Stability and stable groups in continuous logic

Continuous first order logic was introduced by A. Usvyatsov and the author in [BU], with the declared purpose of providing a setting in which classical local stability theory could be developed for metric structures. The actual development of stability theory there is fairly limited, mostly restricted to the definability of $\varphi$-types for a stable formula $\varphi$, the properties of $\varphi$-independence, and in case the theory is stable, properties of independence. Many fundamental results of classical stability theory, and specifically those related to stable groups, are missing there, and it is this gap that the present article proposes to fill.

We assume familiarity with [BU] and follow the notation used therein. Throughout $T$ denotes a continuous theory in a language $\mathcal{L}$. We do not assume that $T$ is complete, so various constants, such as $k(\varphi,\varepsilon)$ of Fact 2.1, are uniform across all completions of $T$ (provided that $\varphi$ is stable in $T$, i.e., in every completion of $T$ separately).

By a model we always mean a model of $T$. Whenever this is convenient, we shall assume that such a model $\mathcal{M}$ is embedded elementarily in a large monster model $\mathfrak{M}$, i.e., in a strongly $\kappa$-homogeneous and saturated model, where $\kappa$ is much bigger than the size of any set of parameters under consideration. Notice that we may not simply choose a single monster model for $T$, as this would consist of choosing one completion.

We shall consider throughout a formula $\varphi(\bar{x},\bar{y})$ whose variables are split in two groups. We recall from [BU] that a definable $\varphi$-predicate is a definable predicate $\psi(\bar{x}$), possibly with parameters, which is equivalent to an infinitary continuous combination of instances of $\varphi$:

$$\displaystyle\psi(\bar{x})\equiv\theta\bigl{(}\varphi(\bar{x},\bar{b}_{n})\bigr{)}_{n\in\mathbb{N}},\qquad\theta\colon[0,1]^{\mathbb{N}}\to[0,1]\text{ continuous}.$$

Equivalently, $\varphi(\bar{x})$ is a $\varphi$-predicate if it can be approximated arbitrarily well by finite continuous combinations of instances of $\varphi$, possibly restricted to the use of the connectives $\neg$, $\frac{1}{2}$, $\mathbin{\mathchoice{\kern 2.77774pt\hbox to0.0pt{\hss\hbox{$\displaystyle-$}\hss}\raise 2.58334pt\hbox to0.0pt{\hss$\displaystyle.$\hss}\kern 2.77774pt}{\kern 2.77774pt\hbox to0.0pt{\hss\hbox{$\textstyle-$}\hss}\raise 2.58334pt\hbox to0.0pt{\hss$\textstyle.$\hss}\kern 2.77774pt}{\kern 2.45831pt\hbox to0.0pt{\hss\hbox{$\scriptstyle-$}\hss}\raise 1.80835pt\hbox to0.0pt{\hss$\scriptstyle.$\hss}\kern 2.45831pt}{\kern 2.29166pt\hbox to0.0pt{\hss\hbox{$\scriptscriptstyle-$}\hss}\raise 1.29167pt\hbox to0.0pt{\hss$\scriptscriptstyle.$\hss}\kern 2.29166pt}}$ alone.

Local types, i.e., $\varphi$-types for a fixed formula $\varphi$, are discussed in [BU, Section 6]. For a model $\mathcal{M}$ and a tuple $\bar{a}$ in some extension $\mathcal{N}\succeq\mathcal{M}$, the $\varphi$-type of $\bar{a}$ over $\mathcal{M}$, denoted $\operatorname{tp}_{\varphi}(\bar{a}/M)$, is the partial type given by $\{\varphi(\bar{x},\bar{b})=\varphi(\bar{a},\bar{b})\}_{\bar{b}\in M}$. The space of all $\varphi$-types over $M$ is denoted $\operatorname{S}_{\varphi}(M)$, and it is a compact Hausdorff quotient of $\operatorname{S}_{n}(M)$. If $\psi(\bar{x})$ is a $\varphi$-predicate over $M$ then $\operatorname{tp}_{\varphi}(\bar{a}/M)$ determines $\psi(\bar{a})$, so we may identify $\psi$ with a mapping $\hat{\psi}\colon\operatorname{S}_{\varphi}(M)\to[0,1]$, sending $p\mapsto\psi^{p}$. Every such mapping is continuous, and conversely, every continuous mapping from $\operatorname{S}_{\varphi}(M)$ to $[0,1]$ is of this form.

For $A\subseteq M$ we define $\operatorname{S}_{\varphi}(A)$ to be the quotient of $\operatorname{S}_{\varphi}(M)$ where two types are identified if all $A$-definable $\varphi$-predicates agree on them. This is again a compact Hausdorff space, a common quotient of $\operatorname{S}_{\varphi}(M)$ and of $\operatorname{S}_{k}(A)$ (for the appropriate $k$), and the continuous mappings $\operatorname{S}_{\varphi}(A)\to[0,1]$ are precisely the $A$-definable $\varphi$-predicates. In particular this does not depend on the choice of $\mathcal{M}$.

