizes anirrational cut over M then no b ∈M(a) \M realizes a rational cut overM.The following lemma is an analogue for non-valuational cuts. It follows from[12], but we provide a succinct proof.

Lemma 2.1. Let M be an o-minimal expansion of an ordered group. Let(C,D) be an irrational non-valuational cut over M and a |= (C,D) anyrealization. ThenM(a) does not realize any valuational cuts over M .

Proof. Let b ∈ M(a) be any element. Then there exists an M-de�nablefunction f such that b = f(a). By o-minimality and the fact that a /∈M (because (C,D) is an irrational cut) there is an M-de�nable interval Icontaining a such that f is continuous and strictly monotone or constant onI. If f is constant on I then b ∈M and there is nothing to prove. So we mayassume without loss of generality that f is strictly increasing. RestrictingI, if needed, we may also assume that I is closed and bounded. So f isuniformly continuous on I.

By [6, Lemma 2.2] the type p ∈ S1(M) of a positive in�nitesimal elementis not realized inM(a). It follows, since (C,D) is non-valuational, that forany c ∈ C(M(a)), d ∈ D(M(a)) and 0 < δ ∈M(a) there are c < c′ < d′ < dwith c′ ∈ C(M) and d′ ∈ D(M) with d′ − c′ < δ. Given any 0 < ε ∈ M(a)take 0 < ε′ < ε with ε′ ∈ M and let δ ∈ M be such that |f(x) − f(y)| < ε′

for any x, y ∈ I with |x− y| < δ. So

inf{f(d)− f(c) : c ∈ C(M(a)), d ∈ D(M(a))} = 0.

Thus, b realizes a non-valuational cut overM. Since b ∈M(a) is arbitrary,this �nishes the proof of the lemma. �

Our main theorem in this section is1:

Theorem 2.2. LetM be an o-minimal expansion of an ordered group. Thenthere existsM≺ N such that M is dense in N and N realizes all irrationalnon-valuational cuts inM.

Proof. Let {(Ci, Di)}i∈I be an enumeration of all the non-valuational cutsinM, and p ∈ S1(M) the type of a positive in�nitesimal. We construct Nby induction as follows. For i = 1 letM1 =M if (C1, D1) is realized in M .Otherwise, letM1 :=M(a1) where a1 |= (C1, D1) is any realization. By [6,Lemma 2.2] p is not realized inM1.

Assume now that we have constructedMi such that (Cj , Dj) is realizedinMi for all j < i and such that p is not realized inMi. If i is a successorordinal we let Mi,0 = Mi−1 and if i is limit we let Mi,0 :=

⋃j<iMi. If

(Ci, Di) is realized in Mi,0 set Mi = Mi,0. Otherwise set Mi := Mi,0(ai)where ai |= (Ci, Di) is any realization.

1We thank Y. Peterzil for pointing out this formulation of the theorem.


We prove that Mi does not realize p. If Mi = Mi+1 this follows fromthe induction hypothesis. So we assume that this is not the case. Thus(Ci(Mi), Di(Mi)) de�nes a cut in Mi. Observe that (Ci(Mi), Di(Mi)) isstill non-valuational. Indeed, if inf{di− ci : ci ∈ Ci(Mi), di ∈ Di(Mi)} doesnot exist, then there are ci ∈ Ci(Mi) and di ∈ Di(Mi) such that di−ci |= p,contradicting the inductive hypothesis. Thus, applying [6, Lemma 2.2] againthe desired conclusion follows.

Let N :=⋃i∈IMi. We claim that N has the desired properties. By

construction N realizes all de�nable non-valuational cuts in M. Now let(a, b) ∈ N be any interval. Since N does not realize p there exists somer ∈ M such that 0 < r < b − a, so in order to show that (a, b) ∩M 6= ∅ itwill su�ce to show that a realizes a non-valuational cut, which is preciselythe conclusion of the previous lemma. �

The construction of the previous proof can be re�ned to give:

Corollary 2.3. LetM be an o-minimal expansion of an ordered group. LetM be the expansion ofM by unary predicates {Ci}i∈I interpreted as distinctirrational non-valuational cuts in M . Then there exists an elementary ex-tensionM≺ N such that M is dense in N and M is precisely the structureinduced onM by all externally de�nable subsets from N .

Proof. Construct N precisely as in the previous theorem, save that this timewe only realize those cuts appearing in {(Ci, Di)}i∈I . ThenM≺ N and Mis dense in N , so (N ,M) is a dense pair. By [3, Theorem 2] the structureinduced onM in the pair (N ,M) is precisely the expansion ofM by unary

predicates for all cuts realized in N . Thus, by construction, we get that Mis a reduct of the structure induced on M from (N ,M). So it remains to

show that any cut over M de�nable in (N ,M) is de�nable in M.So let a ∈ N be any element. We have to show the (−∞, a)∩M is de�nable

in M. By construction there are a1, . . . , an realizing the cuts Ci1 , . . . , Cin andan M -de�nable continuous function, f , such that f(a1, . . . , an) = a. Choosea1, . . . , an and f so that n is minimal possible. For every η ∈ {−1, 1}n saythat f is of type η at a point c ∈ Nn if

fi(xi) := f(c1, . . . , ci−1, xi, ci+1, . . . , cn)

is strictly monotone at ci and for all 1 ≤ i ≤ n and fi(xi) is increasing at ci ifand only if η(i) = 1. By the minimality of n there is some η ∈ {−1, 1}n suchthat f is of type η at (a1, . . . , an). In particular, the set Fη of points x suchthat f is of type η at x is M -de�nable and open. Let Li := Ci if η(i) = −1

and Li := Di otherwise, then L :=∏ni=1 Li ∩Fη is open and de�nable in M ,

and for any x ∈ L we have that f(x) < f(a1, . . . , an) = a. Since M is densein N and L(N) is not empty also L(M) is not empty. Thus

x ∈ (−∞, a) ⇐⇒ ∃y(y ∈ L ∧ x < f(y))

and the right hand side is M -de�nable. �


3. No external limits

In this section we introduce the notion of having no external limits andprove Theorem 2 from the introduction.

De�nition 3.1. A weakly o-minimal structure,M has no external limits ifwhenever (a, b) is an interval in M (with a, b ∈M ∪ {±∞}) and γ : (a, b)→M a de�nable function such that either limt→a γ(t) or limt→b γ(t) exist inM , then the limit is in M . Otherwise, we say thatM has external limits.

By [11, Lemma 1.3] for every M and γ as above the limit limt→b γ(t)exists in M ∪ {±∞}. SoM has no external limits precisely when the limitsare always in M ∪ {±∞}.

Dense pairs provide us with the basic example of weakly o-minimal struc-tures with no external limits.

Lemma 3.2. Let (N ,M) be a dense pair of o-minimal expansions of orderedgroups. Let M∗ be the induced structure on M. Then M∗ has no externallimits.

Proof. By [3, Theorem 3] every function f :M→M de�nable in (N ,M) ispiecewise given byM-de�nable functions f1, . . . , fk (some k). I.e., for everyx ∈ M there is 1 ≤ i ≤ k such that f(x) = fi(x). Let γ : (a, b) → Mbe a de�nable bounded function. By de�nition, γ is de�nable in (N ,M)so γ is given, piecewise by γ1, . . . , γk with all γi M-de�nable. As M∗ isweakly o-minimal the set x ∈ M such that f(x) = fi(x) is M∗-de�nable,and therefore a �nite union of de�nable convex sets. In particular, there issome 1 ≤ i ≤ k such that f = fi on a �nal segment (a′, b) ⊆ (a, b). SinceMis o-minimal and γ is bounded limt→b γ(t) ∈M . �

As an application we get a generalization of a theorem of Shaw [9].

