January 01, 1970
For an \(\aleph_0\)-saturated weakly o-minimal expansion of an ordered group \({\mathcal{M}}\), it is shown that \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness if and only if the collection of definable convex subgroups of \({\mathcal{M}}\) has uniform finiteness. If \({\mathcal{M}}\) expands an ordered field, then considering definable convex valuation subrings is sufficient. The results use a criterion of Johnson for uniform finiteness in \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\), [1].
In addition, it is shown that uniform finiteness in \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) may fail for weakly o-minimal expansions of fields.
A definable (or interpretable) set \(X\) in a first-order structure \({\mathcal{M}}\) has uniform finiteness if for any definable family of subsets \(\{D_t\subseteq X: t\in T\}\) there is a natural number \(n\in \mathbb{N}\) for which \(|D_t|=\infty\iff |D_t|>n\). A structure \({\mathcal{M}}\) has uniform finiteness if all its definable subsets have uniform finiteness. Many well-behaved structures have uniform finiteness; but not all, for example \((\mathbb{Z},+,<)\).
Uniform finiteness in \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) is a different matter (see Definition 4 for a precise meaning of the term): The p-adic field \(\mathbb{Q}_p\) has uniform finiteness, but its interpretable value group does not.
In the present note we study uniform finiteness in weakly o-minimal theories. Recall that a linearly ordered structure \({\mathcal{M}}\) is weakly o-minimal if every definable subset of \(M\) is a finite union of convex sets. Throughout we will be working only with \(\aleph_0\)-saturated weakly o-minimal structures, so we assume that the theory of the structure is weakly o-minimal. Weakly o-minimal theories, essentially by definition, have uniform finiteness. So we focus on uniform finiteness in imaginary sorts. Our main results are:
Theorem 1 (Theorems 2 and 4). Let \({\mathcal{M}}\) be an \(\aleph_0\)-saturated weakly o-minimal expansion of an ordered group. Then
\({\mathcal{M}}^{eq}\) has uniform finiteness if and only if the collection of definable convex2 subgroups of \({\mathcal{M}}\) has uniform finiteness.
If, moreover, \({\mathcal{M}}\) is an expansion of an ordered field, then \({\mathcal{M}}^{eq}\) has uniform finiteness if and only if the collection of definable convex valuation subrings of \({\mathcal{M}}\) has uniform finiteness.
Our proof builds on a general criterion of Johnson, [1], for a structure \({\mathcal{M}}\) to have uniform finiteness in all imaginary sorts. Specifically, Johnson proves that \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness if and only if every definable set of unary imaginaries has uniform finiteness. A definable set \(X\) in \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) is a set of unary imaginaries if \(X\) is a collection (of codes) of pairwise distinct subsets of \(M\).
Using this result, we also prove an Ax-Kochen-Ershov style transfer result, namely that for a weakly o-minimal ordered field \({\mathcal{M}}\), \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness if and only if for some (equivalently, any) definable valuation \(v\) (possibly trivial) the collections of all \(M\)-definable convex subgroups of \(\Gamma_v\) (the value group) and of the additive group of \(\boldsymbol{k}_v\) (the residue field) have uniform finiteness (see Corollary 2 for a more precise statement). This result extends our proof from [3], asserting that power bounded T-convex expansions of o-minimal fields have uniform finiteness.
To show that the above results are not vacuous, we give in the appendix an example of a weakly o-minimal expansion of a real closed valued field \({\mathcal{K}}\) where uniform finiteness fails in \({\mathcal{K}}^{\mathop{\mathrm{eq}}}\). It is a consequence of the following theorem together with [4].
Theorem 2 (Theorem 12). Let \({\mathcal{K}}\) be a pure real closed valued field. Let \(\tilde{k}\), \(\tilde{\Gamma}\) be weakly o-minimal expansions of the \({\mathcal{K}}\)-induced structure on the residue field and value group sorts. Then the resulting expansion \(\widetilde{{\mathcal{K}}}\) of \({\mathcal{K}}\) is weakly o-minimal.
Throughout, given a structure \({\mathcal{M}}\), unless specified otherwise, by a definable set we mean a definable set in \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) with parameters.
The first set of definitions and facts are mostly known and taken, for example, from [5], [6]. We repeat them for the reader’s convenience, and in order to fix some notation.
Let \((M,<,+)\) be a linearly ordered group. By a cut in \(M\) we mean \(\mathcal{C}=(C_1,C_2)\), where \(C_1\) is a non-trivial initial segment of \(M\) and \(C_2\) is its complement. Abusing terminology, we will identify the cut \({\mathcal{C}}\) and the corresponding initial segment \(C_1\).
Definition 1. Given a cut \({\mathcal{C}}=(C_1,C_2)\) in \(M\), the stabilizer group of \({\mathcal{C}}\), also known as the invariance group of \({\mathcal{C}}\) is \(G(C_1)=G({\mathcal{C}})=\{\delta\in M: \delta+C_1=C_1\}\).
The invariance group \(G({\mathcal{C}})\) is a convex (possibly trivial) subgroup of \((M,+)\). The following is standard:
Definition 2. Let \({\mathcal{C}}=(C_1,C_2)\) be a cut.
\({\mathcal{C}}\) is valuational if \(G({\mathcal{C}})\neq \{0\}\) and non-valuational, otherwise.
A non-valuational cut in \(M\) is rational if it is realized (i.e., \(C_1\) has a supremum) in \(M\), and irrational otherwise.
We say that a non-empty initial segment of \(M\) is (non-)valuational or (ir)rational if its corresponding cut is such.
If \({\mathcal{C}}\) is a valuational cut in \(M\) expanding a real closed field, then \(\{x: xG\subseteq G\}\) is a non-trivial valuation ring, [2].
For a convex set \(D\) we write, as usual, \(x<D\) to mean \(x<c\) for all \(c\in D\). We also write \(x\le D\) for \(x<D\vee x\in D\). With this notation the following is immediate:
Lemma 1. Let \({\mathcal{C}}=(C_1,C_2)\) be a cut and \(H:=G(C)\). Then exactly one of the following holds:
\({\mathcal{C}}\) is irrational and non-valuational.
There is a coset \(S\) of \(H\), possibly a singleton, such that \(C_1=\{x\in M:x\,\Box\, S\}\), with \(\Box\in \{<,\leq\}\).
\({\mathcal{C}}\) is valuational and for \(\pi:M\to M/H\) the natural quotient map, \(\pi(C_1)\) is an irrational non-valuational initial segment.
