Locally o-minimal structures

Tomohiro KAWAKAMI^{∗}, Kota TAKEUCHI^{†}, Hiroshi TANAKA^{‡}and Akito TSUBOI^{§}

Abstract

In this paper we study (strongly) locally o-minimal structures. We first give a characterization of the strong local o-minimality. We also investigate locally o-minimal expansions of (ℝ, +, <).

1 Introduction

Toffalori and Vozoris [8] introduced the notion of local o-minimality and that of strong local o-minimality, by weakening the definition of o-minimality. A typical example of locally o-minimal structure is (ℝ, +, <, sin), which is not ominimal (see [8, Theorem 2.7]). They systematically investigated the notions, and, among many others, showed that any weakly o-minimal structure is locally o-minimal.

In this paper we first give a characterization of the strong local ominimality. This characterization shows that a strongly locally o-minimal structure really resembles an o-minimal structure if it is seen locally. In [4], [9], [7], several generalizations of the cell decomposition theorem were studied in the weakly o-minimal context. In this paper, using the characterization, we show that the local version of cell decomposition holds for strongly locally o-minimal structures.

∗Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani Wakayama 640-8510, Japan.

†Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan.

‡Anan National College of Technology, 265 Aoki Minobayashi, Anan, Tokushima 774-0017, Japan

§Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan. Partially supported by Kakenhi(22540110)

2010 Mathematics Subject Classification. Primary 03C64.

Key Words and Phrases. local o-minimality, cell decomposition, simple products.

We then introduce the notion of simple products of two structures. This notion is already implicit in [8], and in the present paper we give an explicit definition. Using the method of taking simple products, a number of structures are shown to be (strongly) locally o-minimal. For example, in Section 4, we show that any structure of the form (ℝ, +, <, P) with P ⊂ ℤ is locally o-minimal. Conversely, we also show that any locally o-minimal structure expanding (ℝ, +, <, ℤ) can be written as a simple product of ℤ and an o-minimal structure.

We only assume the reader’s familiarity with a few basic model theoretic notions. In Section 2, we recall some definitions and results on (local) o-minimality. The notion of local structures are introduced here. For a structure M and its subset A, the local structure A_{def} is defined roughly as the set A with M-definable subsets. A_{def} is an important tool in our characterization.

In Section 3, we give a characterization of strong local o-minimality, using local structures (see Theorem 9). The local monotonicity theorem and the local cell decomposition theorem (for strongly locally o-minimal structures) are easily obtained from our characterization. In this section, we also introduce the notion of uniform local o-minimality, and study the relation between this notion and (strong) local o-minimality. Several examples will be given.

Section 4 is the section for simple products. Let M and N be two structures. If the product M ×N is simple in our sense then every definable subset of M × N has the form A × B, where A ⊂ M is M-definable and B ⊂ N is N-definable. Simple products play important roles in constructing locally o-minimal structures (see Theorem 18). As an application, we can show the following:

• Let ℝ^{∗} be a nonstandard real closed field elementarily extending ℝ. Then (ℝ^{∗}, +, <, P) is locally o-minimal, where P is a unary predicate whose interpretation is ℝ.

In Section 5, we concentrate on expansions of the additive structure (ℝ, +, <). For an expansion M of (ℝ, +, ·, <) we easily have that M is locally o-minimal if and only if M is o-minimal. So the restriction to additive structures seems natural. The main result (Theorem 24) of this section is the following:

• Let M be a locally o-minimal expansion of (ℝ, +, <, ℤ). Then M is expressed as a simple product of ℤ and I = [0, 1)_{def}.

General references on o-minimal structures are [1], [2], [5], see also [6].

2 Preliminaries

Our notations are standard. L denotes a language. M, N,... are used to denote L-structures. The universe of M is also denoted by M. A, B,... are used to denote subsets of some structures. We use x, y, ... for variables. Formulas are denoted by φ, ψ, .... We simply say that A is definable in M (or M-definable) if it is definable in M using parameters from M. So, if A ⊂ M^{n} is definable, then there is an L-formula φ(x_{1}, ..., x_{n}, y_{1}, ..., y_{m}) and parameters b_{1}, ..., b_{m} ∈ M such that A = φ(x_{1}, ..., x_{n}, b_{1}, ..., b_{m})^{M} (the set of all tuples satisfying φ(x_{1}, ..., x_{n}, b_{1}, ..., b_{m})). A family F, consisting of M-definable sets, will be called uniformly definable if F has the form F = {φ(x_{1}, ..., x_{n}, b_{1}, ..., b_{m})^{M} : b_{1}, ..., b_{m} ∈ M}.

Definition 1 Let M be an L-structure and A a subset of M.