Theorem 3.3. LetM be an o-minimal expansion of an ordered group,M∗any proper expansion of M by non-valuational cuts. Then M∗ is weaklyo-minimal and does not admit de�nable Skolem functions.

The theorem will follow immediately from Corollary 2.3 and Lemma 3.2once we prove:

Lemma 3.4. Let M be a weakly o-minimal expansion of an ordered group.Then M is o-minimal if and only if it has no external limits and admitsde�nable Skolem functions.

Proof. The left-to-right direction is well-known. For the right-to-left direc-tion, let (C,D) be any de�nable cut in M. Fix some a ∈ M and considerthe de�nable set

S := {(x, y) ∈M2 : 0 < x < a, y ∈ C, x+ y /∈ C}.Since M admits de�nable Skolem functions, there is a de�nable functionf : (0, a) → M such that (x, f(x)) ∈ S for all x ∈ (0, a). So limt→0 f(x) =


sup(C). Since M has no external limits, supC ∈ M i.e., sup(C) = inf(D)implying that (C,D) is a rational cut. Since (C,D) was arbitrary, it followsthat all de�nable cuts inM are rational, so weak o-minimality implies that,in fact,M is o-minimal. �

We point out that both properties of having no external limits and of beingnon-valuational are preserved under ordered reducts (the latter requiring thepreservation of the group structure as well). So we get:

Corollary 3.5. Let (N ,M) be a dense pair of o-minimal expansions of an

ordered group. Let M be an any expansion ofM by sets de�nable in (N ,M).

Then M does not admit de�nable Skolem functions.

These observations combine with [10] to give:

Corollary 3.6. Let M be a valuational weakly o-minimal expansion of anordered group. ThenM has external limits.

Proof. Since weakly o-minimal ordered groups are divisible abelian ([5, Sec-tion 5]) the reductM+ := (M,<,+) is o-minimal. LetMU := (M,<,+, U)where U is a unary predicate interpreted as a valuational cut de�nable inM.ThenMU is an expansion ofM+ by a valuational cut and by [10, Theorem4.12] it has de�nable Skolem functions. By Lemma 3.4 MU has externallimits. Therefore so doesM, as it is an expansion ofMU . �

We give an example of a weakly o-minimal structure (but not an expansionof an ordered group) which does have external limits.

Example 3.7. LetR be the structure obtained by appending two real closed�elds one "on top" of the other. More precisely, the language is given by(≤, R1, R2,+1, ·1,+2, ·2) and the theory of R is axiomatised by:

(1) R1, R2 are unary predicates such that (∀x)(R1(x) ↔ ¬R2(x)) and(∀x, y)(R1(x) ∧R2(y)→ x < y).

(2) +i, ·i are ternary relations supported only on triples of elements inRi. They are graphs of functions on their domains, and Ri is a realclosed �eld with respect to these operations.

(3) ≤ is an order relation compatible with the �eld ordering of R1 andR2 together with (1) above.

It follows immediately from quanti�er elimination for real closed �elds, thatthe above theory is complete and has quanti�er elimination (after addingconstants for 0, 1 in both �elds, and relation symbols for the inverse functionin both �elds). Thus R is weakly o-minimal, and the only de�nable cut inR not realized in R is (R1(R), R2(R)). It follows that R = R ∪ ∗ where ∗ isa new point realizing the above cut.

However, R does have external limits. Take the function x 7→ x−1 in the�eld structure on R1 on the interval (0, 1). Clearly its limit, as x→ 0+ is ∗.

Remark 3.8. In [11] Wencel introduces the notion of strong cell decompo-sition (see De�nition 4.3 below) and proves that in the context of expansions


of ordered groups this notion is equivalent to notion of being non-valuational.We point out that the previous example, though not expanding a group, hasthe strong cell decomposition.

Indeed, by quanti�er elimination any de�nable subset of Rn is the disjointunion of a de�nable subset of Rn1 and a de�nable subset of Rn2 both de�nablein the language of ordered �elds. Thus, by cell decomposition in R1 and inR2 (as stably embedded o-minimal sub-structures) any de�nable set in Rnis a union of cells in Rn1 and in Rn2 . Note that the iterative convex hull (see4.3) of any cell C ⊆ Rni is C itself. Thus, any such cell is a strong cell, andtherefore R has strong cell decomposition.

Conjecture 3.9. LetM = 〈M,<,+, · · ·〉 be a weakly o-minimal non-valuationalexpansion of an ordered group. ThenM has no external limits.

3.1. Notions of de�nable compactness. De�ning a notion of �de�nablecompactness" in the weakly o-minimal setting is problematic, since the topol-ogy is de�nably totally disconnected, unless the structure is, in fact, o-minimal. In the present sub-section we explore several variants of de�nablecompactness, showing they all fail to satisfy some of the basic properties ofcompact sets. Our examples are not pathological, they are simple and nat-ural. This failure to �nd a satisfactory analogue of de�nable compactnessin the present setting will lead us in the next section to suggest a di�erentapproach to the problem.

The direct analogue of the notion of de�nable compactness from the o-minimal setting would be the following.

De�nition 3.10. LetM be an ordered structure. We say that a de�nableset X ⊆ Mn is curves closed if for any de�nable curve γ : (a, b)→ X (witha, b ∈M ∪ {±∞}) the limit lim

t→bγ(t) exists and is an element of X.

Remark 3.11. If M is a weakly o-minimal non-valuational expansion ofan ordered group, then in the de�nition of curves closed we may restrict tocontinuous de�nable functions.

We next give a few observations supporting the view that curves closednesscaptures some sort of compactness, as well as some examples demonstratingits shortcomings.

Lemma 3.12. Let M be a weakly o-minimal non-valuational expansion ofan ordered group with no external limits. Then any closed and bounded setis curves closed.

Proof. Let S ⊆ Mn be a de�nable closed and bounded set. Let γ(a, b) →S be a de�nable curve. We may assume, using the strong monotonicitytheorem [5, Proposition 6.4] and passing to a �nal segment, that � writingγ := (γ1, . . . , γn) � γi is monotone for all i. Since S is bounded, it follows thatlimt→b γ(t) exists. AsM has no external limits it follows that limt→b γ(t) ∈Mn, and as S is closed limt→b γ(t) ∈ S. �


Question 3.13. Is "non-valuational" really needed in the above?

Remark 3.14. In the above proof we have used Proposition 6.4 of [5]. Fromnow on we will use this result freely without reference.

The converse of the previous lemma is, however, not true:

Example 3.15. LetM be the �eld of real algebraic numbers expanded bya predicate for the irrational cut at π. This is an expansion of a real closed�eld by a non-valuational cut, and therefore has no external limits. We willconstruct an unboundedM-de�nable set which is curves closed.

Let f(x) = 1π−x and g(x) = 1

π−2x . Let

S := {(x, y) ∈M2 : 0 < x <π

2, f(x) < y < g(x)}.

Then S isM-de�nable. It is unbounded, and we will show that it is curvesclosed. Observe that Γf ∩M2 = ∅ and Γg ∩M2 = ∅, that {(π/2, y) : 2

π ≤y} ∩M2 = 0 and that (0, 1

π ) /∈M2. It follows that S is closed in M2. Thus,if we can show that there are no de�nable unbounded curves in S, the proofof the previous lemma will give the desired conclusion. So assume towards acontradiction that there is an unboundedM-de�nable curve γ : (a, b)→ S.Let γ1, γ2 be the components of γ. Then, obviously, limt→b(γ1(t)) = π. Bythe strong monotonicity theorem, there is a �nal segment (c, b) ⊆ (a, b) suchthat γ|(c, d) is the graph of an M-de�nable function f : (d, π) → M withlimt→π f(t) =∞. We now conclude using the following general lemma.

