On expansions of weakly o-minimal non-valuational structures by convex predicates

Roman Wencel^{1}

Mathematical Institute, University of Wrocław

ABSTRACT

We prove that if M = (M, ≤, +, …) is a weakly o-minimal non-valuational structure expanding an ordered group (M, ≤, +), then its expansion by a family of ‘non-valuational’ unary predicates remains non-valuational. The paper is based on the author’s earlier work on strong cell decomposition for weakly o-minimal non-valuational expansions of ordered groups.

0 Introduction

Several examples of weakly o-minimal structures are obtained as expansions of o-minimal structures by predicates interpreted as certain convex sets [MMS]. Among these we have an expansion of a real closed field by a valuation ring and an expansion of the ordered field of real algebraic numbers by a predicate interpreted as (−α, α), where α is a transcendental number. Structures of the first sort were investigated by L. van den Dries and A. H. Lewenberg (see [DL] and [D2]) who showed for instance that if R is an o-minimal expansion of a real closed field and V is a proper non-empty convex subring closed under 0-definable continuous functions, then the expansion (R, V) is weakly o-minimal. The phenomenon occuring in all the mentioned examples was addressed in general by Y. Baisalov and B. Poizat (see [BP]) who proved that an expansion of any o-minimal structure by a family of convex predicates has weakly o-minimal theory (so in particular it is a weakly o-minimal structure). The result of [BP] was generalized by B. Baizhanov (see [Bz]) who proved that an expansion of a model of a weakly o-minimal theory by a family of convex predicates has weakly o-minimal theory. It is worth mentioning that Baizhanov’s theorem has also an easy proof when one uses the fact that weakly o-minimal theories do not have the independence property and the result of S. Shelah (see [Sh783]) concerning quantifier elimination for the theory of an expansion of a sufficiently saturated model of a theory without the independence property by all externally definable sets. The question of G. Cherlin whether an expansion of a weakly o-minimal structure (not necessarily with weakly o-minimal theory) by a family of convex predicates is also weakly o-minimal still remains an open problem.

This paper is a sequel to the study of expansions of weakly o-minimal structures by convex predicates. In [We07] I introduced weakly o-minimal non-valuational expansions of ordered groups as a natural generalization of weakly o-minimal non-valuational expansions of real closed fields considered in [MMS]. A weakly o-minimal expansion of a real closed field is said to be nonvaluational iff it does not define a non-trivial valuation. Similarly, a weakly o-minimal expansion M of an ordered group (M, ≤, +) is called non-valuational (or of non-valuational type) iff there is no proper and non-trivial subgroup of (M, +) definable in M. Being of non-valuational type is equivalent to several conditions which are discussed in [We07]. One of them says that the distance between the two parts of a definable cut is zero (the precise definition appears in §1).

1This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme (contract number: MEIF-CT-2003-501326) and by a Marie Curie European Reintegration Grant (contract number MERG-CT-2005-031095)

Assume that M = (M, ≤, +, …) is a weakly o-minimal non-valuational expansion of an ordered group. If P ⊆ M is a finite union of convex sets, then P in a natural way determines a finite family of cuts in (M, ≤). If the parts of all these cuts are ‘close to’ each other, then P is said to be non-valuational. Moreover, the expansion (M, P) is interdefinable with some expansion of M by a family of convex predicates. By [We07], T h(M) is weakly o-minimal. Therefore by [Bz] also any expansion of M by a family of non-valuational predicates has weakly o-minimal theory. The main result of this paper is Theorem 2.11, which says that every expansion of M by a family of non-valuational predicates is of non-valuational type. We also show that the theory of such an expansion is weakly o-minimal without using Baizhanov’s theorem (cf. Corollary 2.10).

The paper is organized as follows. In §1 we fix our notation and terminology, and recall some particularly useful results, mainly from [MMS] and [We07]. In §2 we outline the proofs of results mentioned above.

Last but not least, I would like to thank the referee whose comments allowed me to improve the quality of the paper.

1 Notation and preliminaries

Let (M, ≤) be a dense linear ordering without endpoints. A set I ⊆ M is called convex in (M, ≤) iff for any a, b ∈ I and c ∈ M with a ≤ c ≤ b we have that c ∈ I. If additionally I ≠ ∅ and inf I, sup I ∈ M ∪ {−∞, +∞}, then I is called an interval in (M, ≤). A maximal convex subset of a non-empty subset of M is called a convex component of it. A pair ⟨C, D⟩ of non-empty subsets of M is called a cut in (M, ≤) iff C < D, C ∪ D = M and D has no lowest element. A first order structure M = (M, ≤, …) expanding (M, ≤) is said to be weakly o-minimal iff every subset of M, definable in M, is a finite union of convex sets. A complete first order theory is called weakly o-minimal iff all its models are weakly o-minimal. The following remark characterizes weakly o-minimal structures in terms of sets definable in them.