1. For n ∈ ω, Def^{n}(A, M) is the set of all subsets of M^{n} of the form A^{n} ∩ D, where D is an M-definable subset of M^{n}. Def(A, M) = ⋃_{n∈ω} Def^{n}(A, M).

2. We simply write Def(M) for Def(M, M) (i.e. the set of all Mdefinable sets).

Definition 2 Let A ⊂ M. We prepare an n-ary predicate symbol P_{X} for each X ∈ Def^{n}(A, M), and let L_{A} be the language {P_{X} : X ∈ Def(A, M)}. The local structure A_{def} of A is the following L_{A}-structure:

• The universe of A_{def} is A;

• The interpretation of P_{X} in A_{def} is X, for all X ∈ Def(A, M).

Remark 3 In general, Def(A_{def}) and Def(A, M) are not equal. However, if A is a definable subset of M, then we have Def(A_{def}) = Def(A, M).

From now on, we assume that M has the form (M, <, ...) and that <^{M} is a dense linear ordering, unless otherwise stated. An open interval of M is a set of the form (a, b), where a ∈ M ∪ {−∞} and b ∈ M ∪ {∞}. Recall that M is said to be o-minimal if every definable subset of M is a finite union of points and open intervals in M. The notion of local o-minimality and that of strongly local o-minimality were defined in [8]. The ordered pair of a and b is usually denoted by ⟨a, b⟩, avoiding confusion of pairs and intervals.

Definition 4 1. M is called locally o-minimal if for any definable set A ⊂ M and a ∈ M there is an open interval I ∋ a such that I ∩ A is a finite union of intervals and points.

2. M is strongly locally o-minimal, if for any a ∈ M there is an open interval I ∋ a such that whenever A is a definable subset of M then I ∩ A is a finite union of intervals and points.

3. M is uniformly locally o-minimal if for any φ(x, y) ∈ L and a ∈ M there is an open interval I ∋ a such that I ∩ φ(M, b) is a finite union of intervals and points for any b ∈ M.

The following facts are proved in [8, Corollaries 2.5 and 3.9].

Fact 5 1. Local o-minimality is preserved under elementary equivalence.

2. Strong local o-minimality is not preserved under elementary equivalence.

Several examples are given below.

Example 6 Let L = {<} ∪ {P_{i} : i ∈ ω}, where P_{i} is an unary predicate. Let M = (ℚ, <^{M}, P_{0}^{M}, P_{1}^{M}, ...) be the structure defined by P_{i}^{M} = {a ∈ M : a < 2^{−i}\sqrt{2}\}. Then M is uniformly locally o-minimal, but it is not strongly locally o-minimal.

If we assume the saturation, we can show the following:

Proposition 7 Let M be a uniformly locally o-minimal structure. Suppose that M is ω-saturated. Then M is strongly locally o-minimal.

Proof: Let a ∈ M. Choose an L-formula φ(x, y) arbitrarily. By the uniformity of M, there is an open interval I ∋ a and numbers n_{b} ∈ ω (b ∈ M) such that I ∩ φ(M, b) is a union of n_{b} many intervals and points. We may assume that each n_{b} is chosen minimum. By the saturation of M, n_{b}’s are uniformly bounded, say by n_{φ} ∈ ω. (Otherwise, by saturation, there would be b ∈ M such that I ∩ φ(M, b) cannot be expressed as a finite union of intervals and points.) Let θ_{φ}(u, v) be the formula saying that for any z the set of x ∈ (u, v) with φ(x, z) is a union of n_{φ} many intervals and points. Then the following set

Γ(u, v) = {u < a < v} ∪ {θ_{φ}(u, v) : φ ∈ L}

is finitely satisfiable in M. So, by saturation, there are c, d ∈ M realizing the set Γ. The open interval I = (c, d) witnesses the strong local o-minimality.

Example 8 We show that there is an ω-saturated locally o-minimal structure that is not uniformly locally o-minimal. For each non-negative q ∈ ℚ, we prepare a binary predicate P_{q}(x, y). L = {<, P_{q}}_{q∈ℚ^{+}} is our language. We define an L-structure M = (ℚ, <^{M}, P^{M}) by the following:

• <^{M} is the standard ordering on ℚ;

• P_{q}(a, b) ⟺ a + \sqrt{2} · q ≤ b (in ℝ).

T = Th_{L}(M) admits elimination of quantifiers. For showing this, let M^{∗} be an ω-saturated model of T. For r ∈ ℝ^{+} ∪ {∞}, let Γ_{r}(x, y) be the following set of quantifier-free formulas.

\{x < y\} ∪ \{P_{q}(x, y) : q ∈ ℚ^{+}, \sqrt{2}q ≤ r\} ∪ \{¬P_{q}(x, y) : q ∈ ℚ^{+}, r < \sqrt{2}q\}.

