Stable formulas in ordered structures

In recent years, we have seen rapid development of the neostability program which aims to extend the ideas of stability to other settings. Efforts have been made toward investigating weaker notions (NIP, simplicity, NSOP₁, NTP₂, etc), considering the stable components (stably dominated types, stables formulas, etc) in unstable theories, or a mix and match between these themes; see [16] for the relevant definitions (e.g. Section 8.2 for the stability related notions). In this paper, we are interested in stable formulas—also called “stable relations”—in unstable theories, in other words, the local stability of these theories. This is an old direction which nevertheless continues to hold relevance with recent applications in combinatorics ([5], [4], [3], [6], [14]). Stable formulas is related to thorn-forking [7] and is the subject of stable forking conjecture for simple theories ([12], [13]). Surprisingly, not much attention have been paid to the down-to-earth problem of classifying stable formulas in frequently seen examples of unstable theory. Our goal here is to fill this gap for the most obvious unstable structures, those that involves an ordering.

We know that ordering gives us unstability. The example below provides us with a slightly more general situation where we have unstability, namely, the formula defines a “large set” with a “slope”. It also points out why “large” and having a “slope” is necessary.

For the rest of the paper, let $T$ be either the theory of dense linear orderings in $L=\{<\}$ (DLO), the theory of divisible ordered abelian groups in $L=\{+,0,<\}$ (DOAG), or the theory of real closed fields in $L=\{+,-,\cdot,0,1,<\}$ (RCF), let $\mathscr{M}$ be a model of $T$ with underlying set $M$, let $\varphi(x;y)$ be an $L(M)$-formula, and let $\dim$ denotes the o-minimal dimension of $T$; see [17] for the basic definition and results. When we say that $\varphi(x;y)$ is stable, we implicitly assume that stability is with respect to the pair $(x;y)$. We say that $\varphi(x;y)$ is rectangular if $\varphi(x;y)$ is $T$-equivalent to $\psi(x)\wedge\theta(y)$ with $\psi(x)$ and $\theta(y)$ being $L(M)$-formulas. It is easy to see that rectangular formulas are stable, and so are their boolean combinations. Propostion 1.2, which combines the later Propostion 3.6 and Proposition 3.8, tell us that Example 1.1 essentially points us in the right direction. Note that the condition $\dim\varphi(\mathscr{M})=|x|+|y|$ is the precise version of what we meant by “large”, and the notion of rectangular formula makes precise the idea of “having no slope”.

Proposition 1.2 is not sufficient for the purpose of classifying stable formulas in $T$ as it says nothing when $\dim\varphi(\mathscr{M})<|x|+|y|$. We say that $\varphi(x;y)$ is order-free if $\varphi(x;y)$ is equivalent over $T$ to quantifier-free $L(M)$-formulas which do not contain $<$. Order-free formulas form another natural class of stables formulas. It is possible to have $\varphi(x;y)$ order-free with $\dim\varphi(\mathscr{M})<|x|+|y|$. If we consider a conjunction of a rectangular formula and an order-free formula, we still obtain a stable formula. Formulas of this form are said to be special stable. A natural guess would be that every stable $L(M)$-formula is equivalent over $T$ to a finite union of special stable formulas. In Theorem 4.1, we show this is the case when the theory under consideration is either the theory of Dense Linear Orders without Endpoints or the theory of Divisible Ordered Abelian Groups:

We expect that a result in this line can be obtained for more general linear orderings in $L=\{<\}$ and linearly ordered abelian groups in $L=\{<,+,0\}$. However, we do not address this question in this paper.

Now, let us move to theory of Real Closed Fields and start with the following example, where we can see that the above description breaks down.