Lemma 3.16. Let M be a weakly o-minimal non-valuational expansion ofan ordered group with no external limits. Suppose that C ⊆ M is a de�n-able bounded convex set. Assume that f : C → M is de�nable and thatlimt→supC f(t) =∞ then supC ∈M .

Proof. By the strong monotonicity theorem, we may assume without lossof generality that f is strictly increasing and continuous on C. Fix somea ∈ C and denote b = f(a). Then f−1 : (b,∞) → C is M-de�nable andlimt→∞ f

−1(t) = supC. SinceM has no external limits, supC ∈M . �

The above has another, not surprising, application:

Corollary 3.17. If M is a weakly o-minimal non-valuational expansion ofan ordered group with no external limits, then the image of any closed andbounded interval in M under a de�nable continuous function is closed andbounded. In particular, the continuous image of a closed and bounded intervalcannot be the whole of M .

Proof. We only have to show that if [a, b] ⊆M and f : [a, b]→M is de�nableand continuous then Fa,b := f([a, b]) is closed. So assume that c ∈ ∂Fa,b. Byweak o-minimality, we may assume that there exists d such that (c, d) ⊆ Fa,b(otherwise replace f with −f). Consider f−1((c, d)). By weak o-minimalityit is a �nite union of de�nable convex subsets of [a, b], restricting to a closedsub-interval, we may assume that f−1((c, d)) is a de�nable convex set C.


By restricting further (changing d if needed) we may assume by the strongmonotonicity theorem that f is strictly monotone or constant on C. If f isconstant then it must be by continuity that f(C) = c. So we assume thatf is, say, strictly decreasing on C. Then inf f(C) = c and f−1 : (c, d) → Chas a limit e at c, and e ∈ [a, b] implying f(e) = c. �

The above does not, however, generalise to higher dimensions. The imageof a closed and bounded set under a continuous function need not be closed:

Example 3.18. LetM and S be as in Example 3.15. Consider the projec-tion π1 : S → M onto the �rst coordinate. Clearly π(S) = (0, π) which isneither curves-closed nor closed.

We �nally consider two alternative ways for de�ning compactness in oursetting which we claim also fail to capture the desired properties. Considerthe algebro-geometric notion of a complete variety, adapted to the presentsetting. Say that an M -de�nable set X is geometrically complete if for anyde�nableM-space Y the projection π : X × Y → Y is a closed map. Thenif M is a strictly weakly o-minimal non-valuational expansion of a �eld nointerval is geometrically complete. Indeed, consider the set S of example3.15, then S ⊆ [0, 4] × R and is closed, but π1(S) = (0, π) which is notclosed. Since any two closed intervals in M are de�nably homeomorphic,this proves our claim.

In [4] yet another notion of de�nable compactness is introduced. A de�n-able set X in a topological structure is types closed if every de�nable typep extending X has a limit in X, i.e., if there exists a point a ∈ X suchthat for any de�nable open set U ∈ p we have a ∈ U (namely, viewed as anulra-�lter on the class of de�nable sets, p converges to a). In the context ofo-minimal expansion of ordered groups Hrushovski and Loeser mention thatthis notion coincides with de�nable compactness. This follows readily fromthe main result of [7]. Indeed, let X be a de�nable closed and bounded set,and let p be a de�nable type extending X. By [7] if a |= p is any realiza-tion then no irrational cut is realized in M(a). In particular stM (a) � thestandard part of a with respect to M � is de�ned (because X is bounded),and is readily seen to be a limit point of p (because p is de�nable and X isclosed). However, it is easy to check that an interval in a weakly o-minimalstructure is types closed if and only if it is o-minimal (with respect to theinduced structure).

4. The canonical o-minimal completion

As we have seen in the previous sub-section, basic topological notions suchas (de�nable) compactness and connectedness are impossible to de�ne inthe present context. As these problems arise due to the underlying topologybeing de�nably totally disintegrated, a natural solution to the problem wouldbe to look for some sort of (canonical) completion, with a better behavedtopology. In order to study the completion model theoretically, we have to


equip it with some structure, making sure that this structure has a tractableinteraction with the structure we started with. In the present section wesuggest how to construct such a completion, and study its basic properties.This is only a �rst step in that direction, as it is not yet clear to us what theright completion of a de�nable space (or even a de�nable set) should be.

In this context, the present section is dedicated to the study of the canoni-cal o-minimal completion M of a weakly o-minimal nonvaluational structure,introduced by Wencel in [11]. Our long term goal is to study the pair (M,M)(the expansion of the structrue M by a unary predicate for the set M , theuniverse of M). Here we develop some of its basic properties. In the nextsection we will give examples showing that this class of structures is strictlylarger than the class of dense pairs.

We start with a simple characterization of the structure M.

De�nition 4.1. LetM be a weakly o-minimal non-valuational expansion ofan ordered group. Let M∗ be the structure whose universe is M and whosede�nable sets are

{ClM (S) : S ⊆Mn,M− de�nable}

Our main result (Proposition 4.8 below) is that M∗ and M have the samede�nable sets. We proceed to de�ne M and establish some results for it.

De�nition 4.2. Let M be a weakly o-minimal non-valuational expansionof an ordered group, C ⊆Mn anM-de�nable set. A function f : C → M isde�nable if the set {(x, y) : y < f(x)} isM-de�nable.

De�nition 4.3. Let M be a weakly o-minimal non-valuational expansionof an ordered group. De�ne recursively:

(1) A 0-strong cell inM is a point in M . Its iterative convex hull in Mis itself.

(2) A 1-strong-cell M is a de�nable convex set (s, t) with s, t ∈ M . Theiterative convex hull of (s, t) in M is {x ∈ M : s < x < t}.

(3) Let C ⊆ Mn be a strong cell. AnM-de�nable function f : C → Mis strongly continuous if it can be extended continuously to C, theiterative convex hull of C.

(4) Let η ∈ {0, 1}n. An η_0-strong-cell in Mn+1 is a set

{(x, f(x)) : x ∈ C}where C ⊆Mn is an η-strong-cell, and f : C →M is anM-de�nablestrongly continuous function. The iterative convex hull of C inMn+1

is {(x, f(x)) : x ∈ C} where f is the (unique) continuous extensionof f to C.

(5) Let η ∈ {0, 1}n. An η_1-strong-cell in Mn+1 is a set

{(x, y) : x ∈ C f(x) < y < g(x)}where C ⊆Mn is an η-strong-cell, and f, g : C → M areM-de�nablestrongly continuous functions. The iterative convex hull of C in


Mn+1 is

{(x, y) : x ∈ C, f(x) < y < g(x)}where f , g are the (unique) continuous extensions of f, g to C.

A weakly o-minimal structureM has the strong cell decomposition prop-erty if whenever X1, . . . , Xk are M-de�nable subsets of Mn there exists adecomposition of Mn into strong cells partitioning each of the Xi.

Fact 4.4. Let M be a weakly o-minimal non-valuational expansion of anordered group. Then:

(1) M admits strong cell decomposition [11, Theorem 2.15].(2) Let M be the structure whose universe is M and whose atomic sets

are

{C : C a strong cell inM}where for anM-de�nable strong cell C we let C be its iterative convexhull in M . Then M is o-minimal [11, Section 3].