Intuitively speaking, Γ_{r}(x, y) asserts that the distance of two points x < y is r. Let A = {a_{1} < ⋯ < a_{n}} and B = {b_{1} < ⋯ < b_{n}} be two finite subsets of M^{∗}. We will write A ≃ B if we have

M^{∗} ⊧ Γ_{r}(a_{i}, a_{j}) ⟺ M^{∗} ⊧ Γ_{r}(b_{i}, b_{j}),

for all i, j ≤ n and r ∈ ℝ^{+} ∪ {∞}. Let c ∈ M^{∗} be any element. We want to find an element d ∈ M^{∗} with Ac ≃ Bd. To simplify our argument, we treat the case when c is bigger than A. Choose r_{1}, ..., r_{n} such that Γ_{r_{i}}(a_{i}, c) holds (i = 1, ..., n). Let us consider the following set Δ(x):

⋃_{1≤i≤n} Γ_{r_{i}}(b_{i}, x).

Since M^{∗} is ω-saturated, we can find d ∈ M^{∗} such that Γ_{r_{n}}(b_{n}, d). Then this d automatically satisfies Δ(x). Now we have Ac ≃ Bd. The above argument shows that T admits elimination of quantifiers. From the elimination of quantifiers, we see that M is locally o-minimal.

Now we show that M is not uniformly locally o-minimal. Let (b, c) be a small interval containing a. Notice that the following sentence is a member of T:

∀x∀x′(x < x′ → ∃y(P_{1}(x, y) ∧ ¬P_{1}(x′, y)).

So we can choose q ∈ M^{∗} such that P_{1}(b, q) ∧ ¬P_{1}(c, q). Then the set X defined by P_{1}(x, q) divides (b, c) into two convexes C_{1} and C_{2}. Neither C_{1} nor C_{2} are intervals.

3 Strong local o-minimality

The following theorem is easy but important.

Theorem 9 The following two conditions are equivalent:

1. M is strongly locally o-minimal.

2. For any finite subset {a_{1}, ..., a_{n}} of M, there are left-open and rightclosed intervals I_{i} with a_{i} ∈ (I_{i})^{∘} such that, by putting I = ⋃_{1≤i≤n} I_{i}, I_{def} is o-minimal. (I^{∘} is the interior of I.)

Proof: 1 → 2: Choose any a_{1}, ..., a_{n} ∈ M. Then, by the strong local o-minimality, there are intervals I_{i} = (b_{i}, c_{i}] with a_{i} ∈ (I_{i})^{∘} (i = 1, ..., n) such that, for any definable set X ⊂ M, X ∩ I_{i} is a finite union of points and open intervals in M (i = 1, ..., n). We may assume that a_{1} < ⋯ < a_{n} and I_{1} < ⋯ < I_{n}.

Let I = ⋃ I_{i} and choose any Y ∈ Def^{1}(I_{def}). Then Y is a definable subset of M and Y = Y ∩ I = ⋃_{i}(Y ∩ I_{i}). By the item 1, there are d_{ik}’s and e_{ik}’s such that

Y ∩ I_{i} = (d_{i1}, e_{i1}) ∪ ⋯ ∪ (d_{im_{i}}, e_{im_{i}}) ∪ {finite points}.

Hence Y is a finite union of convex sets. Using the fact that <^{M} is dense, we may assume that d_{i2}, ..., d_{im_{i}}, e_{i1}, ..., e_{im_{i}} ∈ I_{i}. The point d_{i1} need not to be an element in I_{i}. However, even if d_{i1} ∉ I_{i}, in I, Y ∩ I_{i} can be written as

Y ∩ I_{i} = (−∞, e_{i1}) ∪ ⋯ ∪ (d_{im_{i}}, e_{im_{i}}) ∪ {finite points} (if i = 1),

Y ∩ I_{i} = (c_{i−1}, e_{i1}) ∪ ⋯ ∪ (d_{im_{i}}, e_{im_{i}}) ∪ {finite points} (if i > 1).

So, in I_{def}, Y is expressed as a finite union of intervals and points in I.

2 → 1: Assume 2. Let {a} be a singleton set in M. Choose an interval I′ = (b, c] witnessing the condition in 2. Notice that I = (b, c) also satisfies the required condition in 2, i.e., I_{def} is o-minimal. Let X ∈ Def^{1}(M). Then we have X ∩ I ∈ Def^{1}(I_{def}). By the o-minimality, we have

X ∩ I = I_{1} ∪ ⋯ ∪ I_{m} ∪ {finite points},

for some open intervals in the sense of I_{def}. Notice that each I_{i} is an interval in M. So X ∩ I is a finite union of intervals and points in M. Thus we are done.

The following definition is taken from [8].