In Example 1.4, the obstacle in expressing $\varphi(x;y)$ as a finite disjunction of special stable formulas comes from the singularity at $(0,0)$ of the curve in the picture. This brings us to the idea of using blowing up, or more precisely, resolution of singularities. It turns out that this is essentially the only obstruction when the stable formulas considered are parameter-free, and every such formula is equivalent over $\operatorname{RCF}$ to disjunctions of special stable formulas up to a certain kind of isomorphism. To be more precise, in Section 2.3, we will define the notion of order-free isomorphism between a relation defined by a formula $\varphi(x;y)$ and a relation defined by a formula $\varphi^{\prime}(x^{\prime};y^{\prime})$. This notion generalizes birational equivalence with a catch, namely, the division of variables must be respected. We obtain in Section 5 our main result in this paper:

One could ask whether the statement of the above theorem also holds for a stable formula with parameters, and we think the answer should be yes. Our current proof for Theorem 1.5, in fact, goes through for the more general case when $\varphi(x;y)$ is a formula with parameters over an Archimedean subfield of $\mathscr{M}$. However, the proof involves topological compactness. It is unclear if this technique transfers to the most general case when the infinite/non-Archimedean parameters occur in $\varphi(x;y)$. In a parallel direction, some natural subsequent questions could be the classification of stable formulas in $\operatorname{\mathbb{R}}_{\mathrm{exp}}$ and $\operatorname{\mathbb{R}}_{\mathrm{an}}$, multi-ordered fields ([11]) or $\mathbb{Q}_{p}$.

We continue to let $T$ be $\text{DLO}=\operatorname{Th}(\operatorname{\mathbb{R}},<)$, $\text{DOAG}=\operatorname{Th}(\operatorname{\mathbb{R}},+,0,<)$, or $\mathrm{RCF}=\text{Th}(\operatorname{\mathbb{R}},+,-,\cdot,0,1,<)$. Let $\boldsymbol{\mathscr{M}}$ be a model of $T$ with underlying set $\boldsymbol{M}$, let $D\subseteq\boldsymbol{M}^{(x;y)}$ be $L(\boldsymbol{M})$-definable, and let $\dim$ denote the o-minimal dimension of $T$. Recall that $D\subseteq\boldsymbol{M}^{(x;y)}$ is unstable (with respect to the variable division $(x;y)$) if some (every) formula defining $D$ is unstable. More explicitly, such $D$ is unstable if there is $(a_{i},b_{j})_{i<\omega,j<\omega}$ such that $(a_{i},b_{j})\in D$ if and only if $i\leqslant j$. We call such $(a_{i},b_{j})_{i<\omega,j<\omega}$ an unstable witness of $D$. As $T$ is complete, classifying stable formulas over $T$ is the same as classifying stable sets over $\boldsymbol{\mathscr{M}}$.

Toward classifying stable sets, we consider several classes of sets which are obviously stable. We say that $D$ is order-free definable if it is defined by an order-free formula as in the introduction. This property is equivalent to being quantifier-free definable in the reduct of $\boldsymbol{\mathscr{M}}$ without the linear order. A typical example of an order-free definable set is an algebraic set in $\boldsymbol{\mathscr{M}}\models\mathrm{RCF}$. We say a function $f:X\to Y$ is order-free definable if the graph of $f$ is an order-free definable. Abusing notation, the restriction of $f$ to a subset $Z$ of $X$ is also called order-free definable.

A definable set is rectangular if it is defined by a rectangular formula as in the introduction. It is easy to see that $D$ is rectangular if and only if $D=X\times Y$ with $X\subseteq\boldsymbol{M}^{x}$ and $Y\subseteq\boldsymbol{M}^{y}$ definable in $\boldsymbol{\mathscr{M}}$. Clearly, order-free definable sets and rectangular sets are stable. A special kind of stable sets which generalizes both order-free definable sets and rectangular set is given in the following definition.

The following easy fact will be later used in the proof of Theorem 5.2.

It is well-known that the stability of formulas is preserved under taking Boolean combinations [15, Lemma 2.1], so Boolean combinations of special stable sets are stable. As it was noted in Example 1.4 from the introduction, not all stable sets are Boolean combinations of special stable sets. We introduce the following notion to remedy this situation.