(3) The structure induced from M on M is M. This follows from [13]Proposition 2.2, the discussion following it, and Proposition 2.13.

The structure M above is called the canonical o-minimal completion ofM. Before we proceed to prove that it has the same de�nable sets as M∗,we give some evidence supporting Wencel's claim for the canonicity of hisconstruction.

Lemma 4.5. Let N be an o-minimal structure, M ⊆ N a dense subset.Assume that N ′ is an o-minimal structure with universe N and the sameorder relation as N , and such that for any N -de�nable set S ⊆ Nn thereexists an N ′-de�nable set S′ such that S ∩Mn = S′ ∩Mn. Then N ′ is anexpansion of N .

Proof. By cell decomposition, in N it will su�ce to show that any N -de�nable open cell is also N ′-de�nable. So let C be such an open cell.Then C = int Cl(C). In addition, because C is open and M is dense in Nwe also get Cl(C ∩Mn) = Cl(C).

Let C ′ beN ′-de�nable such that C ′∩Mn = C∩Mn. Since Cl(C) = Cl(C∩Mn) we get that Cl(C ′) ⊇ Cl(C), implying that C = int Cl(C) ⊆ int Cl(C ′).

On the other hand, if ∅ 6= int Cl(C ′) \ Cl(C) then Mn ∩ (int Cl(C ′) \Cl(C)) 6= ∅. By o-minimality of N ′ this implies that there exists a de�nableopen box B ⊆ C ′ such that B ∩Cl(C) = ∅. This contradicts the assumptionthat C ∩Mn = C ′ ∩Mn.

It follows that C = int Cl(C ′). Sinc the left hand side is N ′-de�nable thelemma is proved. �

Corollary 4.6. Let N , N ′ be o-minimal structures with universe N and thesame underlying order. Assume that for some dense M ⊆ N , the trace ofN on M and the trace of N ′ on M are the same. Then N and N ′ have thesame de�nable sets.


Specialising this last corollary to the case whereM is a weakly o-minimalnon-valuational expansion of an ordered group, and N = M , we get:

Corollary 4.7. LetM be a weakly o-minimal non-valuational expansion ofan ordered group. Then M is the unique � up to a change of signature �o-minimal structure on M whose trace on M isM.

We now return to the proof of Proposition 3 from Introduction.

Proposition 4.8. LetM be a weakly o-minimal non-valuational expansionof an ordered group. Then M and M∗ have the same de�nable sets.

Proof. SinceM admits strong cell decomposition, it will su�ce to show thatif C ⊆ M is a strong cell then ClM (C) is de�nable in M, and that C isde�nable in M∗.

The �rst implication is obvious. Indeed, it is easy to check that C is densein C. So ClM (C) = ClM (C), and since M is o-minimal the right hand sideis M-de�nable.

The other implication is proved by induction. For 0-cells and for 1-cells,there is nothing to prove. So assume we have proved the result for all strongcells in Mn and we prove it for (n + 1)-strong-cells in Mn+1. Let C ⊆ Mn

be a strong cell. Let f : C → M be an M-de�nable strongly continuousfunction, and let D be Γf , the graph of f . Then D := {(x, f(x)) : x ∈ C}.We have to show that D is M∗-de�nable. As f is strongly continuous weget that D = ClM (Γf ) ∩ (C × M), which is M-de�nable by the inductivehypothesis.

Now let f, g : C → M beM-de�nable strongly continuous functions withf < g. We have to show that the iterative convex hull of D := (f, g) isM∗-de�nable. By de�nition

D = {(x, y) : x ∈ C, f(x) < y < f(g)}.Since, by induction C is M∗-de�nable, it will su�ce to show that f is M∗-de�nable. If f is the constant function −∞, there is nothing to prove. Sowe assume this is not the case. By de�nition, the set

F := {(x, y) : x ∈ C, y < f(x)}isM-de�nable. For every c ∈ C let

s(c) := sup{y : (c, y) ∈ ClM (F )}.Since f is strongly continuous, s(c) is well de�ned, and by de�nition it co-incides with f on C. Since C is dense in C and f is the unique continuousextension of f to C, necessarily s = f , and as s is M∗-de�nable, we aredone. �

We note that since the structure induced on M from M∗ isM, we obtainthe following.

Corollary 4.9. The closed and boundedM-de�nable subsets of Mn (any n)are precisely the traces of de�nably compact M∗-de�nable sets in Mn.


4.1. De�nable functions. We next use Proposition 4.8 to obtain someinformation about de�nable functions in the canonical o-minimal completion.

De�nition 4.10. Let S ⊆ Mn be a de�nable set. A de�nable functionf : S → M is strongly continuous if there exists anM∗-de�nable set S ⊆ Bsuch that f extends continuously to B.

Note the above de�nition is weaker than De�nition 4.3. E.g., if Rπalg is theextension of the �eld of real algebraic numbers by a predicate for the interval

(0, π) then the function f(x) =

{1, if x < π

0 otherwiseis strongly continuous in

the sense of the above de�nition, but not in that of De�nition 4.3. Withthe present de�nition, however, we can show that any continuous de�nablefunction is strongly continuous on a large set.

Lemma 4.11. Let S ⊆ Mn be a de�nable set and f : S → M a de�nablecontinuous function. Then there exists a de�nable set S′ ⊆ S such that f isstrongly continuous on S′ and dim(ClM(S) \ ClM (S′)) < dim ClM (S)

Proof. Let Γ := ClM Γf . By Proposition 4.8 we know that Γ isM∗-de�nable.Let Γ0 := {x ∈ Γ : |π−1(π(x)) ∩ Γ| = 1}, where π : Mn+1 → Mn is theprojection onto the �rst n-coordinates. Let S0 := π(Γ0). Let Γ1 := Γ \ Γ0

and S1 := π(Γ1). Note that dim(S1) < dim(ClM (S)). Otherwise there mustbe s ∈ S1 ∩ S 6= ∅ and t1, t2 ∈ M such that (s, ti) ∈ Γ1. Assume, withoutloss that t1 = f(s), implying that for all δ > 0 there exists s′ ∈ Bδ(s) ∩ Ssuch that |f(s′)− t2| < 1

2 |t1 − t2|, contradicting the continuity of f on S.

So let S2 := S ∩ S0 and Γ2 := π−1(S2) ∩ Γ. Then Γ2 is the graph ofa function, f : S2 → M , and by o-minimality f is continuous on a largeset. �

On the local level, the previous lemma becomes a little trickier. Considerthe following example:

Example 4.12. Let Rarg := (R,≤,+, ·, arg) where arg is the argumentfunction on R2 \ [0,∞). Then Rarg is o-minimal as a reduct of Ran. LetR be a countable elementary sub-structure with universe R, and α ∈ R \Rwith 1

4 < α < 34 , and de�ne

f(t) =

t, if t ∈ [0, 1

4 ]14 , if t ∈ [1

4 , α)

1, if t ∈ (α, 34 ]

4(1− t), if t ∈ [34 , 1]

Then f is a continuous function de�nable in Rα, the expansion of R by apredicate for x < α. Note that Rα is an o-minimal trace, whence weaklyo-minimal and non-valuational. Finally, observe that f(0) = f(1) = 0 so the

function F (x, y) = (x2 + y2)f(arg(x,y)2π ) (with F (0, 0) = 0) is also de�nable

and continuous. Clearly, F does not have a continuous extension to R2 (or


even to R(α)2), but it does have a continuous extension F to R2 \ arg−1(α).Note, however, that F is continuous at the origin. Thus, (0, 0) is a pointwhere F is continuous, but there is no (relatively) open neighbourhood of(0, 0) in R2 where F can be extended.