Definition 10 We say that a definable unary (possibly partial) function f has local monotonicity if, for every point a ∈ M, there exists some open interval I containing a such that domf ∩ I can be broken up into a finite union of points and open intervals, on each of which f is constant, strictly increasing, or strictly decreasing. We say that M has local monotonicity if every definable unary function f of M has local monotonicity.

In [8], it was shown that a strongly locally o-minimal structure satisfies local monotonicity. In o-minimal case, we can add the local continuity in the monotonicity theorem. As we will see later, this is not the case of local o-minimality. However, by Theorem 9, we can prove the following:

Proposition 11 Let M be strongly locally o-minimal. Let D be a definable set of M and f : D → M a definable function. Then, for any a ∈ D, there are open intervals I ⊂ M containing a and J ⊂ M containing f(a) such that, by putting f^{∗} = f ∩ (I × J), the domain of f^{∗} can be broken up into a finite union of points and open intervals, on each of which f^{∗} is constant, strictly increasing and continuous, or strictly decreasing and continuous.

The following example shows that the replacement of f by f^{∗} in the above proposition is necessary.

Example 12 Let M be any o-minimal structure and let a ∈ M. Let f : {a} × M → M^{2} be the function defined by ⟨a, b⟩ ↦ ⟨b, a⟩. Then N = (M^{2}, <_{lex}, f) is an M-definable structure (in eq-sense), where <_{lex} is the lexicographic ordering on M^{2}. So, N is strongly locally o-minimal. However, f is discontinuous at any point.

As in the o-minimal setting, we can define cells and cell decompositions of definable sets in the locally o-minimal setting, see [3]. We have the following proposition by Theorem 9:

Proposition 13 Assume that M = (M, <, ...) is a strongly locally ominimal structure. Let a ∈ M^{n}. Then, the following results hold.

1. Let X_{1}, ..., X_{m} be definable subsets of M^{n}. Then there is an open box B ∋ a and a finite decomposition P of B into cells partitioning X_{1} ∩ B, ..., X_{m} ∩ B.

2. Let X ⊂ M^{n} be a definable set and f : X → M a definable function. Then there is an open box B ∋ ⟨a, f(a)⟩ such that for the restriction f^{∗} = f ∩B, the domain of f^{∗} admits a finite decomposition P into cells so that for any Y ∈ P, f^{∗}|Y is continuous.

3. Let X ⊂ M^{n+1} be a definable set and b ∈ M. Suppose that X_{c} = {d ∈ M : ⟨c, d⟩ ∈ X} is finite for any c ∈ M^{n}. Then, there is an open box B ∋ a, an open interval I ∋ b and K ∈ ω such that |X_{c} ∩ I| ≤ K for all c ∈ X ∩ B.

4 Simple Products

Let L_{1}, L_{2} and L be languages. For simplicity, we assume that these languages are relational. Under this assumption, a binary function will be treated as a ternary relation. Let M_{i} be an L_{i}-structure (i = 1, 2).

Definition 14 1. Let A ⊂ M_{1}^{n} and B ⊂ M_{2}^{n}. Then A ∗ B is the subset of N^{n}, N = M_{1} × M_{2}, defined by:

A ∗ B := {⟨⟨a_{1}, b_{1}⟩, ..., ⟨a_{n}, b_{n}⟩⟩ ∈ N^{n} : ⟨a_{1}, ..., a_{n}⟩ ∈ A , ⟨b_{1}, ..., b_{n}⟩ ∈ B}

2. Let N be an L-structure whose universe is the product M_{1}×M_{2}. We say that N is a simple product of M_{1} and M_{2} if for any P (x_{1}, ..., x_{n}) ∈ L there are M_{1}-definable sets A_{1}, ..., A_{k} ⊂ M_{1}^{n} and M_{2}-definable sets B_{1}, ..., B_{l} ⊂ M_{2}^{n} such that P^{N} is a boolean combination of the following sets

• A_{i} ∗ M_{2}^{n} (i = 1, ..., k),

• M_{1}^{n} ∗ B_{i} (i = 1, ..., l).

Many important structures can be expressed using simple products.

Example 15 1. Let M_{1} and M_{2} be two ordered sets. The lexicographic order <^{N} on the product N = M_{1} × M_{2} can be expressed as

<^{N} = [(<^{M_{1}}) ∗ M_{2}] ∪ [(( =^{M_{1}}) ∗ M_{2}) ∩ (M_{1} ∗ (<^{M_{2}}))].

So (N, <^{N}) is a simple product.

2. Let M_{1} and M_{2} be two groups. The product group of M_{1} and M_{2} is a simple product.

3. Let I = ([0, 1), <, +) be the additive group of reals modulo 1. Let N = ℤ × I be the simple product defined by:

+^{N} = (P ∗ Q) ∪ (P′ ∗ Q′),

where P = {⟨m, n, k⟩ ∈ ℤ^{3} : m + n = k}, P′ = {⟨m, n, k⟩ ∈ ℤ^{3} : m + n + 1 = k}, Q = {⟨a, b, c⟩ ∈ [0, 1)^{3} : a, b < a +^{I} b = c} and Q′ = {⟨a, b, c⟩ ∈ [0, 1)^{3} : c = a +^{I} b < a, b}. Then N is isomorphic to ℝ = (ℝ, +, <) by the mapping ⟨n, a⟩ ↦ n + a.