Definition 3. A valuational cut \({\mathcal{C}}=(C_1,C_2)\) is called of type (iii), if \(\pi(C_1)\) is irrational non-valuational in the quotient group \(M/G(C_1)\).
Remark 1. The literature contains finer classification of cuts, for example see [5]. In the terminology of [5], type (iii) cuts are non-ball cuts with non-trivial invariance group. In the terminology of [6], type (iii) cuts are cuts with sign \(0\) and non-trivial invariance group.
Example 1. Consider \(M=(\mathbb{Q},+)^2\) with the lexicographic ordering. The cut \(\mathcal{C}\) induced by \((\sqrt{2},0)\) is of type (iii). Indeed, \(G(\mathcal{C})=0\times\mathbb{Q}\) and the induced cut on \(M/G(\mathcal{C})\) is irrational (see [5]).
An example of a cut of type (iii) in a weakly o-minimal expansion of a field can be found in [7].
Since every definable subset of a weakly o-minimal structure is a finite boolean combination of definable initial segments we conclude:
Lemma 2. If \({\mathcal{M}}=(M,+,<,\dots)\) is a weakly o-minimal expansion of an ordered group then every definable subset of \(M\) is a boolean combination of definable initial segments as in Lemma 1 (i)-(iii).
The next lemma will not be used in the sequel, but may be of independent interest. It is a generalization of [4]. For a convexly valued ordered field \({\mathcal{K}}\) we say that the value group \(\Gamma\) (resp. the residue field \(\boldsymbol{k}\)) is definably complete if any \({\mathcal{K}}\)-bounded definable subset of \(\Gamma\) (resp. \(\boldsymbol{k}\)) has a supremum. With this terminology we have:
Lemma 3. Assume that \(({\mathcal{K}},+,\cdot,<,{\mathcal{O}},\dots)\) is an expansion of a convexly valued ordered field, such that \(\boldsymbol{k}\) and \(\Gamma\) are definably complete.
Every non-trivial definable convex subgroup of \((K,+)\) is a ball.
For any valuational cut \({\mathcal{C}}=(C_1,C_2)\) in \(K\) of type (iii), \(G({\mathcal{C}})\) is a closed ball.
Proof. Let \(H\) be a non-trivial definable convex subgroup of \((K,+)\). Consider balls of the form \(B_{\geq r}(0)\) and \(B_{>r}(0)\) for \(r\in \Gamma\). As \(H\) is a convex subgroup it is comparable by inclusion to every ball (open or closed). If \(B_{\geq r}(0)\subseteq H\) for every \(r\in \Gamma\) then \(H=K\), so the set \(\{r:B_{\geq r}(0)\subseteq H\}\) must be bounded below. By definable completeness of \(\Gamma\), it has an infimum \(r_0\). If \(r_0\) is not a minimum then it is easy to see that \(H=B_{>r_0}(0)\) an open ball, so assume \(r_0\) is a minimum. Namely, \(B_{>r_0}(0)\subsetneq H\subseteq B_{\geq r_0}(0)\).
As \(B_{\ge r_0}(0)/ B_{> r_0}(0)\cong \boldsymbol{k}\), \(H\) is mapped via this isomorphism onto a non-zero convex subgroup of \((\boldsymbol{k}, +)\), which by the definable completeness of \(\boldsymbol{k}\) must equal to \(\boldsymbol{k}\), hence \(H=B_{\geq r_0}(0)\). Thus, \(H\) is either a closed or an open ball.
Now assume that \(C_1\) is a valuational initial segment of type (iii). So \(H=G(C_1)\) is nontrivial convex and \(\pi(C_1)\subseteq K/H\) is irrational and non-valuational. After rescaling we may assume that \(H\) is a ball of radius \(0\). We claim that \(H={\mathcal{O}}\).
Indeed, if \(H=\boldsymbol{m}\), the maximal ideal, then there exists \(z\in {\mathcal{O}}^\times\) which does not lie in \(H\), i.e. there are elements \(x\in C_1\) and \(y\notin C_1\) such that \(v(y-x)=0\). So there is a closed ball of radius \(0\) containing both \(x\) and \(y\). Translating, we may assume that this closed ball is \({\mathcal{O}}\). Thus, \(\pi:K\to K/\boldsymbol{m}\) sends \({\mathcal{O}}\) onto \(\boldsymbol{k}\) and sends \(C_1\cap {\mathcal{O}}\) to a nontrivial cut in \(\boldsymbol{k}\). By definable completeness of \(\boldsymbol{k}\), \(\pi(C_1)\) has a supremum, contradicting the assumption that it is irrational. ◻
Keeping the notation of the lemma, if \({\mathcal{K}}\) is weakly o-minimal, so are \(\Gamma\) and \(\boldsymbol{k}\). In this setting, \(\Gamma\) and \(\boldsymbol{k}\) are definably complete if and only if they are o-minimal (and therefore stably embedded, see [8]).
Below, we say that a cut \({\mathcal{C}}\) is called a ball-cut if there exists a ball \(B\subseteq K\) such that \(C=\{x\in K: x\, \square\, B\}\), where \(\square \in \{<,\leq \}\). 3 By what we have seen thus far, we may conclude:
Corollary 1. If \({\mathcal{K}}\) is a weakly o-minimal expansion of a valued field and both \(\Gamma\) and \(\boldsymbol{k}\) are o-minimal then every definable subset of \(K\) is a boolean combination of non-valuational cuts, ball-cuts and valuational cuts of type (iii), with stabilizers a closed ball.
In this section we prove the main theorem stated in the introduction. For that, we first refine the definition of uniform finiteness from the introduction; it is based on Johnson’s [1].
Definition 4. Let \({\mathcal{M}}\) be a sufficiently saturated structure.
A definable (by our convention, possibly interpretable) set \(X\) has uniform finiteness if for every definable family \(\{Y_t\subseteq X:t\in T\}\) there is a natural number \(n\in \mathbb{N}\) such that for all \(t\in T\): \(|Y_t|=\infty \iff |Y_t|>n.\)
Let \({\mathcal{D}}\) be a (not necessarily definable) collection of definable subsets of \(\bigcup_n M^n\). A definable set \(X\) is a set of imaginaries from \({\mathcal{D}}\) if there is a definable relation \(R\subseteq X\times M^n\) such that
\(x\mapsto R_x:=\{m\in M^n:(x,m)\in R\}\) is injective, and
for every \(x\in X\), \(R_x\in {\mathcal{D}}\).