A local analysis of the proof of Lemma 4.11 shows that the above exampleis, in a way, the worst case scenario. To formulate the local version of Lemma4.11 we need a couple of de�nitions:

De�nition 4.13. (1) LetM be an o-minimal structure and S ⊆Mn ade�nable set. The dimension of S at a ∈Mn is

dima(S) := min{dim(U ∩ S) : a ∈ U,U open}.

This is the dimension of the germ of S at a.(2) Let M be a weakly o-minimal non-valuational expansion of an or-

dered group. Let S ⊆ Mn and f : S → M be de�nable with fcontinuous on S. Let domM f := dom f , where f is the continuousextension of f to M as provided by Lemma 4.11.

Corollary 4.14. Let S ⊆ Mn and f : S → M a de�nable continuousfunction then for any a ∈ S the function f is strongly continuous at a, i.e.,a ∈ domM f and dima(domM f) = dima(ClM (S)). In fact, we even havedima(ClM (S) \ domM f) < dima(ClM (S)).

Proof. Since f is continuous at a it is bounded in some neighbourhood ofa, so we may assume without loss of generality that f is bounded on S.Consider, as above, Γ := ClM Γf and Γ0 := {x ∈ Γ : |π−1(π(x)) ∩ Γ| = 1}.Let B0 := π(Γ0) where π is the projection onto the �rst n-coordinates.Then Γ0 is the graph of a function, f with domain B0. By curve selection inM∗, if f is not continuous at some point b ∈ B0 there are de�nable curvesγ1, γ2 ⊆ B such that lim γi(t) = b but lim(f(γ1(t))) 6= lim(f(γ2(t))). Since fis bounded on S, it must also be that f is bounded on B0. Thus, the abovelimits exist. But then lim(f(γi(t))) ∈ ClM Γf , contradicting the assumtionthat b0 ∈ B0.

Since by continuity of f we must have a ∈ B0, and � as we proved in theprevious lemma � B0 has small co-dimension, the corollary is proved. �

4.2. dimension. In the above results the notion of dimension in the canon-ical o-minimal completion of a weakly o-minimal non-valuational structureplayed an important role. In fact, to de�nable sets in such structures threenatural notions of dimension can be associated:

De�nition 4.15. LetM be a weakly o-minimal non-valuational expansionof an ordered group, S ⊆Mn a de�nable set:

(1) The topological dimension of S, dimτ (S), is the largest r ≤ n suchthat there exists a projection π : Mn → M r such that π(S) hasnon-empty interior.


(2) The combinatorial dimension of S, dimcl(S), is the dcl-dimension ofS.

(3) The o-minimal dimension of S, dimo(S), is dim(ClM (S)).

We will need the following easy observation:

Lemma 4.16. LetM be a weakly o-minimal structure, S ⊆Mn a de�nableopen set. If S′ ⊆ S is a de�nable dense subset of S then dimτ (S) = dimτ (S′).

Proof. If dimτ (S) = k then there is a projection π : S → Mk such thatπ(S) has non-empty interior. It follows that π(S′) is dense in π(S), and bydecomposing π(S) into cells, we are reduced to the case that S′ is an opencell. Now if U ⊆ Mn and f : U → M is a de�nable continuous functionthen, since the underlying order is dense, the graph of f is nowhere densein Mn+1. So partition S′ into strong cells. If no cell is open then eachcell is, with respect to some choice of projection onto Mn−1, the graph of ade�nable function, whence nowhere dense in Mn. Since the �nite union ofnowhere dense sets is nowhere dense, this leads to a contradiction. �

As could be expected, these three notions of dimension are, essentially,the same:

Lemma 4.17. Let M be a weakly o-minimal non-valuational expansion ofan ordered group, S ⊆ Mn a de�nable set. Then dimo(S) = dimτ (S) ≥dimcl(S), and ifM is ω1-saturated, then equality holds throughout.

Proof. First, we prove the equality of the o-minimal and topological dimen-sions. If S ⊆ Mn is a de�nable open set then there exists a de�nable M -closed box B ⊆ S. Since M is dense in M we know that B := ClM (B) ⊆ClM (S) and B is a box in Mn. If follows that n = dim(M) ≤ dim(ClM (S)) ≤n, so equality holds. If follows that if S contains an open set then dimτ (S) =dimo(S) = n. Now, if S ⊆ Mn and dimτ (S) = r let π : Mn → M r witnessthis. Then r = dimτ (S) = dimτ (π(S)) = dimo(π(S)) ≤ dimo(S). So itremains to prove the other inequality. So assume that dimo(S) = r′. Sincein the o-minimal setting the o-minimal dimension is equal to the topologicaldimension there is some projection π : Mn → M r′ such that π(ClM (S)) con-

tains an open box B. Denote S := ClM (S) and let B := B∩M r′ . Let b ∈ Bbe any point and c ∈ π−1(b). Then, as S = ClM (S) for any M -de�nable boxD 3 c there exists c′ ∈ S ∩ D, and choosing D small enough we concludethat π(c′) ∈ B. Since B and b were arbitrary, it follows that π(S) ∩ B isdense in B and therefore in B. By the previous lemma this implies that π(S)contains an open set.

It remains to show that dimτ (S) ≥ dimcl(S) and that equality holds ifM is saturated enough. Since for any de�nable sets S1, S2 we have thatClM(S1 ∪ S2) = ClM(S1) ∪ClM(S2) it follows form what we proved, andfrom the corresponding equality in the o-minimal setting that

(∗) dimτ (S1 ∪ S2) = max{dimτ (S1), dimτ (S2)}.


Thus, it will su�ce to prove the claim under the assumption that S is acell. In that case, it follows immediately from the de�nition that dimτ (S) ≥dimcl(S).

From the de�nition we observe that in order to prove the remaining partof the claim it su�ces to assume that S is open. In that case, however, theresult follows from a standard compactness argument. �

The above results allow us to reduce the proofs of properties of the dimen-sion in weakly o-minimal non-valuational expansions of an ordered group tothe analogous properties in the o-minimal setting. We give two basic exam-ples illustrating this:

Corollary 4.18. LetM be a weakly o-minimal non-valuational expansion ofan ordered group. Then the dimension function is preserved under de�nablebijections. I.e., if S ⊆ Mn is de�nable and f : S → Mm is a de�nablebijection, then dim(S) = dim(f(S)).

Proof. Partition S into strong cells such that f |S is continuous. Restrictingto a strong cell of maximal dimension, it thus su�ces to prove the resultunder the assumption that f is continuous. By further restricting S we mayassume that f |S is strongly continuous (either use Lemma 4.11 or just thede�nition of strong cell decomposition with respect to the graph of f). LetB be M∗-de�nable such that f extends continuously to B. Without lossof generality, we may assume that S ⊆ B ⊆ ClM (S), so that B is dense inClM (S) and dim(B) = dim(ClM (S)). Then

dim(B) = dim(ClM(S)) = dim(ClM(S) ∩B) = dim(f(B))

where the last equality is true because o-minimal dimension is preserved un-der bijections. Now f(B) ⊇ f(S), and f(B) isM∗-de�nable, so dim(f(B)) ≥dim(ClM (f(S)). The other inequality is obtained by replacing f with f−1,and the result now follows. �

We now turn to the addition formula:

Lemma 4.19. Let M be a weakly o-minimal non-valuational expansion ofan ordered group. Let S ⊆Mn×Mm be de�nable, π the standard projectionon Mm. Assume that dim(π−1(a) ∩ S) = k for all a ∈ T := π(S). Thendim(S) = k + dim(T ).