Because of the preservation of stability under Boolean combinations, one can deduce the following easy fact: if the definable sets $D_{1},\ldots,D_{n}\subseteq\boldsymbol{M}^{(x;y)}$ are stable and $D\subseteq D_{1}\cup\ldots\cup D_{n}$ is definable, then $D$ is stable if and only if $D\cap D_{i}$ is stable for each $i\leqslant n$. This easy fact leads to the following, more important, observation:

Our strategy of classifying stable sets $D$ relies on an induction on $\dim(D)$. The goal of this section is to obtain a description of stable sets $D\subseteq\boldsymbol{M}^{(x;y)}$ with $\dim(D)=|x|+|y|$. See Proposition 3.6 and 3.8 for such a description. Eventually, they will enable us to carry out the induction.

Next, we will obtain a description for large stable sets up to sets of smaller dimension. A definable set $E$ is definably connected by codimension $2$ if for all definable $E^{\prime}\subseteq E$ with $\dim(E\setminus E^{\prime})\leq\dim E-2$, the set $E^{\prime}$ is definable connected in the o-minimal structure $\boldsymbol{\mathscr{M}}$. Before moving forward, let us recall a few basic facts from o-minimal structures, see [17] for details on this subject.

Suppose $X$ is a definable subset of $\boldsymbol{M}^{x}$. For $a\in\boldsymbol{M}^{x}$, the local dimension at $a$ of $X$ is defined as

where the “$\dim(U\cap X)$” refers to the o-minimal dimension and “open” refers to being open in the topology generated by open boxes defined by the ordering (here and in the rest of this paper an open box is just the Cartesian product of open intervals - similarly for a closed box). Suppose $E$ is an order-free definable subset of $\boldsymbol{M}^{x}$ with nonempty $X\subseteq E$. As usual, the closure $\mathrm{cl}_{E}(X)$ of $X$ is the smallest closed subsets in $E$ containing $X$, the interior $\mathrm{int}_{E}(X)$ of $X$ in $E$ is the largest open subset of $E$ that $X$ contains, and the boundary $\mathrm{bd}_{E}(X)$is defined as $\mathrm{cl}_{E}(X)\setminus\mathrm{int}_{E}(X)$. The essence of $X$ in $E$ is the set

The essential interior of $X$ in $E$ is the set

The essential boundary of $X$ in $E$ is the set

Note that $\operatorname{ess}(X)=\operatorname{\operatorname{int}^{\operatorname{ess}}}_{E}(X)\,\cup\,\operatorname{\operatorname{bd}^{\operatorname{ess}}}_{E}(X)$. We will omit “$E$” if $E=\boldsymbol{M}^{n}$ for some natural number $n$.

The following example illustrates the above notions. This was pointed out to us by the anonymous referee leading to a correction of a mistake in the earlier proof of Lemma 3.5, where a wrong definition of essence was used.

We collect some well known facts about o-minimal theories.

Statement (1) can be found in [17, Theorem 2.9, p.130]. Statement (2) follows easily from [17, Chapter 4], and (3) is a consequence of (2). Item (4) can be proven easily using cell decomposition. In fact, a much more general phenomenon is true; see [8, Corollary 1.6].

The following fact is well-known. A proof can be found, for example, from [10].

The next lemma gives us a model-theoretic counterpart of Fact 3.3.

The next lemma is the key ingredient in proving Proposition 3.6.

We now prove the main result of this section.