Remark 1.1 Assume that (M, ≤) is a dense linear ordering without endpoints and for m ∈ ℕ_{+}, D_{m} is a family of subsets of M^{m} for which the following conditions are satisfied.

(a) If I ⊆ M is an interval, then I ∈ D_{1}.

(b) If X ∈ D_{1}, then X is a union of finitely many convex sets.

(c) {⟨x, y⟩ ∈ M^{2} : x < y} ∈ D_{2}.

(d) D_{m} with the usual set-theoretic operations is a Boolean algebra.

(e) If X ∈ D_{m}, then X × M, M × X ∈ D_{m+1}.

(f) If X ∈ D_{m+1} and π : M^{m+1} ⟶ M^{m} is the projection dropping the last coordinate, then π[X] ∈ D_{m}.

(g) If 1 ≤ i < j ≤ m, then {(x_{1}, …, x_{m}) ∈ M^{m} : x_{i} = x_{j}} ∈ D_{m}. Then there is a weakly o-minimal structure M expanding (M, ≤) such that for every X ⊆ M^{m}, X is definable in M iff X ∈ D_{m}.

Note that if one replaces ‘convex sets’ in (b) with ‘intervals’, then (a)-(g) in Remark 1.1 imply that there is an o-minimal expansion M of (M, ≤) such that for every X ⊆ M^{m}, X is definable in M iff X ∈ D_{m} (cf. [D1], Chapter 1).

Assume that M = (M, ≤, …) is a weakly o-minimal structure. A cut ⟨C, D⟩ in (M, ≤) is called definable in M iff the sets C, D are definable in M. The set of all such cuts will be denoted by \overline{M}^{M}. The set M can be regarded as a subset of \overline{M}^{M} by identifying an element a ∈ M with the cut ⟨(−∞, a], (a, +∞)⟩. After such an identification, \overline{M}^{M} is naturally equipped with a dense linear ordering without endpoints extending that of (M, ≤), and (M, ≤) is dense in (\overline{M}^{M}, ≤). For a definable set X ⊆ M^{m}, a function f : X ⟶ \overline{M}^{M} is said to be definable in M iff the set \{⟨\overline{x}, y⟩ ∈ X × M : f(\overline{x}) > y\} is definable in M.

Now assume that M = (M, ≤, + …) is a weakly o-minimal structure expanding an ordered group (M, ≤, +). A cut ⟨C, D⟩ is called non-valuational iff inf{y − x : x ∈ C, y ∈ D} = 0. We will say that M is non-valuational (or of non-valuational type) iff all cuts definable in M are non-valuational. If M is non-valuational, then \overline{M}^{M} can be naturally equipped with an ordered group structure extending that of (M, ≤, +). The ordered groups (M, ≤, +) and (\overline{M}^{M}, ≤, +) are divisible, abelian and torsion free. For details we refer the reader to §1 of [We07].

Assume that M = (M, ≤, …) is a weakly o-minimal structure. Below, for every m ∈ ℕ_{+} we inductively define strong cells in M^{m} and their completions in (\overline{M}^{M})^{m}. The completion in (\overline{M}^{M})^{m} of a strong cell C ⊆ M^{m} will be denoted by \overline{C}.

(1) A one-element subset of M is a strong ⟨0⟩-cell in M and is equal to its completion.

(2) A non-empty convex open definable subset of M is a strong ⟨1⟩-cell in M. If C ⊆ M is a strong ⟨1⟩-cell in M, then \overline{C} := \{x ∈ \overline{M}^{M} : (∃a, b ∈ C)(a < x < b)\}.

Assume that m ∈ ℕ_{+}, i_{1}, …, i_{m} ∈ {0, 1} and suppose that we have already defined strong ⟨i_{1}, …, i_{m}⟩-cells in M^{m} and their completions in (\overline{M}^{M})^{m}.

(3) If C ⊆ M^{m} is a strong ⟨i_{1}, …, i_{m}⟩-cell in M^{m} and f : C ⟶ M is a continuous definable function which has a continuous extension \overline{f} : \overline{C} ⟶ \overline{M}^{M}, then Γ(f), the graph of f, is a strong ⟨i_{1}, …, i_{m}, 0⟩-cell in M^{m+1}. The completion of Γ(f) in (\overline{M}^{M})^{m+1} is defined as Γ(\overline{f}).

(4) If C ⊆ M^{m} is a strong ⟨i_{1}, …, i_{m}⟩-cell in M^{m} and f, g : C ⟶ \overline{M}^{M} ∪ \{−∞, +∞\} are continuous definable functions which have continuous extensions \overline{f}, \overline{g} : \overline{C} ⟶ \overline{M}^{M} such that \overline{f}(\overline{x}) < \overline{g}(\overline{x}) for \overline{x} ∈ \overline{C}, then the set

(f, g)_{C} := \{⟨\overline{a}, b⟩ ∈ C × M : f(\overline{a}) < b < g(\overline{a})\}

is called a strong ⟨i_{1}, …, i_{m}, 1⟩-cell in M^{m}. The completion of (f, g)_{C} in (\overline{M}^{M})^{m+1} is defined as

\overline{(f, g)_{C}} := (\overline{f}, \overline{g})\overline{C} := \{⟨\overline{a}, b⟩ ∈ \overline{C} × \overline{M}^{M} : \overline{f}(\overline{a}) < b < \overline{g}(\overline{a})\}.