Proof. Let S be as in the statement of the lemma, S0 := ClM (S) and T0 itsprojection onto Mm. Then dim(S0) = dim(S) and as

ClM (T ) = ClM (π(S)) ⊇ π(ClM (S)) = T0 ⊇ Talso dim(T0) = dim(T ). Note also that for any a ∈ T we have thatClM (S(a)) ⊆ S0(a), where S(a) := {x ∈ Mn : (x, a) ∈ S}. Since dimensionis de�nable in the o-minimal setting, T1 := {a ∈ T0 : dim(S0(a)) ≥ k} isM∗-de�nable, and contains T . Thus dim(T1) = dim(T0) = dim(T ).

By the addition formula in the o-minimal setting dim(S0) ≥ k+ dim(T1),and by the previous equality dim(S) ≥ k+dim(T ). This �nishes the proof in


case k = n. So we now assume that k < n. It follows that for all a ∈ T thereis some projection σa : Mn → Mk such that σ(S(a)) contains an open set.Since there are only �nitely many projections there must be, by (∗) above,a projection σ : Mn → Mk such that dim(T1) = dim(T ), where T1 := {a ∈T : dim(σ(S(a)) = k}. Restricting ourselves to π−1(T1)∩S, we may assumethat T1 = T . For simplicity, assume that σ is the projection onto the �rst kcoordinates. It follows that for all a ∈ T , and b := (b1, . . . , bn) ∈ S(a) thereis a de�nable function fa(x, a) : Mk → Mn such that fa(b1, . . . , bk, a) =(bk+1, . . . , bn). By compactness (alternatively, by strong cell decomposition),and reducing S further, we may assume that the function fa(x, y) does notdepend on a.

Consider the set S1 ⊆Mk ×Mm de�ned by

S1 := {(σ(b), a) : b ∈ S(a), a ∈ T}.

By assumption dim(S1(a)) = k for all a ∈ T . So by what we have alreadyshown we know that dim(S1) = dim(T ) + k. But S is in de�nable bijectionwith S1 given by F (b, a) := (b, f(b, a), a), so by the previous corollary theresult now follows. �

Lemma 4.20. Let M be a weakly o-minimal non-valuational expansion ofa group, S ⊆Mn a de�nable set. Then dim(ClM(S) \ S) < dimS.

Proof. Let S := ClM (S). By Proposition 2.3(1) of [13] there is a decom-position of S into basic cells (cf. for the de�nition). Let P be such a de-composition of S. For each cell C ∈ P let ∂C := ClM (C) \ C. Denote∂S :=

⋃C∈P

∂C. Note that ∂S contains the boundary of S, but probably con-

tains more points. Then dimM∗(∂S) < dimM∗ S = dimM S. It will su�ceto show that ClM(S)\S ⊆ ∂S. By the discussion preceding [13, Proposition2.3] (and, essentially, from the de�nition) C∩Mn is either empty or a strongcell. So P induces a decomposition PM of ClM(S) into re�ned strong cells.Now, if p ∈ ClM(S)\S then there is some C ∈ P such that p ∈ ClM(C∩M)and p /∈ C ∩M . Thus, p ∈ ∂M(C ∩M). Clearly, p /∈ C, and p ∈ ClM (C).So p ∈ ∂C. �

Remark 4.21. Since we have shown that dimension inM is the geometricdimension, and since M is weakly o-minimal, the proof of the analogousclaim for o-minimal theories goes through essentially unaltered to the presentcontext. However, the present proof is shorter.

Remark 4.22. (1) The results of the present sub-section do not dependon the structure being an expansion of an ordered group, but onlyon the strong cell decomposition property.

(2) The results in the present sub-section are most probably all known.We include them for ease of reference and to demonstrate the useful-ness of the canonical o-minimal completion as a tool in the study ofweakly o-minimal non-valuational expansions of ordered groups.


5. O-minimal traces

Using the canonical o-minimal completion, we see that any weakly o-minimal expansion of an ordered group is obtained as the trace of an o-minimal structure on a dense subset. Of course, given an o-minimal structureN and a dense subset M , the structure induced on M can be wild (takeQ ⊆ R, the latter equipped with the full �eld structure).

As already mentioned, one case where this construction does give a wellbehaved structure is in the case where M is the universe of an elementarysub-model ofN . One is then naturally led to ask whether this is the only wayweakly o-minimal non-valuational structures are obtained. More precisely:

De�nition 5.1. A weakly o-minimal expansion of an ordered group,M isan o-minimal trace if there is exists a dense pair of o-minimal structures(N ,M′) such that M′ has universe M and M is the structure induced onM from (N ,M′).

In the present section we provide an example showing that ordered reductsof o-minimal traces need not themselves be o-minimal traces. We then pro-vide an example of a weakly o-minimal non-valuational expansion of an or-dered group that is not a reduct of an o-minimal trace.

Before we proceed to the example itself, one word concerning De�nition5.1 is in place. The theory of dense pairs, as developed in [3] requires thatthe o-minimal structure expands a group. Thus it seems natural to requirein De�nition 5.1 that the group structure on M be part of the languageof N . We point out that this assumption is harmless. As we have seen inCorollary 4.7 ifM is a trace, then � up to a change of signature � M is theonly possible candidate for N . Moreover, by Wencel's work, if M expandsa group (�eld) then so does M. Of course, a priori, the group structureneed not be part of the signature of N . But there is a ∅-de�nable formulain ϕ(x, y, z) and a parameter a ∈ N such that ϕ(x, y, a) makes N into anordered abelian group. Since M ≺ N one can also �nd such a parametera′ ∈ M in M , and there is no harm introducing ϕ(x, y, a′) as an atomicfunction in the signature of N .

In view of the above, we will henceforth assume that ifM is an o-minimaltrace then the group operation is part of the signature of the o-minimalstructure witnessing it. Now we proceed to the construction of the example.Let R be the �eld of real numbers, Ralg the real closure of Q. Then (R,Ralg)

is a dense pair. Let Ralg be the structure on Ralg induced from R, and Rπalg

the reduct (Ralg,≤,+, π·), i.e., the additive group of the �eld of real algebraicnumbers equipped with an additional binary relation y > πx (which, forconvenience we denote as the unary function x 7→ πx).

The structure Rπalg is a weakly o-minimal non-valuational expansion of an

ordered group with no external limits, being the reduct of Ralg, which hasthese properties by virtue of being an o-minimal trace. First we prove:


Lemma 5.2. For all α ∈ Q(π) the relation αx < y is ∅-de�nable in Rπalg.

Moreover, for α1, . . . , αn ∈ Q(π) the relation∑n

i=1 αixi < 0 is ∅-de�nable.

Proof. We will show that if α, β ∈ Q(π) are de�nable in Rπalg then so isα+β and that πn is de�nable for all n ∈ Z. Since the domain of all of thesefunctions is Ralg and all these functions are strongly continuous, it will followimmediately from the de�nition of the canonical o-minimal completion thatthese functions are de�nable in the completion, which will �nish the proof.

Indeed, if α, β are de�nable by Sα, Sβ then α+ β is de�ned by

(∃z1, z2)(Sα(x, z1) ∧ Sβ(x, z2) ∧ y > z1 + z2).

If x 7→ πnx is de�nable by Pn(x, y) then x 7→ πn+1x is de�ned by

(∃z)(Pn(x, z) ∧ P1(z, y).

So to conclude the �rst part of the lemma it remains only to note thatP−1(x, y) is given by P1(y, x).