We say that \({\mathcal{D}}\) has uniform finiteness if any definable set of imaginaries from \({\mathcal{D}}\) has uniform finiteness.
With this definition, Johnson’s theorem [1] states that \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness if
and only if the collection of all definable subsets of \(M=M^1\) has uniform finiteness. When \({\mathcal{M}}\) is weakly o-minimal we can reduce to an even simpler collection of sets, as
discussed below.
From now on we fix \({\mathcal{M}}=(M,+,<,\dots)\), a sufficiently saturated expansion of a weakly o-minimal ordered group.
Lemma 4. If the collection of initial segments of \(M\) has uniform finiteness then \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness.
Proof. By Johnson’s theorem, it suffices to show that the collection of definable subsets of \(M\) has uniform finiteness. To each definable \(X\subseteq M\) we let \(C(X)\) denote the (finite) collection of initial segments determined by its convex components (each convex set determines two initial segments). For a set of unary codes \(Y\) we let \(C(Y):=\bigcup\limits_{X\in Y} C(X)\).
Let us verify that \(Y\) is finite if and only if \(C(Y)\) is finite. Indeed, a set \(C(Y)\) of \(n\) initial segments gives rise to at most \(f(n)=2^{2^n}\) boolean combinations of these segments, thus there are at most \(f(n)\)-many definable sets \(X\subseteq M\) whose code belongs to \(Y\), namely \(|Y|\leq f(n)\). The other direction is obvious.
This implies that uniform finiteness of the collection of initial segments implies uniform finiteness of the collection of definable subsets of \(M\), as claimed. ◻
Our goal here is to establish which definable sets in \({\mathcal{M}}^{eq}\) necessarily have uniform finiteness. An obvious obstruction is the existence of a definable infinite discrete linear order. We start by noting that this is, in fact, the only obstacle to \({\mathcal{M}}^{eq}\) having uniform finiteness.
Lemma 5. \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) does not have uniform finiteness if and only if there exists a definable convex equivalence relation \(E\) on \(M\) such that \((M/E,<)\) infinite and discrete.
Proof. Infinite discrete linear orders fail uniform finiteness (as witnessed by the family of intervals). So interpretable discrete linear orders contradict uniform finiteness for \({\mathcal{M}}^{eq}\).
Conversely, if \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) does not have uniform finiteness, we prove that \({\mathcal{M}}\) interprets an infinite discrete linear order. By Lemma 4, failure of uniform finiteness appears in some definable family of sets of initial segments of \(M\). The initial segments are linearly ordered by inclusion. If \(\{Y_t\}_{t\in T}\) is a family of codes such that each \(Y_t\) consists of codes of initial segments, then each \(Y_t\) is linearly ordered, uniformly in \(t\). In particular, if \(Y_t\) is finite it is discrete with respect to its natural ordering.
Thus, if \(\{Y_t\}_{t\in T}\) fails uniform finiteness, by compactness and saturation, there is some \(t_0\in T\) such that \(Y_{t_0}\) is infinite and discretely ordered. Given such a set \(Y_{t_0}\), denote for \(x\in M\) the set \(C(x):=\{y\in Y_{t_0}: x\in y\}\). The equivalence relation \(E(x_1,x_2)\) on \(M\) given by \(C(x_1)=C(x_2)\) has the desired properties ◻
Lemma 4 reduces uniform finiteness in \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) to uniform finiteness of the collection of all initial segments of \(M\). The aim of this section is to reduce it further to the collection of all convex subgroups. We first prove this important observation.
Lemma 6. The collection of all non-valuational (possibly rational) definable cuts has uniform finiteness.
Proof. Assume that \(\{Y_t:t\in T\}\) is a definable family of sets of unary codes, each consisting of (codes of) non-valuational initial segments. For each \(t\in T\) denote \(I_t:=\{y\in Y_t: y \text{ is a code of a rational cut}\}\) and \(J_t\) its complement in \(Y_t\). By uniform finiteness in \(M\) there is \(n\in {\mathbb{N}}\) such that \(|I_t|<n\) if and only if \(|I_t|<\infty\). So we may assume that \(Y_t=J_t\) for all \(t\).
As in the proof of Lemma 5, compactness and saturation imply that if there are finite \(Y_t\) of unbounded size, then there is \(t_0\) such that \(Y_{t_0}\) is discrete with respect to inclusion. Let us see that this contradicts weak o-minimality.
Let \(\{y_i\}_{i\in {\mathbb{N}}}\subseteq Y_{t_0}\) be such that for all \(i\in {\mathbb{N}}\) \(y_{{i+1}}\) is an immediate successor in \(Y_{t_0}\) of \(y_{i}\). By compactness, there is \(\alpha>0\) in \(M\), such that for all \(i\) there are \(y_i<x_1<x_2<y_{i+1}\) with \(x_2-x_1>\alpha\). For \(y\in Y_{t_0}\) let \(s(y)\) denote the successor of \(y\) in \(Y_{t_0}\). By our choice of \(\alpha\), the set of \(y\in Y_{t_0}\), such that there are \(y<x_1<x_2<s(y)\) with \(x_2-x_1>\alpha\) is infinite (since it contains \(\{y_i\}_{i\in {\mathbb{N}}}\)).
For every code \(y\in Y_{t_0}\), let \(D_y\) be the convex subset of \(M\) given by \(\{x\in M: x<y< x+\alpha \}\). Since \(y\) is non-valuational, \(D_y\neq \emptyset\). Then \(\bigcup\{D_y: y\in Y_{t_0}\}\) is a definable set consisting of infinitely many convex components. This contradicts weak o-minimality. ◻
Lemma 7. Assume that the collection of definable convex subgroups of \(M\) has uniform finiteness. Then the collection of definable valuational initial segments of type (iii) has uniform finiteness.
Proof. Assume that we have a definable family \(\{Y_t:t\in T\}\), with each \(Y_t\) consisting of (codes of) initial segments, \(C\subseteq M\), with stabilizers \(H=G(C)\), such that \(\pi(C)\subseteq M/H\) is irrational non-valuational.