(5) We say that C ⊆ M^{m} is a strong cell in M^{m} iff there are i_{1}, …, i_{m} ∈ {0, 1} such that C is a strong ⟨i_{1}, …, i_{m}⟩-cell in M^{m}.

If C ⊆ M^{m} is a strong cell, then a definable function f : C ⟶ \overline{M}^{M} is called strongly continuous iff f has a continuous extension \overline{f} : \overline{C} ⟶ \overline{M}^{M}. In a standard way we define the notion of decomposition of a definable set into strong cells partitioning a given definable set (for details we refer the reader to §2 of [We07]). We will say that M has the strong cell decomposition property iff for any m, k ∈ ℕ_{+} and any definable sets X_{1}, …, X_{k} ⊆ M^{m}, there is a decomposition of M^{m} into strong cells partitioning each of the sets X_{1}, …, X_{k}.

Fact 1.2 [MMS, Ar] Let M = (M, ≤, …) be a weakly o-minimal structure and A ⊆ M. If U ⊆ M is an infinite A-definable set and f : U ⟶ \overline{M}^{M} is an A-definable function, then there is a partition of U into A-definable sets X, I_{0}, …, I_{m} such that X is finite, I_{0}, …, I_{m} are non-empty convex open sets, and for every i ≤ m, f ↾ I_{i} is locally constant or locally strictly increasing or locally strictly decreasing.

The following fact easily follows from the definition of strong cells.

Fact 1.3 Assume that M = (M, ≤, …) is a weakly o-minimal structure with the strong cell decomposition property and A ⊆ M. If U ⊆ M is a non-empty A-definable set and f : U ⟶ \overline{M}^{M} is an A-definable function, then there is a partition of U into A-definable sets X, I_{0}, …, I_{m} such that X is finite, I_{0}, …, I_{m} are non-empty convex open sets, and for every i ≤ m, f ↾ I_{i} is strongly continuous and strictly monotone or constant.

Proof. By assumption, there is a decomposition C of M^{2} into A-definable strong cells partitioning the set {⟨x, y⟩ ∈ U ×M : y < f(x)}. This yields a decomposition of U into finitely many A-definable convex open sets J_{0}, …, J_{k} and a finite set X such that

{J_{0}, …, J_{k}} ∪ {{a} : a ∈ X} = {π[C] : C ∈ C},

where π : M^{2} ⟶ M is the projection dropping the second coordinate. By our definition of strong cells, f ↾ J_{i} is strongly continuous for i ≤ m. In such a situation, weak o-minimality of M implies that each of the J_{i}’s could be decomposed into finitely many A-definable convex open sets and some finite set so that on each of the open sets, f is strictly monotone or constant. This finishes the proof.

Theorem 1.4 [We07] Let M = (M, ≤, +, …) be a weakly o-minimal structure expanding an ordered group (M, ≤, +). Then M is of non-valuational type iff M has the strong cell decomposition property.

2 The main result

Throughout this section we shall work in a weakly o-minimal structure M = (M, ≤, …), usually assuming that M has the strong cell decomposition property or is a non-valuational expansion of an ordered group. By C we will denote a non-empty, convex and non-definable (in M) proper subset of M such that inf C = −∞. Then ⟨C, M \ C⟩ is a cut in (M, ≤) (according to the terminology of [Bz] and [BVT] such cuts are called irrational). By Corollary 2.15 from [We07], we know that if M is a non-valuational expansion of an ordered group, then T h(M) is weakly o-minimal. Hence, by [Bz], also T h(M, C) is weakly o-minimal. In a series of lemmas below we will show that if M is a weakly o-minimal non-valuational expansion of an ordered group and the cut ⟨C, M \ C⟩ is non-valuational, then also the expansion (M, C) is of non-valuational type. Moreover, we will give a direct proof of the weak o-minimality of T h(M, C).

For m ∈ ℕ_{+} and i ∈ {1, …, m + 1}, by π_{i}^{m+1} we will denote the projection from M^{m+1} to M^{m} dropping the i-th coordinate and by D_{m}(M) the family of all subsets of M^{m} definable in M.

Lemma 2.1 Assume that M = (M, ≤, …) is a weakly o-minimal structure and I ⊆ M is a convex open definable set such that I ∩ C ≠ ∅ and I \ C ≠ ∅. If f : I ⟶ \overline{M}^{M} is a definable function such that (∀x ∈ I)(f(x) > c) [(∀x ∈ I)(f(x) < c)] for some c ∈ M, then there are an open interval J ⊆ I and α > c [α < c], α ∈ M, such that J ∩ C ≠ ∅, J \ C ≠ ∅ and (∀x ∈ J)(f(x) > α) [(∀x ∈ J)(f(x) < α)].