For the second part of the lemma we will show that if n > 1 and α1, . . . , αnare linearly independent over Q then:

n∑i=1

αixi < 0 ⇐⇒ (∀x′1, . . . x′n)

(n∧i=1

x′i < αix→∑i=1n

x′i < 0

)unless xi = 0 for all i. The lef-to-right direction is clear, so we have to showthe other implication. The assumption implies

∑ni=1 αixi ≤ 0, so we only

have to check that equality cannot hold. First, observe that since the xiare not all 0, we may assume that the xi are linearly independent over Q.Indeed, if x1 =

∑ni=2 qixi for qi ∈ Q we get

n∑i=1

αixi = α1

n∑i=2

qixi +n∑i=2

αixi =n∑n=2

(qiα1 + αi)xi

and {(qiα1 + αi)}ni=2 are still independent over Q. Now αi are polynomialsin π with rational co-e�cients and xi are real algebraic numbers, so we

can write∑n

i=1 αixi =∑k

i=0 βiπi = 0 where βi are Q-linear combinations

of x1, . . . , xn. So βi = 0 for all i if and only if the αi are all 0, which isimpossible, since they are linearly independent. �

Let Rπalg be the set of de�nable cuts in Rπalg. By [1] those are precisely

the cuts de�ned by∑n

i=1 αixi for α1, . . . , αn ∈ Q(π) and x1, . . . , xn ∈ Ralg

(we do not claim, of course, that any two such cuts are distinct). Theabove lemma implies that the mapping αx 7→ x ⊗ α extends to a bijectionRπalg → Ralg ⊗ Q(π) and this bijection is an isomorphism of ordered Q-

vector spaces (in fact, it is automatically an isomorphism of Q(π)-vectorspaces). Let T0 be the theory asserting that for an ordered Q-vector spaceR the mapping de�ned above (from R onto R⊗Q(π)) is an isomorphism ofordered Q(π)-vector spaces.


Proposition 5.3. The theory T0 |= Th(Rπalg) has quanti�er elimination in

the language of ordered Q-vector spaces expanded by predicates∑n

i=1 αixi < 0for all α1, . . . , αn linearly independent over Q.

Proof. Let Q1,Q2 be two saturated models of T0 of the same cardinality. Wewill show that if Ai ⊆ Qi are small divisible subgroups, and f : A1 → A2

is a partial isomorphism (in the expanded langauge), then for every a ∈ Q1

the isomorphim f can be extended to a.Since f is an isomorphism in the expanded language and Qi |= T0 we

can extend f to an isomorphism of A1 ⊗ Q(π) with A2 ⊗ Q(π). Identify awith a⊗1. By quanti�er elimination in the theory of ordered divisible abeiangroups and since A1⊗Q(π) is a divisible abelian sub-group, tp(a/A1⊗Q(π))is determined by the cut it realises. Thus, it will su�ce to show that thesame cut over A2⊗Q(π) is realised in Q2⊗Q(π). But since Q2 is a saturatedmodel of T0 we automatically get that Q2 ⊗ Q(π) is saturated (for ≤), asrequired. �

As a corollary we get:

Corollary 5.4. The canonical o-minimal completion of Rπalg is Ralg ⊗Q(π)

as an ordered Q(π)-vector space.

Proof. We identify Ralg with {r⊗1}r∈Ralg. Since this mapping extends to an

isomorphism of Q-vector spaces (on Rπalg) it follows that Ralg⊗Q(π) induceson the image of Ralg a structure of an ordered Q-vector space. Moreover,for all α1, . . . , αi ∈ Q(π) and r1. . . . rn ∈ Ralg the cut

∑ni=1 αiri is externally

de�nable in the image of Ralg by the set x <∑n

i=1 ri ⊗ αi. More generally,the relation

∑ni=1 αixi < 0 is de�nable in the Q(π)-vector space structure on

Ralg ⊗Q(π). Thus every atomically de�nable set of Rπalg is de�nable in the

structure induced from Ralg⊗Q(π). Since, by the previous proposition, Rπalg

has quanti�er elimination, this implies that the structure induced on Ralg

from Ralg⊗Q(π) expands Rπalg. Since the theory of ordered Q-vector spaceshas quanti�er elimination, we get that � in fact � the induced structure onRalg is precisely Rπalg.

On the other hand, the completion of Rπalg is, as a set, Ralg⊗Q(π) and as a

structure it is an expansion of theQ(π)-vector space structure onRalg⊗Q(π),since the function x 7→ πx is a strongly continuous function on Ralg (intoRalg ⊗ Q(π), and therefore its unique extension to Ralg ⊗ Q(π) is part ofthe completion. By Corollary 4.7 and the previous pragraph, the conclusionfollows. �

We will then show:

Lemma 5.5. Let S be any set de�nable in Rπalg, but not in (Ralg,≤,+).

Then the reduct of Rπalg given by (Ralg,≤,+, S) is not o-minimal.

Proof. Any reduct (Ralg,≤,+, S) is a weakly o-minimal non-valuational ex-pansion of an ordered group (as these properties are preserved under taking


ordered reducts keeping the group structure). Therefore (Ralg,≤,+, S) hasthe strong cell decomposition. Thus, it will su�ce to prove the lemma underthe assumption that S is a re�ned strong cell.

The lemma is obvious in the case of 1-cells (and is vacuously true for 0-cells). Assume the lemma is true for strong cells in (Rπalg)n, and we prove

it for cells in (Rπalg)n+1. So assume that S is a strong (n + 1)-cell. Theprojection Sn on the �rst n-coordinates is a strong n-cell, and de�nable in(Ralg,≤,+, S). Thus, if Sn is not de�nable in (Ralg,≤,+), by induction(Ralg,≤,+, S) is not o-minimal. So we are reduced to the case where Sn isde�nable in (Ralg,≤,+). It will su�ce to check the case where S = Γf wheref : Sn → R is continuous and de�nable. If f(Sn) ⊆ R \Ralg the structure isnot o-minimal, so we may assume that f(Sn) ⊆ Ralg, and we have to showthat in that case f is de�nable in (Ralg,≤,+).

Since f is strongly continuous f , the iterative convex hull of f is continuousand de�nable in the canonical o-minimal completion, which by the previouslemma is Rπvs, implying that f is piecewise-linear. Let us assume, �rst, thatf is linear. Then f(v1, . . . , vn) =

∑aivi where ai ∈ Q(π). We will show that

the assumption that f(Sn) ⊆ Ralg implies that ai ∈ Q for all i. Assume not,then for a �xed v = (v1, . . . , vn) ∈ Sn we have that f(v) is a polynomial inπ with coe�cients in Ralg. Since π is transcendental, the assumption thatf(v) ∈ Ralg implies that this is the zero polynomial. The coe�cients of thepolynomial f(v) (as a polynomial in π) are linear combinations of v1, . . . , vnoverQ. Since Sn is open, we can choose (v1, . . . , vn) linearly independent overQ assuring that f(v) is a non-zero polynomial in π, leading to a contradiction.Thus f is de�nable in (Ralg,≤,+). Now let f1, . . . , fk be the linear pieces off . The sets Sn,i := {v ∈ Sn : f(v) = fi(v)} are thus de�nable in (Ralg,≤,+)for all 1 ≤ i ≤ k. By the inductive assumption Sn,i is de�nable in (Ralg,≤,+)for all i, implying that S is de�nable in (Ralg,≤,+) as claimed. �

In view of the discussion following De�nition 5.1 these two lemmas com-bine to prove:

Theorem 5.6. Rπalg is a weakly o-minimal non-valuational expansion of anordered group which is not an o-minimal trace.