For each \(t\in T\), let \(H(t)=\{G(C):C\in Y_t\}\). Then \(H(t)\) is a set of convex subgroups of \((K,+)\) and \(|H(t)|\leq |Y_t|\). By the assumption, there is some \(n\in \mathbb{N}\), such that \(H(t)\) is finite if and only if \(|H(t)|\leq n\). Thus, since we want to bound the size of the finite \(Y_t\), we may assume that for every \(t\in T\), \(|H(t)|\leq n\). Since the groups in \(H(t)\) are linearly ordered by inclusion, we may partition each \(Y_t\) according to the associated group, and assume that for every \(t\in T\), \(|H(t)|=1\).
Assume now that the result fails. Then, there exists a strictly increasing sequence of natural numbers \(n_k\), and for every \(k\), some \(t_k\in T\), such that \(n_k=|Y_{t_k}|\). Thus, there is some \(t_0\) for which \(Y_{t_0}\) is infinite and discretely ordered, and there is some fixed \(H\) such that for all \(C\in Y_{t_0}\), \(G(C)=H\).
Consider now the family of non-valuational initial segments inside \(M/H\): \(\{\pi(C):C\in Y_{t_0}(H)\}\). Since each such \(C\) consists of cosets of \(H\), the map \(C\mapsto C/H\) is injective and order preserving on \(Y_{t_0}(H)\), hence it contains a relatively convex subsequence of order \(\omega\).
This in turn gives rise to a definable family of sets of irrational non-valuational cuts in the weakly o-minimal group (so dense) \(M/H\), whose finite sets are unbounded in size, contradicting Lemma 6. ◻
Lemma 8. Assume that the collection of definable convex subgroups of \(M\) has uniform finiteness. Then the collection of cosets of definable convex subgroups of \(M\) also has uniform finiteness.
Proof. Let \(\{Y_t: t\in T\}\) be a definable family with each element in \(Y_t\) a coset of a definable convex subgroup. We can order cosets of convex subgroups by: \(a+D<b+E\) if \(D<E\) and if \(D=E\) then \(a+D<b+D\). Since we want to bound the size of those finite \(Y_t\), we may assume that for all \(t\), \(Y_t\) is discretely ordered with respect to this order.
The definable map, mapping \(C\in Y_t\) to \(C-C\) maps each \(Y_t\) to the set \(S_t\) of definable convex subgroups associated to the cosets in \(Y_t\). By the assumption, there is some natural number \(n\) such that if \(|S_t|\) is finite then \(|S_t|<n\). Thus assume that \(S_t\) is finite for all \(t\).
By saturation we find some \(t_0\) for which \(Y_{t_0}\) is infinite and as \(|S_{t_0}|\) is finite there is some \(H\in S_{t_0}\) with infinitely many cosets as elements of \(Y_{t_0}\). This gives a definable, infinite, discrete subset of the weakly o-minimal \(M/H\), contradiction. ◻
Theorem 2. Let \({\mathcal{M}}=(M,+,<,\dots)\) be a sufficiently saturated weakly o-minimal expansion of an ordered group. If the collection of all definable convex subgroups of \(M\) has uniform finiteness then \({\mathcal{M}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness.
Proof. By Lemma 4, it suffices to show that the collection of initial segments of \(M\) has uniform finiteness.
In Lemma 2 we classified the different types of initial segments in \({\mathcal{M}}\). Given a definable family \(\{Y_t: t\in T\}\) where each \(Y_t\) is a set of (codes of) definable initial segments, we have to show that there is \(n\in {\mathbb{N}}\) such that for all \(t\in T\) such that \(|Y_t|<\infty\) it follows that \(|Y_t|< n\). For \(t\in T\) partition \(Y_t\) (definably, uniformly in \(t\)) into \(Y_t^{(i)}\), \(Y_t^{(ii)}\) and \(Y_t^{(iii)}\) according to whether \(y\in Y_t\) is a code for an initial segment of type (i), (ii) or (iii), respectively. It will suffice to show that there is \(n\) such that for all \(t\) if any one of \(Y_t^{(i)}\), \(Y_t^{(ii)}\) and \(Y_t^{(iii)}\) is finite, it has size at most \(n\).
For \(\{Y_t^{(i)}: t\in T\}\) and \(\{Y_t^{(iii)}: t\in T\}\) we use Lemma 6 and Lemma 7, respectively. For \(\{Y_t^{(ii)}: t\in T\}\), to each initial segment coded by an element \(y\in Y_t^{(ii)}\) corresponds a coset \(C(y)\) of a definable convex subgroup of \(M\). The map \(y\mapsto C(y)\) is at most two-to-one, so the conclusion follows from Lemma 8. ◻
Remark 3. Although Theorem 2 is stated for weakly o-minimal structures, the theorem also holds for weakly o-minimal definable sets in sufficiently saturated structures, i.e., definable ordered abelian groups \((D,+,<)\) for which every \({\mathcal{M}}\)-definable subset of \(D\) is a finite union of convex definable sets. Namely, we do not need to assume that \(D\) is stably embedded.
More concretely: Let \(D\) be a weakly o-minimal definable ordered abelian group in a sufficiently saturated structure \({\mathcal{M}}\). If the collection of all definable convex subgroups of \(D\) has uniform finiteness then every \({\mathcal{M}}\)-definable imaginary sort in \(D\) has uniform finiteness.
Let \({\mathcal{K}}=(K,+,\cdot,<,\dots)\) be a sufficiently saturated weakly o-minimal expansion of an ordered field.
Below, by a discrete linear order we mean one where every non-extremal element has an immediate successor and an immediate predecessor. The following is then straightforward.
Lemma 9. Let \((X,<)\) be a definable set with a definable linear order in some sufficiently saturated structure \({\mathcal{M}}\) and \(\{X_t\subseteq X:t\in T\}\) be a definable family. Let \((t_n)_{n<\omega}\) be such that \(X_{t_n}\) are finite and \(|X_{t_n}|\to \infty\).
For any definable subset \(Y\subseteq X\), there exists \(t_0\in T\) such that either \(X_{t_0}\cap Y\) or \(X_{t_0}\cap (X\setminus Y)\) are infinite and discretely ordered.
Theorem 4. Let \({\mathcal{K}}\) be as above. Then \({\mathcal{K}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness if and only if the collection of all definable convex valuation subrings of \({\mathcal{K}}\) has uniform finiteness.
Proof. Obviously if \({\mathcal{K}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness then the collection of all definable convex valuation subrings has uniform finiteness. We prove the reverse implication. We
first show the following:
\((\dagger)\) Uniform finiteness of the collection of definable convex valuation rings implies that for every definable valuation \(v\) on \(K\), the
collection of all definable convex subgroups of \(\Gamma_v\) has uniform finiteness.