Proof. Let f : I ⟶ \overline{M}^{M} be a definable function such that (∀x ∈ I)(f(x) > c), where c ∈ M. By Fact 1.2 and the non-definability of C, there is an open interval I_{1} ⊆ I such that I_{1} ∩ C ≠ ∅, I_{1} \ C ≠ ∅ and f ↾ I_{1} is either strictly monotone or constant. Fix a ∈ I_{1} ∩ C, b ∈ I_{1} \ C and α ∈ M such that c < α < min{f(a), f(b)}. It is clear that f(x) > α whenever x ∈ (a, b). The other case is proved in a similar way.

Lemma 2.2 Let M = (M, ≤, …) be a weakly o-minimal non-valuational expansion of an ordered group. Assume that I ⊆ M is a non-empty convex open definable set such that I ∩ C ≠ ∅ and I \ C ≠ ∅, and f, g : I ⟶ \overline{M}^{M} are definable functions such that (∀x ∈ I)(f(x) < g(x)). There are an element a ∈ M and an open interval J ⊆ I such that J ∩ C ≠ ∅, J \ C ≠ ∅ and f(x) < a < g(x) for x ∈ J.

Proof. By Theorem 1.4 and Fact 1.3, without loss of generality we can assume that the functions f, g are strongly continuous. By Lemma 2.1, there are an open interval I_{1} ⊆ I and an element α ∈ M, α > 0, such that I_{1} ∩ C ≠ ∅, I_{1} \ C ≠ ∅ and (∀x ∈ I_{1})(g(x) − f(x) > α). Fix ε > 0, ε ∈ M, such that 2ε < α. For x_{0} ∈ I_{1} define

δ_{f}(x_{0}) = min(ε, sup{d ∈ M : d > 0, (∀x ∈ (x_{0} − d, x_{0} + d) ∩ I_{1})(|f(x) − f(x_{0})| < ε)})
δ_{g}(x_{0}) = min(ε, sup{d ∈ M : d > 0, (∀x ∈ (x_{0} − d, x_{0} + d) ∩ I_{1})(|g(x) − g(x_{0})| < ε)}).

Again, by Lemma 2.1, there are an open interval I_{2} ⊆ I_{1} and an element β ∈ M, β > 0, such that I_{2} ∩ C ≠ ∅, I_{2} \ C ≠ ∅ and min(δ_{f}(x_{0}), δ_{g}(x_{0})) > β for x_{0} ∈ I_{2}. Fix c_{1} ∈ I_{2} ∩ C and c_{2} ∈ I_{2} \ C such that c_{2} − c_{1} < β. For x_{1}, x_{2} ∈ (c_{1}, c_{2}) we have that

g(x_{2}) − f(x_{1}) = (g(x_{2}) − g(x_{1})) + (g(x_{1}) − f(x_{1})) > −ε + 2ε = ε.

Consequently,

inf{g(x) : x ∈ (c_{1}, c_{2})} − sup{f(x) : x ∈ (c_{1}, c_{2})} ≥ ε.

If a ∈ M is such that sup{f(x) : x ∈ (c_{1}, c_{2})} < a < inf{g(x) : x ∈ (c_{1}, c_{2})}, then for x ∈ (c_{1}, c_{2}) we have that f(x) < a < g(x).

Lemma 2.3 Let M be a weakly o-minimal non-valuational expansion of an ordered group. Assume that X ⊆ M^{2} is a set definable in M such that for any a, b ∈ M, if ⟨a, b⟩ ∈ X, then there are a_{1}, a_{2} ∈ M such that a_{1} < a < a_{2} and (a_{1}, a_{2}) × {b} ⊆ X. The following conditions are equivalent.

(a) There are a_{1} ∈ C, a_{2} ∈ M \ C and b ∈ M such that (a_{1}, a_{2}) × {b} ⊆ X.

(b) There are a_{1} ∈ C and a_{2} ∈ M \ C such that (a_{1}, a_{2}) ⊆ π_{2}^{2}[X].

Proof. The implication from (a) to (b) is obvious, so assume that (b) holds. Let C be a decomposition of M^{2} into strong cells in M^{2} partitioning the set X. There is a convex open definable set I ⊆ M such that I ∩ C ≠ ∅, I \ C ≠ ∅ and for every D ∈ C, we have that either I = π_{2}^{2}[D] or π_{2}^{2}[D] ∩ I = ∅. Below we consider two cases.

Case 1. The set (I ×M)∩X has empty interior. The following claim is a consequence of Lemma 2.1 from [We06] but for the sake of completeness we give a proof in our particular situation.