Lemma 16 Suppose that N = M_{1} × M_{2} is a simple product. Let D be an N-definable subset of N^{n}. Then there are M_{1}-definable sets A_{1}, ..., A_{k} ⊂ M_{1}^{n} and M_{2}-definable sets B_{1}, ..., B_{l} ⊂ M_{2}^{n} such that D is a boolean combination of A_{i} ∗ M_{2}^{n} (i = 1, ..., k) and M_{1}^{n} ∗ B_{i} (i = 1, ..., k).

Proof: For simplicity, we assume D is ∅-definable. Choose an L-formula φ(x_{1}, ..., x_{n}) defining D. If φ is an atomic formula, the lemma follows from the definition of simpleness. Our proof proceeds by induction on the complexity of φ. The case when φ has the form ψ ∧ χ, ψ ∨ χ, ¬ψ or ψ → χ is clear. So we assume that φ has the form ∃yψ. Further, for simplicity of the notation, we assume ψ = ψ(x, y), where x and y are single variables. By the induction hypothesis, ψ^{N} has the form

⋃_{1≤i≤k}(A_{i} ∗ M_{2}^{2}) ∩ (M_{1}^{2} ∗ B_{i}),

where all A_{i} ⊂ M_{1}^{2} and all B_{i} ⊂ M_{2}^{2} are definable sets. Then φ^{N} is the following set.

⋃_{⟨a,b⟩∈M_{1}×M_{2}} ⋃_{1≤i≤k}{⟨a_{1}, b_{1}⟩ : ⟨⟨a_{1}, b_{1}⟩, ⟨a, b⟩⟩ ∈ (A_{i} ∗ M_{2}^{2}) ∩ (M_{1}^{2} ∗ B_{i})}.

This set is equal to

⋃_{1≤i≤k} ⋃_{⟨a,b⟩∈M_{1}×M_{2}}{⟨a_{1}, b_{1}⟩ : ⟨a_{1}, a⟩ ∈ A_{i}, ⟨b_{1}, b⟩ ∈ B_{i}}.

Finally, notice that the set ⋃_{⟨a,b⟩∈M_{1}×M_{2}}{⟨a_{1}, b_{1}⟩ : ⟨a_{1}, a⟩ ∈ A_{i}, ⟨b_{1}, b⟩ ∈ B_{i}} is equal to {⟨a_{1}, b_{1}⟩ : a_{1} ∈ proj(A_{i}), b_{1} ∈ proj(B_{i})}, where proj is the projection map to the first coordinate. Since proj(A_{i}) and proj(B_{i}) are definable sets, the induction step is complete.

Lemma 17 Let N be a simple product of M_{1} and M_{2}. For every a ∈ M_{1} and every definable set D ⊂ N, the section D_{a} = {b ∈ M_{2} : ⟨a, b⟩ ∈ D} is definable in M_{2}.

Proof: We can find definable sets A_{i} ⊂ M_{1} and B_{i} ⊂ M_{2} such that D = ⋃_{1≤i≤k}(A_{i} ∗ M_{2}^{n}) ∩ (M_{1}^{n} ∗ B_{i}). Then D_{a} can be written as

D_{a} = ⋃{B_{i} : a ∈ A_{i}}.

So D_{a} is a definable subset of M_{2}.

Theorem 18 For i = 1, 2, let M_{i} = (M_{i}, <^{M_{i}}, ...) be an expansion of a linear order. Let N = (N, <^{N}, ...) be a simple product of M_{1} and M_{2}, where <^{N} is given by the lexicographic ordering.

1. Suppose that M_{2} is a (strongly) locally o-minimal structure without endpoints. Then N is (strongly) locally o-minimal.

2. Suppose that M_{2} is an o-minimal structure possibly with endpoints. Suppose also that M_{1} is a discrete order. Then N is strongly locally o-minimal.

Proof: We prove 2, since 1 can be proven similarly. We assume that M_{1} has the form (−∞, m], where m is the maximum element. Let ⟨a, b⟩ ∈ N be any point. First assume that b ∈ M_{2} is not an endpoint. Let I be an open M_{2}-interval with I ∋ b. Then I′ = {a} × I is an N-interval containing ⟨a, b⟩. Let D ⊂ N be any definable set. Then D ∩ I′ = {a} × (D_{a} ∩ I). By Lemma 17, D_{a} is a definable subset of M_{2}. So D_{a} ∩I is a finite union of intervals and points. Hence D ∩ I′ is a finite union of intervals and points in the sense of N. This shows the o-minimality of I′_{def}, and hence we have the strong local o-minimality of N.