Indeed, by Theorem 2 (and Remark 3) it suffices to show that the collection of definable convex subgroups of \(\Gamma\) has uniform finiteness, and since for every convex subgroup of \(\Gamma\) there corresponds a unique convex valuation subring of \(K\) (the corresponding coarsening of \({\mathcal{O}}_v\)), the result follows.
To prove the theorem, we use Theorem 2 again. Assume that \(\{Y_t:t\in T\}\) is a definable family, with each \(Y_t\) a definable set of (codes of) definable (convex) subgroups of \((K,+)\). The collection of all convex subgroups is linearly ordered by inclusion, and since we want to bound the size of those \(Y_t\) which are finite, we may assume that for all \(t\), \(Y_t\) is discretely ordered by inclusion.
We assume towards a contradiction that there is no \(n\) which bounds the size of all finite \(Y_t\). By saturation there exists \(t_0\), such that \(Y_{t_0}\) is infinite and by our assumption discretely ordered by inclusion.
For each \(H\in \bigcup_{t\in T} Y_t\), let \(R(H)=\{a\in K: aH\subseteq H\}\) be the corresponding (convex) valuation ring. The family \(\{R(H):H\in \bigcup_{t\in T}Y_{t}\}\) is a uniformly definable chain (with respect to inclusion) of valuation rings and therefore its intersection \({\mathcal{O}}\) is a definable convex valuation ring. Let \(\Gamma\) be the corresponding value group and \(v:K^\times\to \Gamma\) the corresponding valuation.
For every convex group \(H\), let \(v(H)=\{v(a):a\in H\}\), a final segment of \(\Gamma\). Each such \(v(H)\) either has a minimal element in \(\Gamma\) or not. By Lemma 9, after possibly replacing \(Y_{t_0}\), either all the elements of \(Y_{t_0}\) have a minimum or none of them have. We claim that in each of the two cases \(H\mapsto v(H)\) is injective. Assuming this, we get that \(\{v(H):H\in Y_{t_0}\}\) is a discrete family of subsets of \(\Gamma\), contradicting \((\dagger)\) (and Theorem 2, Remark 3). So we prove injectivity in each of these cases.
Assume first that no \(v(H)\), \(H\in Y_{t_0}\) has a minimum. Note that in that case we have \(H=v^{-1}(v(H))\), for assume that \(v(a)\in v(H)\) for some \(a\in K\), which we may assume to be strictly positive. Then there is \(b\in H\), \(b>0\), with \(v(b)<v(a)\). It follows that \(a<b\), so by convexity \(a\in H\). Thus, the map \(H\mapsto v(H)\) is injective.
Now assume that for all \(H\in Y_{t_0}\), \(v(H)\) has a minimum, denoted by \(\gamma_H\). Note that if \(\gamma_H=r\) (and recall \(H\) is convex) then \(B_{>r}(0)\subsetneq H\subseteq B_{\geq r}(0)\) and therefore \(R(H)\subseteq {\mathcal{O}}\). By minimality of \({\mathcal{O}}\), \(R(H)={\mathcal{O}}\) and hence \(H=B_{\geq r}(0)\). Hence \(H\mapsto v(H)\) is injective and we are done. ◻
Remark 5. As in Remark 3, the theorem also holds for weakly o-minimal definable fields (which are not necessarily stably embedded). Namely, let \(K\) be weakly o-minimal definable field in a sufficiently saturated structure \({\mathcal{M}}\). If the collection of all definable convex valuation subrings of \(K\) has uniform finiteness then every \({\mathcal{M}}\)-definable imaginary sort in \(K\) has uniform finiteness.
In [4] we proved that if \({\mathcal{K}}\) is power bounded \(T\)-convex then \({\mathcal{K}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness. The following corollary generalizes this result:
Corollary 2. The following are equivalent for \({\mathcal{K}}\) as before:
\({\mathcal{K}}^{\mathop{\mathrm{eq}}}\) has uniform finiteness.
For any definable valuation \(v\) on \(K\) (possibly trivial), the collection of all definable convex subgroups of \(\Gamma_v\) and \(\boldsymbol{k}_v\) has uniform finiteness.
There exists a definable valuation \(v\) on \(K\) (possibly trivial) such that the collection of all definable convex subgroups of \(\Gamma_v\) and \(\boldsymbol{k}_v\) has uniform finiteness.
For any definable valuation \(v\) on \(K\) (possibly trivial), the collection of all definable convex subgroups of \(\Gamma_v\) has uniform finiteness.
For any definable valuation \(v\) on \(K\) (possibly trivial), the collection of all definable convex subgroups of \(\boldsymbol{k}_v\) has uniform finiteness.
Proof. \((1)\) easily implies the rest, (2) easily implies (3) by taking the trivial valuation and similarly (5) implies (1). We show the rest.
\((3)\implies (1)\). Let \(v\) be the given valuation, with valuation ring \({\mathcal{O}}_v\). If \(v\) is trivial then the collection of definable convex subgroups of \(K\) has uniform finiteness so we are done by Theorem 2. So assume that \(v\) is not trivial.
We use Theorem 4 and show that the collection of definable convex valuation subrings has uniform finiteness. Let \(\{X_t: t\in T\}\) be a definable family of definable convex valuation subrings and let \((t_n)_n\) be such that \(|X_{t_n}|\to\infty\). For any \(t\in T\), and \(R\in X_t\) either \(R\subseteq {\mathcal{O}}_v\) or \({\mathcal{O}}_v\subseteq R\). Thus we may assume that in addition either \(R\subseteq {\mathcal{O}}_v\) for every \(R\in X_{t_n}\) and natural number \(n\) or the other inclusion for all \(R\in X_{t_n}\) and natural number \(n\).
If \(R\subseteq {\mathcal{O}}_v\) for every such \(R\), then the induced valuation rings on \(\boldsymbol{k}_v\) form a family of definable convex subgroups of \(\boldsymbol{k}_v\) contradicting \((3)\). If \({\mathcal{O}}_v\subseteq R\) for every such \(R\) then the induced definable convex subgroups in \(\Gamma_v\) contradict \((3)\).
\((4)\implies (1)\). This is a repeat of the proof of Theorem 4, see \((\dagger)\). ◻
We still do not know the answer to the following:
Question 6. Let \({\mathcal{K}}\) be a T-convex valued field (not necessarily power-bounded). Does \({\mathcal{K}}^{\mathop{\mathrm{eq}}}\) have uniform finiteness?