Claim. For every x ∈ I, the set {y ∈ M : ⟨x, y⟩ ∈ X} is finite.

Proof of the Claim. Suppose for a contradiction that for some a ∈ I, the set {y ∈ M : ⟨a, y⟩ ∈ X} is infinite, so it contains an open interval J. For b ∈ J define

f(b) = sup{c ∈ M : c > a and {a} × (b, c) ⊆ (I × M) ∩ X}.

The function f assumes values greater than a in \overline{M}^{M} ∪ \{+∞\}. By Fact 1.3, there is an open interval J′ ⊆ J such that f ↾ J′ is strongly continuous and strictly monotone or constant. It is clear that the set {⟨x, y⟩ : a < x < f(y), y ∈ J′} is contained in (I × M) ∩ X and contains an open box itself. This means that (I × M) ∩ X has non-empty interior, a contradiction.

Using the Claim, for x ∈ I, we can define f(x) = min{y ∈ M : ⟨x, y⟩ ∈ X}. Our assumptions guarantee that f is constant (say f(x) = b whenever x ∈ I), so for any a_{1} ∈ I ∩ C and a_{2} ∈ I \ C, we have that (a_{1}, a_{2}) × {b} ⊆ X.

Case 2. There are definable strongly continuous functions f : I ⟶ \overline{M}^{M} ∪ \{−∞\} and g : I ⟶ \overline{M}^{M} ∪ \{+∞\} such that \overline{f}(a) < \overline{g}(a) for a ∈ \overline{I} and D := (f, g)_{I} ⊆ X ∩ (I × M). By Lemma 2.2, there are a_{1} ∈ I ∩ C, a_{2} ∈ I \ C and b ∈ M such that (a_{1}, a_{2}) × {b} ⊆ D ⊆ X.

For a weakly o-minimal structure M and a set X ∈ D_{m+1}(M) let

I(X, C) = \{\overline{a} ∈ M^{m} : (∃b ∈ C)(∃c ∈ M \ C)(\{\overline{a}\} × (b, c) ⊆ X)\}.

Define also E_{m}(M, C) = {I(X, C) : X ∈ D_{m+1}(M)}. Below we will show that if M is a nonvaluational expansion of an ordered group, then E_{m}(M, C) is exactly the family of subsets of M^{m} definable in (M, C).

Lemma 2.4 Let M be a weakly o-minimal structure with the strong cell decomposition property. (a) If X ∈ D_{2}(M) is a strong cell, then I({⟨x, y⟩ ∈ M^{2} : ⟨y, x⟩ ∈ X}, C) is a convex set.

(b) If Y ∈ E_{1}(M, C), then Y is a finite union of convex sets.

(c) C ∈ E_{1}(M, C).

Proof. (a) is obvious from the definition of strong cells. (b) follows from (a) and the strong cell decomposition property of M. For the proof of (c), note that

C = I({⟨x, y⟩ ∈ M^{2} : y > x}, C).

Here the strong cell decomposition property in not needed.

Lemma 2.5 Assume that M is a weakly o-minimal structure and m ∈ ℕ_{+}.

(a) D_{m}(M) ⊆ E_{m}(M, C).

(b) E_{m}(M, C) is closed under Boolean operations.

(c) If X ∈ E_{m}(M, C), then X × M, M × X ∈ E_{m+1}(M, C).

Proof. (a) If m ∈ ℕ_{+} and X ∈ D_{m}(M), then X = I(X × M, C).

(b) Fix m ∈ ℕ_{+} and X, Y ∈ E_{m}(M, C). There are X_{1}, Y_{1} ∈ D_{m+1}(M) such that X = I(X_{1}, C) and Y = I(Y_{1}, C). Clearly, X ∪ Y = I(X_{1} ∪ Y_{1}, C), X ∩ Y = I(X_{1} ∩ Y_{1}, C) and M^{m} \ X = I(M^{m+1} \ X_{1}, C).

(c) Let m ∈ ℕ_{+} and X ∈ E_{m}(M, C). Then X = I(X_{1}, C) for some X_{1} ∈ D_{m+1}(M). Hence M × X = I(M × X_{1}, C) and X × M = I(\{⟨\overline{x}, y, z⟩ ∈ M^{m+1} : ⟨\overline{x}, z⟩ ∈ X\}, C).

Lemma 2.6 Let M be a weakly o-minimal expansion of an ordered group. If X ∈ E_{m+1}(M, C), then π_{m+1}^{m+1}[X] ∈ E_{m}(M, C).

Proof. Fix m ∈ ℕ_{+} and X ∈ E_{m+1}(M, C). There is X_{1} ∈ D_{m+2}(M) such that X = I(X_{1}, C). Let

X_{2} = ⋃\{\{\overline{a}\} × (b, c) : \overline{a} ∈ M^{m+1}, b ∈ C, c ∈ M \ C and \{\overline{a}\} × (b, c) ⊆ X_{1}\}.