In the remainder of this section, we will supose that $T$ is either DLO or DOAG and prove an analogue of Proposition 3.6. In a model of RCF, we have two natural topologies, namely, the Zariski topology and Euclidean topology, with the former coarser than the latter. In the same fashion, we can define on $\boldsymbol{M}^{x}$ a topology coarser than the Euclidean topology. Let $\mathscr{C}$ denote the collection of sets defined by lattice combinations (i.e. positive boolean combinations) of sets defined by subsets of $\boldsymbol{M}^{x}$ defined by equations. As $T$ is complete, we can arrange that $\boldsymbol{\mathscr{M}}$ is a reduct of a model of $\mathrm{RCF}$. Recall that the Zariski topology in RCF is noetherian, so for any decreasing sequence ${(C_{n})}_{n\in\operatorname{\mathbb{N}}}$ of elements in $\mathscr{C}$, there is $N\in\operatorname{\mathbb{N}}$ such that $C_{N}=\bigcap_{n}C_{n}$. Hence, $\mathscr{C}$ is the collection of closed sets of certain noetherian topology which we call the linear topology on $\boldsymbol{M}^{x}$. As usual, an element $X$ of $\mathscr{C}$ is irreducible if $X$ cannot be written as a nontrivial union of two elements of $\mathscr{C}$. It is then a standard fact about noetherian topology that every element of $\mathscr{C}$ can be uniquely (up to permutation) decomposed into a finite union of irreducible elements of $\mathscr{C}$ such that there is no containment between any two distinct elements.

With $T$ still either DLO or DOAG, we will define suitable versions of tangent spaces and smoothness in this setting. Suppose that $X\subseteq\boldsymbol{M}^{x}$ is definable. Let $X^{\mathrm{ccl}}$ be the closure of $X$ with respect to the (coarser) linear topology on $\boldsymbol{M}^{x}$, and let $\{V_{i}\}_{i\leqslant n}$ lists all irreducible components of $X^{\mathrm{ccl}}$. The tangent space of $X$ at $a\in X$ is the smallest irreducible closed set containing $\bigcup\{V_{i}\;|\;i\leqslant n,\,a\in V_{i}\}$. Remark 3.7 tells us that this a reasonable definition. Moreover, we say that $a\in X$ is smooth if there is unique $V_{i}$ such that $a\in V_{i}$. The proof of Proposition 3.6 with this new definition of smoothness and tangent space yields the Proposition 3.8 below. Note that this is in accordance with the notion of smoothness in algebraic geometry; again, see [2, Definition 3.3.4]. More precisely, in the case of DOAG, the $V_{i}$’s are all affine subspaces, hence algebraic varieties. All affine subspaces are in particular smooth varieties. Hence, a point $a\in\bigcup_{i}V_{i}$ is a smooth point if and only if there is a unique $i$ such that $a\in V_{i}$; a point lying on the intersection of two distinct irreducible components is not regular, hence not smooth.

In this very short section, we classify the stable formulas in the case where $T$ is either DLO or DOAG.

One can observe that the above proof cannot be carried out when $T$ is $\operatorname{RCF}$ because if $Z$ is an irreducible variety, one cannot choose subtuples $x^{\prime}$ of $x$ and $y^{\prime}$ of $y$ such that the projection map $\pi_{x^{\prime},y^{\prime}}$ as defined in the proof induces an isomorphism.

Finally, we are heading to the classification of stable formulas when $T$ is $\operatorname{RCF}$. With caveats, the strategy is the same as the proof of Theorem 4.1, namely, taking projections to get largeness and then use Proposition 3.6. This is quite similar to the proof of Theorem 4.1 except that we need to work harder to arrange for the projection maps to be bijective. To this end, we will need to decompose the original sets into finitely many disjoint pieces. Hence, the need for topological compactness, which we get by passing to projective space through Lemma 5.1 and working with a fixed copy of $\operatorname{\mathbb{R}}$ in the monster model. Another obstruction to finite decomposition comes from singularities, and this can be avoided by the use of Hironaka’s resolution of singularities at the cost of obtaining a classification only up to order-free isomorphisms.

In this section, we will use real algebraic geometry in the classical sense (i.e. zeros of systems of polynomial); see [2, Section 2, 3] for the precise set up. We let $\mathbb{P}^{x}$ denote of the $|x|$-dimensional projective space over $\boldsymbol{\mathscr{M}}$ corresponding to the tuple of variables $x$. We will identify $\boldsymbol{M}^{x}$ in the usual way with a Zariski open subset of $\mathbb{P}^{x}$.