In this section we give an example of a weakly o-minimal expansion of a valued field which does not have uniform finiteness in its imaginary sort. To build such an example, we start with a weakly o-minimal expansion \(\Gamma\) of an ordered abelian group, admitting an infinite discrete imaginary sort (see, e.g., [4]). We then consider the real closed valued field \(\mathbb{R}((t^\Gamma))\) together with an expansion of RCVF by the extra structure on \(\Gamma\), and show that it is weakly o-minimal. The key observation is a weakly o-minimal variant of known resplendence results for henselian valued fields. Explicitly, we show that weakly o-minimal expansions of the value group and the residue field of a (pure) real closed valued field preserve weak o-minimality of the valued field sort.
We first recall: Let \({\mathcal{M}}\) be a \(|T|^+\)-saturated structure in a language \({\mathcal{L}}\) and \(T=\mathrm{Th}(M)\) its theory. Assume that \(D\) is a stably embedded sort in \({\mathcal{M}}\). Let \(\mathcal{D}\) be \(D\) with its \(\emptyset\)-induced structure and \(\widetilde{\mathcal{D}}\) an arbitrary expansion of \(\mathcal{D}\). It is folklore (see, e.g., [9] for more details) that letting \(\widetilde{L}\), \(\widetilde{{\mathcal{M}}}\) be the corresponding expansions, then \(D\) is stably embedded in \(\widetilde{M}\) and the \(\widetilde{L}\) induced structure on \(D\) is \(\widetilde{\mathcal{D}}\). Below we apply this without further mention to expansions of the RV-sort of (pure) real closed valued fields.
Let \({\mathcal{K}}=(K,\cdot,+,0,1,{\mathcal{O}})\) be a sufficiently saturated model of RCVF. We denote by \(\mathbf{RV}=(\mathrm{RV},\cdot,0,1,\oplus)\) the interpretable RV-sort, augmented by an element \(0\) for which we set \(\mathop{\mathrm{rv}}(0)=0\) and the ternary relation \[\oplus(x,y,z):=(\exists a,b)(\mathop{\mathrm{rv}}(a)=x\land \mathop{\mathrm{rv}}(b)=y\land \mathop{\mathrm{rv}}(a+b)=z).\]
Fact 7. For a fixed \(x,y\in \mathrm{RV}\) the following are equivalent:
There is a unique \(z\in \mathrm{RV}\) such that \(\oplus(x,y,z)\).
\(x=0\) or \(y=0\) or \(x\neq \mathop{\mathrm{rv}}(-1)y\).
\(x=0\) or \(y=0\) or \(v(x+y)=\min\{v(x),v(y)\}\).
The above allows us to define a partial function:
Notation 8. For \(x, y\in \mathrm{RV}\) we write \(x\oplus y\) for the unique element \(z\in \mathrm{RV}\) satisfying \(\oplus(x,y,z)\), if such a \(z\) is unique. If such a \(z\) is not unique \(x\oplus y\) is undefined.
It is well known that if \((K,v)\) is a henselian valued field of equi-characteristic \(0\) then in the structure \({\mathcal{K}}=(K,\mathbf{RV},\mathop{\mathrm{rv}})\), every formula is equivalent to one without field-quantifiers (i.e., quantifiers ranging over variables from the valued field sort), [10]. As a direct consequence, \(\mathbf{RV}\) is stably embedded, see for example [11]. These two results hold resplendently, i.e. they remain valid under arbitrary expansions of the \(\mathbf{RV}\)-sort, see the discussion before [10]. As a consequence:
Fact 9. [10] Let \(\widetilde{{\mathcal{K}}}\) be \({\mathcal{K}}\) together with some added structure on \(\mathbf{RV}\). Any definable set \(X\subseteq K\) is of the form \(\{x\in K: (\mathop{\mathrm{rv}}(x-a_1),\dots, \mathop{\mathrm{rv}}(x-a_n))\in S\}\) for some definable subset \(S\subseteq \mathrm{RV}^n\) and \(a_1,\dots,a_n\in K\).
Since \({\mathcal{K}}\) is real closed, the order on \(K\) is definable, and therefore so is the induced order on \(\mathrm{RV}\). Therefore, there is no harm expanding both structures by these orders. From now we let \({\mathcal{K}}=(K,\cdot,+,0,1,<,{\mathcal{O}})\) and \(\mathbf{RV}=(\mathrm{RV},\cdot,0,1,<,\oplus)\).
Lemma 10. Let \(\widetilde{K}\) be \(K\) together with a weakly o-minimal enrichment of \(\mathbf{RV}\) which we denote by \(\widetilde{\mathbf{RV}}\). Then \(\widetilde{K}\) is still weakly o-minimal.
Proof. Let \(X\subseteq K\) be a definable set in \(\widetilde{K}\), by Fact 9 it is of the form \[\{x\in K: (\mathop{\mathrm{rv}}(x-a_1),\dots, \mathop{\mathrm{rv}}(x-a_n))\in S\},\] for some definable subset \(S\subseteq \mathrm{RV}^n\).
For any \(1\leq i\leq n\) let \(U_{i}=\{x\in K: \bigwedge_j v(x-a_{i})\geq v(x-a_j)\}\). This is a finite union of convex sets and \(K=\bigcup_i U_i\).
Let \(D_{i}=\{y\in \mathrm{RV}: (y\oplus \mathop{\mathrm{rv}}(a_{i}-a_1),\dots, y\oplus \mathop{\mathrm{rv}}(a_{i}-a_n))\in S\}\). Let us verify that \[X=\bigcup_{i} U_{i}\cap \{x\in K: \mathop{\mathrm{rv}}(x-a_{i})\in D_{i}\}.\]
Right to left inclusion. Let \(x\in U_i\) with \(\mathop{\mathrm{rv}}(x-a_i)\in D_i\). In particular, \(\mathop{\mathrm{rv}}(x-a_i)\oplus \mathop{\mathrm{rv}}(a_i-a_j)\) is defined for all \(j\) and thus is necessarily equal to \(\mathop{\mathrm{rv}}(x-a_j)\). This gives that \((\mathop{\mathrm{rv}}(x-a_1),\dots, \mathop{\mathrm{rv}}(x-a_n))\in S\).