Clearly, X = I(X_{2}, C) and X_{2} ∈ D_{m+2}(M). We claim that π_{m+1}^{m+1}[X] = I(π_{m+1}^{m+2}[X_{2}], C).

In order to prove that π_{m+1}^{m+1}[X] ⊆ I(π_{m+1}^{m+2}[X_{2}], C), fix \overline{a} ∈ π_{m+1}^{m+1}[X]. There is b ∈ M such that ⟨\overline{a}, b⟩ ∈ X = I(X_{2}, C). So there are c ∈ C and d ∈ M \ C such that \{⟨\overline{a}, b⟩\} × (c, d) ⊆ X_{2}. Hence \{\overline{a}\} × (c, d) ⊆ π_{m+1}^{m+2}[X_{2}] and \overline{a} ∈ I(π_{m+1}^{m+2}[X_{2}], C).

For the reverse inclusion, let \overline{a} ∈ I(π_{m+1}^{m+2}[X_{2}], C). There are c ∈ C and d ∈ M \ C such that \{\overline{a}\} × (c, d) ⊆ π_{m+1}^{m+2}[X_{2}]. Let Z = \{⟨x, y⟩ ∈ M^{2} : ⟨\overline{a}, y, x⟩ ∈ X_{2}\}. Clearly, (c, d) ⊆ π_{2}^{2}[Z]. By Lemma 2.3, there are c′ ∈ C, d′ ∈ M \ C and e ∈ M such that (c, d) × {e} ⊆ Z. Consequently, \{⟨\overline{a}, e⟩\} × (c, d) ⊆ X_{2}. The latter implies that ⟨\overline{a}, e⟩ ∈ X and \overline{a} ∈ π_{m+1}^{m+1}[X].

Lemmas 2.4, 2.5 and 2.6 imply that if M is a weakly o-minimal non-valuational expansion of an ordered group, then

(a) all intervals in (M, ≤) belong to E_{1}(M, C);

(b) every set belonging to E_{1}(M, C) is a union of finitely many convex sets;

(c) {⟨x, y⟩ ∈ M^{2} : x < y} ∈ E_{2}(M, C);

(d) for every m ∈ ℕ_{+}, ∅, M^{m} ∈ E_{m}(M, C) and (E(M, C), ∩, ∪,^{c}) is a Boolean algebra;

(e) if X ∈ E_{m}(M, C), then X × M, M × X ∈ E_{m+1}(M, C);

(f) if 1 ≤ i < j ≤ m, then {⟨x_{1}, …, x_{m}⟩ ∈ M^{m} : x_{i} = x_{j}} ∈ E_{m}(M, C);

(g) if X ∈ E_{m+1}(M, C) and i ∈ {1, …, m}, then π_{i}^{m+1}[X] ∈ E_{m}(M, C);

(h) C ∈ E_{1}(M, C). Therefore, by Remark 1.1, there is a weakly o-minimal structure M′ expanding M such that a set X ⊆ M^{m} is definable in M′ iff X ∈ E_{m}(M, C). In the following lemma, D_{m}(M, C) denotes the family of all subsets of M^{m} which are definable in the structure (M, C).

Lemma 2.7 Let M be a weakly o-minimal non-valuational expansion of an ordered group. For every m ∈ ℕ_{+}, E_{m}(M, C) = D_{m}(M, C).

Proof. The inclusion ⊆ is obvious. That D_{m}(M, C) ⊆ E_{m}(M, C) follows easily by induction from the above remark.

Corollary 2.8 If M is a weakly o-minimal non-valuational expansion of an ordered group, then (M, C) is weakly o-minimal.

Lemma 2.9 Let M = (M, ≤, +, …) be a weakly o-minimal non-valuational expansion of an ordered group (M, ≤, +). Assume that I is a non-empty convex open definable (in M) set with I ∩ C ≠ ∅ and I \ C ≠ ∅. Let f : I ⟶ \overline{M}^{M} be a definable strongly continuous and strictly monotone function.

(a) If f is strictly increasing, then

⟨{a ∈ M : (∃c ∈ C)(f(c) > a)}, {a ∈ M : (∃d ∈ M \ C)(f(d) < a)}⟩

is a non-definable and non-valuational cut in (M, ≤, +).

(b) If f is strictly decreasing, then

⟨{a ∈ M : (∃d ∈ M \ C)(f(d) > a)}, {a ∈ M : (∃c ∈ C)(f(c) < a)}⟩

is a non-definable and non-valuational cut in (M, ≤, +).

Proof. As both cases are similar, we will only prove (a). Assume that f is strictly increasing and let

C′ = {a ∈ M : (∃c ∈ C)(f(c) > a)} and D′ = {a ∈ M : (∃d ∈ M \ C)(f(d) < a)}.

It is clear that C′ and D′ are both convex definable sets with inf C′ = −∞ and sup D′ = +∞. The sets C′, D′ are disjoint since otherwise we would have f(d) < a < f(c) for some a ∈ M, c ∈ C and d ∈ M \ C.