Then we treat the case that b is the maximum element m. Choose any c ∈ M_{2} ∖ {b} and let I = ({a} × (c, b]) ∪ ({a + 1} × (−∞, c)), where a + 1 is the successor of a in the discrete structure M_{1}. (If a + 1 does not exist, we can put I = {a} × (c, b].) As in the previous case, I_{def} is o-minimal, hence N is strongly locally o-minimal.

Example 19 Let A ⊂ ℤ and P a new unary predicate symbol. Then the structure (ℝ, +, <, P^{ℝ}) with P^{ℝ} = A is locally o-minimal. Proof: Let I = ([0, 1), +, <) be the additive group of reals modulo 1. Let P_{0} be a unary predicate symbol and P_{0}^{ℤ} = A. There is a simple product N = ℤ × I such that N ≅ (ℝ, +, <). We give a P -structure on N by

P^{N} = P_{0}^{ℤ} ∗ {0}.

Then (N, P^{N}) is a simple product, hence it is locally o-minimal by Theorem 18. It is easy to see that (N, P^{N}) ≅ (ℝ, +, <, A).

Example 20 Let (ℝ^{∗}, +, ·, <, ℤ^{∗}) be a saturated elementary extension of (ℝ, +, ·, <, ℤ). Let P be a new unary predicate symbol such that P^{ℝ^{∗}} = ℚ. Then (ℝ^{∗}, +, <, P^{ℝ^{∗}}) is locally o-minimal. To see this, using the saturation, choose a positive infinitesimal h ∈ ℝ^{∗} such that hℤ^{∗} = {hn | n ∈ ℤ^{∗}} ⊃ ℚ. Then, for a similar reason as in the previous example, (ℝ^{∗}, +, <, ℚ) is given by a simple product of ℤ^{∗} and [0, h)^{∗}.

Example 21 Let ℝ^{∗} be a nonstandard real closed field extending ℝ. Then (ℝ^{∗}, +, <, P^{ℝ^{∗}}) is locally o-minimal, where P^{ℝ^{∗}} = ℝ. This is a corollary of the following more general statement:

Let (G, 0, +, −, <) be a divisible ordered abelian group and G_{0} ⊂ G a subgroup. Suppose there is an h ∈ G such that nh < |a| for all n ∈ ℕ and a ∈ G_{0}\{0}. Then (G, 0, <, +, −, P^{G}) is locally o-minimal, where P^{G} = G_{0}.

Proof: First notice that every ordered divisible abelian group with the language L = {0, +, −, <} has quantifier elimination. Let H = {a ∈ G | ∃n ∈ ℕ, |a| < nh}. Then H is also a divisible ordered abelian group. So H is an ominimal structure with the language L. Let G′ ⊃ G_{0} be a maximal divisible subgroup of G such that G′ ∩ H = {0}. Then G splits as the direct sum of G′ and H. It is easy to check that (G, 0, +, −, <, G_{0}) is given by a simple product of G′ and H.

5 Locally o-minimal structures on ℝ

As is shown in the last section, the structure (ℝ, +, <, ℤ) is locally o-minimal. On the other hand, for an expansion M of (ℝ, +, ·, <), M is locally o-minimal if and only if it is o-minimal. So, in the study of local o-minimality, it may be important to consider structures without multiplication. In this section we show that any locally o-minimal expansion R of (ℝ, +, <, ℤ) is given by a simple product of ℤ and I = [0, 1).

We start with some basic remarks on local o-minimality.

Remark 22 1. For any a ∈ ℝ, the structure M = (ℝ, +, <, aℤ) is locally o-minimal. If a ∈ ℝ is an irrational number, then the structure N = (ℝ, +, <, ℤ, aℤ) is not locally o-minimal, since 0 is a limit of the set {m + x : m ∈ ℤ, x ∈ aℤ}.

2. Let M be locally o-minimal. Let K ⊂ M be a (nonempty) compact definable subset of M. Then K_{def} is an o-minimal structure. (K is possibly not dense, but it is a finite union of dense subsets.)

Proof: Let A be a definable subset of K_{def}. First notice that A is definable in M also. We show that A is a finite union of intervals (in the sense of K_{def}) and points. Let a ∈ K_{def}. By the local o-minimality, we can choose an open interval I ⊂ M with a ∈ I such that K ∩ I has one of the following form:

(a) (b, c), (b, c], [b, c),

(b) {a},

where b < c and b ≤ a ≤ c. But, by the closedness of K, the endpoints b and c must belong to K. So K ∩ I is an interval (or a point) in K_{def}.