Left to right inclusion. Let \(x\in X\), then \(x\in U_{i}\) for some \(i\). To show that \(\mathop{\mathrm{rv}}(x-a_{i})\in D_{i}\) it is enough to show that for all \(j\), \(\mathop{\mathrm{rv}}(x-a_{i})\oplus \mathop{\mathrm{rv}}(a_i-a_j)\) is defined, for then as above it is equal to \(\mathop{\mathrm{rv}}(x-a_j)\) and the result follows since \(x\in X.\)
We show that \(\mathop{\mathrm{rv}}(x-a_{i})\oplus \mathop{\mathrm{rv}}(a_i-a_j)\) is defined using Fact 7. Assume that \(\mathop{\mathrm{rv}}(x-a_{i})\oplus \mathop{\mathrm{rv}}(a_i-a_j)\) is undefined for some \(j\): necessarily \(x- a_{i}\neq 0\), \(a_j-a_i\neq 0\) and \(\mathop{\mathrm{rv}}(x-a_{i})=\mathop{\mathrm{rv}}(a_{j}-a_i)\). By [10], \(v(a_j-a_i-(x-a_i))>v(x-a_i)\), i.e. \(v(x-a_j)>v(x-a_{i})\) contradicting \(x\in U_{i}\).
By weak o-minimality of \(\widetilde{\mathbf{RV}}\), \(D_{i}\) is a finite union of convex sets, thus \(\{x\in K:\mathop{\mathrm{rv}}(x-a_{i})\in D_{i}\}\) is also a finite union of convex sets. We conclude that \(X\) is a finite union of convex sets. ◻
Transferring weak o-minimality from the \(\mathrm{RV}\) sort to the valued field is the first step, the second is to transfer from \(\Gamma\) and \(\boldsymbol{k}\) to \(\mathrm{RV}\). For that we will need to deal with short exact sequences. The following is proved in [12] but taken in this form from [13]. Note that as \(A\) is divisible, it is pure in \(B\).
Fact 10. Let \({\mathcal{M}}=(0\to A\xrightarrow[]{\iota} B\xrightarrow{v}C\to 0)\) be a short exact sequence of abelian groups, with \(A\) divisible, allowing expansion of the group structure on \(A\) and \(C\).
Then \(A\) and \(C\) are stably embedded in \({\mathcal{M}}\) and every definable subset of \(B\) is a boolean combination of definable sets of the following forms:
\(\{x\in B: (v(t_1(x)),\dots v(t_n(x))\in V\}\), where the \(t_i\) are terms in the group language and \(V\) is a definable subset of \(C^n\).
\(\{x\in B: (\rho(t_1(x)),\dots \rho(t_n(x))\in U\}\), where the \(t_i\) are terms in the group language, \(U\) is a definable subset of \(A^n\) and \(\rho(a)=\iota^{-1}(a)\) if \(a\in \iota(A)\) and \(0\), otherwise.
Using this result we prove a general weak o-minimality transfer result.
Proposition 11. Let \({\mathcal{M}}=(0\to A\xrightarrow[]{\iota} B\xrightarrow{v}C\to 0)\) be a short exact sequence of abelian groups, with \(A\) divisible.
Let \(\widetilde{{\mathcal{M}}}=((B,+,0),\widetilde{A}=(A,+,0<,\dots),\widetilde{C}=(C,+,0,<,\dots),\iota,v)\) be \({\mathcal{M}}\) together with weakly o-minimal expansions of \(A\) and \(C\). Then,
There is a unique (definable) order \(<\) on \(B\) such that with respect to this order, \(\iota\) and \(v\) are order preserving homomorphisms, \(\iota(A)\) is a convex subgroup and \((B,+,<)\) is an ordered abelian group.
The sort \((B,+,0,<)\) (together with its induced structure) is weakly o-minimal.
Proof. (1) The order on \(B\) defined by \(b>0\) if and only if \(v(b)>0\) or \(v(b)=0\) and \(\iota^{-1}(b)>0\), is the unique order on \(B\) rendering it an ordered abelian group such that \(\iota\) and \(v\) are order preserving.
(2) Applying Fact 10, every definable subset of \(B\) is a boolean combination of definable sets of the form
\(\{x\in B: (v(a_1+m_1x),\dots, v(a_k+m_kx))\in V\}\) for \(a_i\in B\), \(m_i\) an integer and \(V\) a definable set in \(\widetilde{C}\) and
\(\{x\in B:(\rho(a_1+m_1x),\dots,\rho(a_k+m_kx))\in U\}\) for \(a_i\in B\), \(m_i\) an integer and \(U\) a definable set \(\widetilde{A}\), where \(\rho(a)=\iota^{-1}(a)\) if \(a\in \iota(A)\) and \(0\in A\) otherwise.
It suffices to show that each kind of these definable sets is a finite union of convex sets.
Let \(X\subseteq B\) be a definable set of the first kind. Written differently, it is the same as \(v(x)\in \{y: (v(a_1)+m_1y,\dots, v(a_k)+m_ky)\in V\}\). The latter is a definable subset of \(\widetilde{C}\) so a finite union of convex sets by weak o-minimality. As \(v\) is order preserving, \(X\) is a finite union of convex subsets of \(B\).
Let \(X\subseteq B\) be a definable set of the second kind. For each \(J\subseteq \{1,\dots,k\}\) let \(P_J=\{x\in B: a_i+m_ix\in \iota(A)\iff i\in J\}\). It is a finite covering of \(B\) so it is enough to show that \(X\cap P_J\) is a finite union of convex sets.
Claim 1. Each \(P_J\) is a finite union of convex sets.
Note that \(a_i+m_ix\in \iota(A)\iff v(a_i)+m_iv(x)=0\). Setting \(Y=\{y\in C:\bigwedge_{i\in J} m_iy=-v(a_i)\wedge \bigwedge_{i\notin J}m_iy\neq-v(a_i)\}\). This is a finite union of convex sets and since \(P_J=\{x\in B:v(x)\in Y\}\) it is a finite union of convex sets as well.
If \(m_i=0\) for every \(i\in J\) then \(X\cap P_J=\{x\in P_J: (\rho(a_1'),\dots,\rho(a_k'))\in U\}\), where \(a_i'=a_i\) if \(i\in J\) and \(a'_i=0\) otherwise. So either empty or all of \(P_J\), and thus either way a finite union of convex sets.
As in the proof of the claim, if there exists \(i\in J\) with \(m_i\neq 0\) then either \(P_J=\emptyset\) and we are done or \(P_J\subseteq v^{-1}(\gamma)\) non-empty for some \(\gamma\in C\). Fix an element \(b\in v^{-1}(\gamma)\), and let \(c_i\in A\) be such that \(\iota(c_i):=a_i+m_ib\).