To show that C′ ∪ D′ = M, suppose for a contradiction that there exists an element a ∈ M \ (C′ ∪ D′). This means that f(c) ≤ a ≤ f(d) whenever c ∈ C and d ∈ M \ C. Note that if there was a c ∈ C with f(c) = a, then there would be also a c′ ∈ C with c′ > c and f(c′) > f(c) = a, a contradiction. So (∀c ∈ C)(f(c) < a) and similarly (∀d ∈ M \ C)(f(d) > a). Now,

C = {x ∈ M : x ≤ inf I} ∪ {x ∈ I : f(x) < a},

which means that C is definable in M, a contradiction. In this way we have shown that ⟨C′, D′⟩ is a cut in (M, ≤). Its non-definability is a consequence of the non-definability of C.

In order to complete the proof, suppose for a contradiction that

inf{z − y : y ∈ C′, z ∈ D′} = inf{f(d) − f(c) : c ∈ C, d ∈ M \ C} > 0

and fix ε > 0, ε ∈ M, such that ε < inf{f(d)−f(c) : c ∈ C, d ∈ M \C}. So clearly f(x_{2})−f(x_{1}) > ε whenever x_{1} ∈ I ∩ C and x_{2} ∈ I \ C. For x_{0} ∈ I define

δ(x_{0}) = min{ε, sup{d ∈ M : d > 0 and |f(x) − f(x_{0})| < ε for x ∈ (x_{0} − d, x_{0} + d) ∩ I}}.

By Lemma 2.1, there are an open interval J ⊆ I and an element α > 0, α ∈ M, such that J ∩C ≠ ∅, J \ C ≠ ∅ and (∀x_{0} ∈ J)(δ(x_{0}) > α). Since the cut ⟨C, M \ C⟩ is non-valuational, we can choose x_{1} ∈ J ∩ C and x_{2} ∈ J \ C so that x_{2} − x_{1} < α. In such a situation we have that δ(x_{1}) > α and |f(x_{2}) − f(x_{1})| < ε, which contradicts our choice of ε.

Corollary 2.10 Let M be a weakly o-minimal non-valuational expansion of an ordered group. (a) The structure (M, C) is of non-valuational type.

(b) T h(M, C) is weakly o-minimal.

Proof. We already know by Corollary 2.8 that (M, C) is weakly o-minimal. To demonstrate that (M, C) is of non-valuational type, fix ⟨D, D′⟩, a cut in (M, ≤) which is definable in (M, C). We have to show that ⟨D, D′⟩ is non-valuational. As there is nothing to do in case D ∈ D_{1}(M), suppose that D is not definable in M. By Lemma 2.7, there exists a set X ∈ D_{2}(M) such that D = I(X, C). Denote by X′ the union of all sets of the form {a} × (b, c), where a, b, c ∈ M, b < c and {a} × (b, c) ⊆ X. Obviously, X′ is definable in M and I(X′, C) = D. Fix C, a decomposition of M^{2} into strong cells partitioning X′ and let I_{0}, …, I_{n} be an enumeration of all convex sets of the form π_{2}^{2}[Y] where Y ∈ C and Y ⊆ X′ such that I_{i} precedes I_{j} whenever i < j ≤ n. Without loss of generality we can assume that the functions appearing in definitions of cells from C are strictly monotone or constant. Non-definability of D guarantees that there is k ≤ n such that I_{k} is a convex open set intersecting both D and M \ D. There are strongly continuous functions f_{0}, g_{0}, …, f_{m}, g_{m} : I_{k} ⟶ \overline{M}^{M} ∪ \{−∞, +∞\} such that

(a) each of f_{0}, g_{0}, …, f_{m}, g_{m} is either strictly monotone or constant;

(b) \overline{f}_{i}(x) < \overline{g}_{i}(x) for i ≤ m and x ∈ \overline{I}_{k};

(c) \overline{g}_{i}(x) < \overline{f}_{i+1}(x) for i < m and x ∈ \overline{I}_{k};

(d) X′ ∩ (I_{k} × M) = (f_{0}, g_{0})_{I_{k}} ∪ … ∪ (f_{m}, g_{m})_{I_{k}}. There is a unique i ≤ m such that sup I((f_{i}, g_{i})_{I_{k}}, C) = sup D ∩ I_{k} and exactly one of the following two conditions holds.

(1) f_{i} is strictly increasing and D ∩ I_{k} = I((f_{i}, +∞)_{I_{k}}, C).

(2) g_{i} is strictly decreasing and D ∩ I_{k} = I((−∞, g_{i})_{I_{k}}, C). Since the reasoning is similar in both cases, we will only consider (1). To simplify notation let f := f_{i}. Fix a ∈ C such that a > inf{f(x) : x ∈ I_{k}} and b ∈ M \C such that b < sup{f(x) : x ∈ I_{k}}, and let J = (a, b). For y ∈ J define

h(y) = sup{x ∈ M : ⟨x, y⟩ ∈ (f, +∞)_{I_{k}}}.