Since other cases can be treated similarly, we assume K ∩ I = [b, c] and b < a < c. Now we consider the set K∩I ∩A. By the local o-minimality of M, there are b_{1}, c_{1} ∈ M such that K ∩ (b_{1}, c_{1}) ∩ A is a finite union of intervals and points. We may assume that b < b_{1} < a < c_{1} < c. So, by letting I_{a} = (b_{1}, c_{1}), K ∩ I_{a} ∩ A is a finite union of intervals in K and points in K. Since ⋃_{a∈K} I_{a} is an open covering of K, by compactness of K, there is a finite set F ⊂ K such that ⋃_{a∈F} I_{a} ⊃ K. Then K ∩A = ⋃_{a∈F} (K ∩I_{a} ∩A) is a finite union of intervals and points in the sense of K_{def}.

Lemma 23 Let M be a locally o-minimal expansion of (ℝ, +, <) and let I = [0, 1). Suppose that a family X ⊂ Def^{n}(I, M) is at most countable. If X is uniformly M-definable, then it is finite.

Proof: We use the fact that any compact subset of M is o-minimal (see Remark 22). So we know that I_{def} is an o-minimal structure.

We proceed by induction on n. First let n = 1 and let X be uniformly definable. By the o-minimality of I_{def}, for each X ∈ X, δ(X) = cl(X) − X^{∘} is finite. So Δ = ⋃_{X∈X} δ(X) is at most countable. Moreover, by the uniform M-definability, Δ ∖ {1} is an I_{def}-definable set. Again, by the o-minimality of I_{def}, Δ must be finite. From this, we see that X is a finite set.

Now we consider the case when X ⊂ M^{n+1} is a uniformly definable countable family. For X ∈ X and a ∈ I^{n}, let X_{a} be the section {b ∈ I : ⟨a, b⟩ ∈ X} and let δ(X_{a}) = cl(X_{a}) − (X_{a})^{∘}. As in the case n = 1, the set Δ_{a} = ⋃_{X∈X} δ(X_{a}) is a finite set. So {Δ_{a} : a ∈ I^{n}} is a uniformly I_{def}- definable family of finite sets in I. By the uniform finiteness (o-minimality of I_{def}), there is a number k such that, for any a ∈ I^{n}, |Δ_{a}| ≤ k.

We enumerate Δ_{a}∪{0, 1} as {d_{0}(a), d_{1}(a), ..., d_{k+1}(a)} in increasing order. For F, G ⊂ {0, .., k+1}, let J_{a,F,G} be the union of all singletons {d_{i}(a)} (i ∈ F) and open intervals (d_{i}(a), d_{i+1}(a)) (i ∈ G). Then, for any X ∈ X and a ∈ I^{n}, we can find F, G with X_{a} = J_{a,F,G}. Using this fact, we define definable sets

Y_{X,F,G} = {a ∈ I^{n} : X_{a} = J_{a,F,G}},

and we put Y = {Y_{X,F,G}}_{X,F,G}. There are only finitely many ⟨F, G⟩’s. So the family Y (consisting of subsets of I^{n}) is a uniformly M-definable family. From this, using the induction hypothesis, we know that Y is a finite family. Now notice that if Y_{X,F,G} = Y_{X′,F,G} for all F, G, then X = X′. So we know that X is a finite family.

Theorem 24 Let M be a locally o-minimal expansion of (ℝ, +, <, ℤ). Then M is expressed as a simple product of ℤ and I = [0, 1)_{def}.

Proof: Let L be the language of M. Let P be an n-ary predicate symbol in L. For each η = ⟨η(1), ..., η(n)⟩ ∈ ℤ^{n}, we define

D_{η} = {⟨d_{1}, ..., d_{n}⟩ ∈ I^{n} : ⟨η(1) + d_{1}, ..., η(n) + d_{n}⟩ ∈ P^{M}}.

Then, using the predicate for ℤ, we can show that X = {D_{η}}_{η} is a uniformly M-definable family. Since X is at most countable, it must be finite, by Lemma 23. So we can enumerate X as X_{0}, ..., X_{k}. For i = 0, ..., k, let A_{i} = {η ∈ ℤ^{n} : D_{η} = X_{i}}. Now we regard ℤ as a {A_{i} : i ≤ k}-structure. We give a simple structure on N = ℤ × I by

P^{N} = A_{0} ∗ X_{0} ∪ ⋯ ∪ A_{k} ∗ X_{k}.

Now it is sufficient to show the following.

Claim A The natural mapping ⟨m, a⟩ ↦ m + a gives an isomorphism of N and M.

Suppose that ⟨⟨m_{1}, a_{1}⟩, ..., ⟨m_{n}, a_{n}⟩⟩ is a member of P^{N}. Then, by the definition of P^{N}, there is i ≤ k such that

⟨⟨m_{1}, a_{1}⟩, ..., ⟨m_{n}, a_{n}⟩⟩ ∈ A_{i} ∗ X_{i}.