Claim 2. \(X\cap P_J=b+\iota(Y_J)\), for \(Y_J=\{y\in A:(d_1(y),\dots,d_k(y))\in U\}\) where \(d_i(y)=c_i+m_iy\) if \(i\in J\) and \(0\) otherwise.
Let \(x\in X\cap P_J\) so \(x=b+\iota(t)\) for some \(t\in A\). We will show that \(t\in Y_J\). Indeed, for every \(i\in J\), \(d_i(t)=c_i+m_it=\iota^{-1}(a_i+m_ib+m_i\iota(t))=\iota^{-1}(a_i+m_ix)=\rho(a_i+m_ix)\). The other direction is similar.
As \(Y_J\) is definable in \(A\), it is a finite union of convex sets and thus so is \(X\cap P_J\), as required. ◻
We wish to apply this result to a short exact sequence coming from \(\mathrm{RV}, \boldsymbol{k}\) and \(\Gamma\).
Let \(\mathbf{RV}_{\boldsymbol{k},\Gamma}=((\mathrm{RV},\cdot, 0,1),\boldsymbol{k}=(\boldsymbol{k},\cdot,+,0,1),\mathbf{\Gamma}=(\Gamma,+,0,<,\infty),\iota,v)\) be the three sorted structure associated together with a \(0\) element to \(\boldsymbol{k}\) and \(\mathrm{RV}\) and an \(\infty\) element to \(\Gamma\). Whenever \((K,v)\) is a henselian equi-characteristic \(0\) valued field, both \(\Gamma\) and \(\boldsymbol{k}\) are stably embedded inside \(\mathbf{RV}_{\boldsymbol{k},\Gamma}\), as a pure ordered abelian group and as a pure field, respectively. As noted in [14] the ternary relation \(\oplus\) on \(\mathrm{RV}\) is also definable in \(\mathbf{RV}_{\boldsymbol{k}, \Gamma}\).
By Proposition 11(1) the induced order on \(\mathrm{RV}\) is definable.
Lemma 11. Let \(\widetilde{\mathbf{RV}_{\boldsymbol{k},\Gamma}}=((\mathrm{RV},\cdot, 0,1,<),\widetilde{\boldsymbol{k}}=(\boldsymbol{k},\cdot,+,0,1,<,\dots),\widetilde{\mathbf{\Gamma}}=(\Gamma,+,0,<,\infty,\dots),\iota,v)\) be \(\mathbf{RV}_{\boldsymbol{k},\Gamma}\) together with an enrichment of \(\boldsymbol{k}\) and \(\Gamma\) still rendering them weakly o-minimal. Then the \(\mathrm{RV}\) sort of \(\widetilde{\mathbf{RV}_{\boldsymbol{k},\Gamma}}\) is weakly o-minimal.
Proof. Let \(\mathrm{RV}_{>0}\) be \(\mathop{\mathrm{rv}}(K_{>0})\). It is enough to show that \(\mathrm{RV}_{>0}\) is weakly o-minimal (there is a definable bijection between \(\mathrm{RV}_{>0}\) and \(\mathrm{RV}_{<0}\)). Note that \(\mathrm{RV}_{>0}\) is a subgroup of \(\mathrm{RV}\). Also, \(\boldsymbol{k}_{>0}\) is a multiplicative subgroup and \(\mathrm{RV}_{>0}/\boldsymbol{k}_{>0}\cong \Gamma\).
We have a short exact sequence \(1\to \boldsymbol{k}_{>0}\to \mathrm{RV}_{>0}\to \Gamma\to 0\), \(\boldsymbol{k}_{>0}\) is divisible because \(\boldsymbol{k}\) is real closed.
We are now in the situation of Proposition 11 where on \(\Gamma\) we take the reverse order in order to make \(v\) order preserving. ◻
Theorem 12. Let \({\mathcal{K}}\) be a pure real closed valued field and let \(\widetilde{K}\) be the structure \({\mathcal{K}}\) together with weakly o-minimal expansions of \(\boldsymbol{k}\) and \(\Gamma\). Then \(\widetilde{K}\) is weakly o-minimal as well.
Proof. The \(\widetilde{\mathbf{RV}_{\boldsymbol{k},\Gamma}}\) reduct is weakly o-minimal by Lemma 11. As \(\oplus\) is definable in this structure (see [14]), \(\widetilde{\mathbf{RV}}\) is a reduct of \(\widetilde{\mathbf{RV}_{\boldsymbol{k},\Gamma}}\). By Lemma 10, \(\widetilde{K}\) is weakly o-minimal. ◻
Corollary 3. There exists a weakly o-minimal valued field \({\mathcal{K}}\) for which \({\mathcal{K}}^{\mathop{\mathrm{eq}}}\) does not have uniform finiteness.
Proof. In [4], we constructed a weakly o-minimal ordered abelian group \(\mathcal{Q}=(Q,+,0,<)\) and a convex equivalence relation \(E\) on \(Q\) such that \(\widetilde{\mathcal{Q}}=(Q,+,0,<,E)\) is still weakly o-minimal but \(Q/E\) is discrete. There is no harm in assuming that \(\mathcal{Q}\) (and \(\widetilde{\mathcal{Q}}\)) is \(|T|^+\)-saturated.
Let \({\mathcal{K}}\) be a \(|T|^+\)-saturated pure real closed valued field with value group \(\mathcal{Q}\), e.g. \(\mathcal{R}((t^{\mathcal{Q}}))\), for some \(\mathbb{R}\prec \mathcal{R}\). By Theorem 12, the structure \(\widetilde{\mathcal{K}}\) obtained from \(\mathcal{K}\) after expanding the value group to \(\widetilde{\mathcal{Q}}\) is still weakly o-minimal. As \(\widetilde{\mathcal{Q}}^{\mathop{\mathrm{eq}}}\) does not have uniform finiteness, \(\widetilde{\mathcal{K}}^{\mathop{\mathrm{eq}}}\) does not have it as well. ◻
The first and second authors were partially supported by ISF grant No. 555/21. The third author was supported by ISF grant No. 290/19.↩︎
By [2] the convexity assumption is redundant. We keep it here and throughout for improved readability.↩︎
In [5] the notion of ball-cut is defined similarly, allowing \(B\) to be any convex subgroup of \((K,+)\).↩︎