Proof. By Corollary 4.7 and Lemma 5.4 if Rπalg is an o-minimal trace, it must

be the trace of Ralg ⊗Q(π) as an ordered Q(π)-vector space � possibly witha di�erent signature. Thus, we are looking for an o-minimal reduct R ofRπalg with the properties that

• R expands (Ralg,≤,+),• The extension of R to R has the same de�nable sets as Ralg ⊗Q(π)(as a Q(π)-vector space).

Since the extension of (Ralg,≤,+) to Ralg ⊗ Q(π) is (Ralg ⊗ Q(π),≤,+),a proper reduct of the Q(π)-vecto space structure, it must be that R is aproper expansion of (Ralg,≤,+). But by Lemma 5.5 no such expansion iso-minimal. �


Thus we obtain:

Corollary 5.7. The class of o-minimal traces is not closed under takingordered reducts.

It is now natural to ask whether any weakly o-minimal non-valuational ex-pansion of an ordered group is a reduct of an o-minimal trace. The followingexample shows that this, too, is not the case.

Let Qπvs := (Q,≤,+, α·)α∈Q(π) where α· is the binary relation y > αx.Clearly, Qπvs |= T0, so by quanti�er elimination Qπvs ≡ Rπalg. In particular,since Rπalg is weakly o-minimal and non-valuational, so is Qπvs.

Theorem 5.8. Qπvs is not a reduct of an o-minimal trace.

Proof. Assume towards a contradiction that there exists an o-minimal struc-ture Q with universe Q, and R � Q such that (R,Q) is a dense pair, andthe structure induced on Q from R expands Qπvs.

The desired conclusion now follows from [8] as follows. First, by Theo-rem A thereof, either R is linearly bounded or there is a de�nable binaryoperation · such that (R,≤,+, ·) is a real closed �eld. Since Q ≺ R, inthe latter case we would have a binary operation ·Q de�nable in Q making(Q,≤,+, ·) into a real closed �eld, which is impossible, because (Q,+) is thestandard addition on Q, and therefore there exists at most one �eld structure(de�nable or not) expanding it, but (Q,+, ·) is not real closed.

So we are reduced to the linearly bounded case. By Theorem B of [8]every de�nable endomorphism of (R,+) is ∅-de�nable. Thus, it will su�ceto show that x 7→ πx is a de�nable endomorphism of (R,+), since then itis ∅-de�nable, contradicting the assumption that Q ≺ R. But this shouldnow be obvious, since x 7→ πx is de�nable as a function from Q to Q andstrongly continuous, so by Corollary 4.7 x 7→ πx is a de�nable continuousfunction in R, which is clearly an endomorphism of (R,+). �

The above proof actually shows more:

Corollary 5.9. If Q ≡ Qπvs then Q is not an o-minimal trace.

However, we do have:

Example 5.10. Qπvs ≡ Rπalg. Thus, there exists a weakly o-minimal non-valuational expansion of an ordered group which is not a reduct of an o-minimal trace, whereas some elementarily equivalent structure is a reduct ofan o-minimal trace.

Question 5.11. Is every weakly o-minimal non-valuational structure ele-mentarily equivalent to a reduct of an o-minimal trace?

The above examples show that weakly o-minimal non-valuational expan-sion of ordered groups need not be o-minimal traces, or even reducts ofo-minimal traces. It is natural, therefore, to ask, given such a structure,M,whether it is necessarily the case that any N ≡M is not an o-minimal trace.We have partial answers to the above question. First we prove:


Lemma 5.12. Let M be a weakly o-minimal non-valuational expansion ofan ordered group. Assume that M is an o-minimal trace. Then so is anyN �M.

Proof. Let M be an o-minimal structure on M witnessing that M is ano-minimal trace. By de�nition this means that M is the universe of an

elementary sub-structure M0 ≺ M such that the structure induced on M

from M isM. It follows thatM has the same de�nable sets asM0 expandedby predicates for allM-de�nable cuts.

Let L0 be the signature ofM0, and let N �M. Then the reduct N0 :=N|L0 is o-minimal. Expand N by predicates for all N -de�nable cuts. ByCorollary 2.3 there exists an o-minimal N � N0 such that N is dense in Nand the structure induced on N0 in the dense pair (N ,N0) is precisely theexpansion of N0 by all cuts realized in N , i.e., by all N -de�nable cuts. Butsince the theory of dense pairs is complete, and N0 ≡ M0, it follows that

(N ,N0) ≡ (M,M0). In particular, M0 with the induced structure from

M is precisely M, and is elementarily equivalent to N0 with the inducedstructure from N . SinceM is also elementarily equivalent to N , it followsthat N0 expanded by all N -de�nable cuts has the same de�nable sets as N ,so N is an o-minimal trace. �

The last lemma implies:

Corollary 5.13. Let M be a weakly o-minimal non-valuational expansionof an ordered group. Assume thatM is an o-minimal trace. Then:

• Any |M |+-saturated N ≡M is an o-minimal trace.• IfM is a prime model then any N ≡M is a o-minimal trace.

Question 5.14. Let M be a weakly o-minimal non-valuational expansionof an ordered group. Assume thatM is a reduct of an o-minimal trace. Isany N �M a reduct of an o-minimal trace?

References

[1] Yerzhan Baisalov and Bruno Poizat. Paires de structures o-minimales. J. Symbolic

Logic, 63(2):570�578, 1998.[2] G. Cherlin and M. A. Dickmann. Real closed rings II. model theory, Annals of Pure

and Applied Logic 25 (3), 213�231, 1983.[3] Lou van den Dries. Dense pairs of o-minimal structures. Fund. Math., 157(1):61�78,

1998.[4] E. Hrushovski and F. Loeser. Non-archimedean tame topology and stably dominated

types. Available at http://de.arxiv.org/pdf/1009.0252.pdf, 2012.[5] Dugald Macpherson, David Marker, and Charles Steinhorn. Weakly o-minimal struc-

tures and real closed �elds. Trans. Amer. Math. Soc., 352(12):5435�5483 (electronic),2000.

[6] David Marker. Omitting types in o-minimal theories. J. Symbolic Logic, 51(1):63�74,1986.

[7] David Marker and Charles I. Steinhorn. De�nable types in o-minimal theories. J.Symbolic Logic, 59(1):185�198, 1994.


[8] Chris Miller and Sergei Starchenko. A growth dichotomy for o-minimal expansions ofordered groups. Trans. Amer. Math. Soc., 350(9): 3505�3521 (1998).

[9] Christopher Shaw. Weakly o-minimal structures and Skolem functions. Ph.D. Thesis,University of Maryland (2008).

[10] M. C. Laskowski and C. Shaw. De�nable choice for a class of weakly o-minimaltheories. Available on http://de.arxiv.org/pdf/1505.02147.pdf, 2015.

[11] Roman Wencel. Weakly o-minimal nonvaluational structures. Ann. Pure Appl. Logic,154(3):139�162, 2008.

[12] Roman Wencel. On expansions of weakly o-minimal non-valuational structures byconvex predicates. Fund. Math. 202 (2):147�159, 2009.

[13] Roman Wencel. On the strong cell decomposition property for weakly o-minimalstructures. MLQ Math. Log. Q., 59(6):452�470, 2013.

Department of Mathematics and Statistics, University of Konstanz, 78457

Konstanz, Germany

E-mail address: [email protected]

Department of mathematics, Ben Gurion University of the Negev, Be'er

Sheva, Israel


Department of mathematics, Ben Gurion University of the Negev, Be'er

Sheva, Israel


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