So we have (1) ⟨m_{1}, ..., m_{n}⟩ ∈ A_{i} and (2) ⟨a_{1}, ..., a_{n}⟩ ∈ X_{i}. From (1) and the definition of A_{i}, we have D_{⟨m_{1},...,m_{n}⟩} = X_{i}. From this and (2), we have ⟨a_{1}, ..., a_{n}⟩ ∈ D_{⟨m_{1},...,m_{n}⟩}. Hence ⟨a_{1} + m_{1}, ..., a_{n} + m_{n}⟩ ∈ P^{M}. The other direction can be shown similarly.

Theorem 24 shows that, if the given locally o-minimal expansion of (ℝ, < , +) has ℤ as a definable set, then it can be expressed as a simple product. The next proposition shows that there is a locally o-minimal expansion M having the properties (1) M has an infinite discrete definable set and (2) M cannot be expressed as a simple product.

Proposition 25 Let E = {e^{n} : n ∈ ω}, where e is the base of the natural logarithm. Then the structure (ℝ, +, <, E) is locally o-minimal.

Proof: Let (ℝ^{∗}, +, <, E^{∗}) be a proper elementary extension of (ℝ, +, <, E) with infinitesimals. Let μ be the monad of 0, i.e. μ = {a ∈ ℝ^{∗} : |a| < r (∀r ∈ ℝ)}. Let D^{∗} ⊂ ℝ^{∗} be the smallest divisible group containing E^{∗}.

Claim A D^{∗} ∩ μ = {0}.

Assume otherwise. We consider ℝ and ℝ^{∗} as ℚ-modules. Then there is an infinitesimal ε ∈ ℝ^{∗}∖{0} and finitely many rationals q_{i} ∈ ℚ and E^{∗}-elements α_{1} < ⋯ < α_{n} such that ε = q_{1}α_{1} + ⋯ + q_{n}α_{n}. By (ℝ, +, <, E) ≺ (ℝ^{∗}, +, < , E^{∗}), for any positive r ∈ ℝ, there are E-elements a_{1} < ⋯ < a_{n} such that 0 ≠ |q_{1}a_{1} + ⋯ + q_{n}a_{n}| < r. We show that this is impossible. For fixed s_{1}, ..., s_{n} ∈ ℚ, let A_{s_{1}...s_{n}} = {|s_{1}e^{m_{1}} + ⋯ + s_{n}e^{m_{n}}| : m_{1} < ⋯ < m_{n} ∈ ω}. Then, by induction on n, we can show that for any s_{1}, ..., s_{n} ∈ ℚ and positive r ∈ ℝ, there are only finitely many elements a ∈ A_{s_{1}...s_{n}} with a ≤ r. (End of Proof of Claim A)

Using Claim A, choose a maximal divisible group X ⊂ ℝ^{∗} extending D^{∗} such that X ∩ μ = {0}. Then we have ℝ^{∗} = X ⊕ μ, and X is a representative set of ℝ^{∗}/μ. X has a natural induced order. On M = X × μ, we can define naturally +^{M} and <^{M} so that M becomes a simple product. We also define E^{M} by

⟨a, b⟩ ∈ E^{M} ⟺ a ∈ E^{∗} and b = 0.

Then the expanded structure M = (M, +^{M}, <^{M}, E^{M}) is still simple. So M is a locally o-minimal structure, by Theorem 18.

Claim B Let σ : ℝ^{∗} → M be the natural mapping defined by α ↦ ⟨a, b⟩, where a ∈ X and b ∈ μ are (unique) elements with α = a + b. Then σ is an isomorphism.

We only need to check σ(E^{∗}) = E^{M}. Let α ∈ E^{∗}. Then α ∈ X and σ(α) = (α, 0). So σ(α) belongs to E^{M}. The other inclusion follows similarly. (End of Proof of Claim B)

By Claim B, we see that (ℝ^{∗}, +, <, E^{∗}) is locally o-minimal. Since the local o-minimality is preserved by elementary equivalence (see Fact 5), we have the local o-minimality of (ℝ, +, <, E).

Remark 26 Let us say that E ⊂ ℝ is a good set if for all n ∈ ω and for all q_{1}, ..., q_{n} ∈ ℚ ∖ {0}, the set {|q_{1}a_{1} + ⋯ + q_{n}a_{n}| : a_{i} ∈ E} has a positive infimum. Then, for any good E, we can prove the local o-minimality of (ℝ, +, <, E), exactly by the same argument as above. Moreover, if P_{0}, P_{1}, ... are relations on E, then the structure (ℝ, +, <, E, P_{0}, P_{1}, ...) is also locally o-minimal.

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