The function h is strictly increasing and strongly continuous. As

D = {x ∈ M : (∃x′ ∈ C)(h(x′) > x)},

by Lemma 2.9, the cut ⟨D, D′⟩ is non-valuational.

The weak o-minimality of T h(M, C) now follows from the fact that (M, C) is a weakly ominimal non-valuational structure and from Corollary 2.15 from [We07].

Before formulating the main result, we will introduce so called unary non-valuational predicates. Assume that M = (M, ≤, …) is a weakly o-minimal non-valuational expansion of an ordered group and let X ⊆ M be a finite union of convex sets. For a ∈ X denote by R(a, X) the convex component of X containing a. Similarly, for a ∈ M \ X, let R(a, X) be the convex component of M \ X containing a. For a ∈ M define

D(a, X) = ⋃_{α∈R(a,X)}(α, +∞).

Clearly, if D(a, X) ≠ M, then ⟨M \ D(a, X), D(a, X)⟩ is a cut in (M, ≤). A cut ⟨C, D⟩ in (M, ≤) is said to be determined by X if it is of the form ⟨M \ D(a, X), D(a, X)⟩ for some a ∈ M.

We say that a set X ⊆ M is a unary non-valuational predicate iff X is a union of finitely many convex sets and all cuts determined by X are non-valuational.

Theorem 2.11 Assume that M is a weakly o-minimal non-valuational expansion of an ordered group and N is an expansion of M by a family of non-valuational unary predicates. Then N is of non-valuational type.

Proof. There is a family of convex open sets C_{i} ⊊ M, i ∈ I, such that

• for every i ∈ I, C_{i} is not definable in M and inf C_{i} = −∞;

• the structures N and (M, C_{i} : i ∈ I) have the same definable sets.

Without loss of generality we can assume that I is finite, in which case the theorem follows easily by induction on |I| from Corollary 2.10(b).

Theorem 2.11 actually shows that for weakly o-minimal expansions of ordered groups, the property of a structure having the strong cell decomposition is preserved under expansions by families of unary non-valuational predicates. Having in mind that non-valuational predicates are those which determine non-valuational cuts, one can speak of valuational/non-valuational cuts in an arbitrary weakly o-minimal structure with the strong cell decomposition property, not necessarily expanding an ordered group. More precisely, if M = (M, ≤, …) is a weakly o-minimal structure with the strong cell decomposition property, then a cut ⟨C, D⟩ in (M, ≤) could be called nonvaluational if the structure (M, C) has the strong cell decomposition property. This gives us notions of ”being close” and ”being far” for the parts of a cut ⟨C, D⟩ in (M, ≤) and it would be interesting to further investigate this topic, probably relating it to the canonical o-minimal extension of a weakly o-minimal structure with the strong cell decomposition property constructed in [We07].

References

[Ar] R. D. Aref’ev, On the monotonicity property for weakly o-minimal models, Algebra and Model Theory (Russian), pp. 8-15, Novosibirsk. Gos. Tekh. Univ., Novosibirsk 1997; Model theory in Kazakhstan (Russian), pp. 7-13, Eco Stydy, Almaty 2006.

[BP] Y. Baisalov and B. Poizat, Paires de structures o-minimales, J. Symbolic Logic 63 (1998), 570-578.

[Bz] B.S. Baizhanov, Expansion of a model of a weakly o-minimal theory by a family of unary predicates, J. Symbolic Logic 66 (2001), 1382-1414.

[BVT] B.S. Baizhanov, V. Verbovskiy and G. Turehanova, Non-orthogonality of 1-types in weakly o-minimal structures of finite depth, preprint (in Russian).

[D1] L. van den Dries, Tame Topology and o-minimal Structures, London Mathematical Society Lecture Notes Series, vol. 248, Cambridge: Cambridge University Press 1998.

[D2] L. van den Dries, T-convexity and Tame extensions, II, J. Symbolic Logic 62 (1997), 14-34.

[DL] L. van den Dries and A.H. Lewenberg, T-convexity and tame extensions, J. Symbolic Logic 60 (1995), 74-102.

[MMS] D. Macpherson, D. Marker, and C. Steinhorn, Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 (2000), 5435-5483.

[Sh783] S. Shelah, Dependent first order theories, continued, Israel J. Math., to appear.

[We06] R. Wencel, Topological properties of sets definable in weakly o-minimal structures, J. Symbolic Logic, submitted.

[We07] R. Wencel, Weakly o-minimal non-valuational structures, Ann. Pure Appl. Logic 154 (2008), 139-162.

Mailing address: Mathematical Institute, University of Wrocław,

pl. Grunwaldzki 2/4, 50-384 Wrocław, POLAND

E-mail:  rwenc@math.uni.wroc.pl