It will also be convenient to adopt the following somewhat non standard terminology:

An essential notion for the study of definability of types and canonical bases in a stable theory is that of imaginary elements and sorts. Let us give a brief reminder of their construction, as given in [BU]. Consider a definable predicate with parameters in some set $A$, let us denote it by $\varphi(\bar{x},A)$. Then we may assume that $A$ is countable, say $A=(a_{n})_{n\in\mathbb{N}}$, and express $\varphi(\bar{x},A)$ as a uniform limit of formulae $\varphi_{n}(\bar{x},a_{<n})$. Furthermore, using a forced limit argument, we may assume that the sequence $\varphi_{n}(\bar{x},y_{<n})$ converges uniformly to some infinitary definable predicate $\varphi(\bar{x},Y)$, giving sense to $\varphi(\bar{x},B)$ for any sequence $B=(b_{n})_{n}$. If $\mathcal{M}$ is any structure, we equip $M^{\mathbb{N}}$ with the pseudo-metric $d_{\varphi}(B,C)=\sup_{\bar{x}}\,|\varphi(\bar{x},B)-\varphi(\bar{x},C)|$ and define the sort of canonical parameters for $\varphi$ in $\mathcal{M}$, denoted $S_{\varphi}^{\mathcal{M}}$, as the complete metric space associated to $(M^{\mathbb{N}},d_{\varphi})$. In other words, we divide $M^{\mathbb{N}}$ by the kernel $d_{\varphi}(Y,Z)=0$, obtaining a true metric on the quotient, and pass to the completion. The predicate $\varphi(\bar{x},Y)$ is uniformly continuous with respect to $Y$ in the metric $d_{\varphi}$, and therefore passes first to the quotient and then to the completion, thus inducing a uniformly continuous predicate $P_{\varphi}(\bar{x},z)$, where $z\in S_{\varphi}$. We may now add $(S_{\varphi},d_{\varphi})$ as a sort to $\mathcal{M}$ and equip the new structure with an additional predicate symbol for $P_{\varphi}$. This does not add structure to the original sorts of $\mathcal{M}$, elementary embeddings of structures commute with this construction (by which we mean, in particular, that an elementary embedding $\mathcal{M}\preceq\mathcal{N}$ extends uniquely to $(\mathcal{M},S_{\varphi}^{\mathcal{M}})\preceq(\mathcal{N},S_{\varphi}^{\mathcal{N}})$), elementary classes and model completeness thereof are respected by this construction, and so. The construction can be slightly simplified when $\varphi$ only uses finitely many parameters, e.g., if it is an honest formula rather than a definable predicate, but we are going to need the general case.

If $c=[A]$ is the image of $A$ in $S_{\varphi}$ then $c$ is indeed a canonical parameter for $\varphi(\bar{x},A)$, in the sense that an automorphism of $\mathcal{M}$ or of an elementary extension thereof (and such an automorphism extends uniquely to $S_{\varphi}$) fixes $c$ if and only if it fixes the predicate $\varphi(\bar{x},A)$. By construction we have $P_{\varphi}(\bar{x},c)=\varphi(\bar{x},A)$, and by a convenient abuse of notation we shall permit ourselves to write $\varphi(\bar{x},c)$ instead of either one.

By an imaginary sort we mean any sort added in this fashion, and by imaginary elements we mean members of such a sort. We may repeat this construction for any other definable predicate $\psi(\bar{x}^{\prime},Y^{\prime})$, or for any family of predicates. A delicate point here is that even with a countable language one can construct continuum many definable predicates for whose canonical parameters imaginary sorts may be added. For the purposes of stability theory, however, no more imaginary sorts than the size of the language are truly required: we need the sort $S_{d\varphi}$ for each formula $\varphi$ (see Fact 2.1 for the predicate $d\varphi$). Therefore, with some intentional ambiguity, by $\mathcal{M}^{eq}$ we shall usually mean “$\mathcal{M}$ along with all the imaginary sorts we are going to need”, e.g., all the sorts $S_{d\varphi}$. By $\operatorname{dcl}^{eq}$, $\operatorname{acl}^{eq}$, etc., we mean the respective operations in the structure $\mathcal{M}^{eq}$.

The following notion and fact also appear (and are used much more extensively) in [Bena, Section 1]:

Notice that the interior of $[Y]$ does depend on $A$ (i.e., if $A^{\prime}\supseteq A$ then $[Y]^{\circ}$ calculated in $\operatorname{S}_{n}(A^{\prime})$ may be larger than the pullback of the interior of $[Y]$ in $\operatorname{S}_{n}(A)$). We may nonetheless choose any parameter set we wish:

Having fixed a theory $T$, we shall call here a formula $\varphi(\bar{x},\bar{y})$ stable if it is stable in $T$, that is, if it does not have the order property in any model of $T$. The order property was defined for continuous logic in [BU], but the reader may simply use Fact 2.1 below as the definition of a stable formula.

Let us introduce some convenient notation. If $\varphi(\bar{x},\bar{y})$ is any formula with two groups of variables, $\tilde{\varphi}(\bar{y},\bar{x})$ denotes the same formula with the groups of variables interchanged. More generally, let us define

$$\displaystyle\tilde{\varphi}^{n}(\bar{y},\bar{x}_{\leq 2n})=\operatorname{med}_{n}\bigl{(}\varphi(\bar{x}_{i},\bar{y})\bigr{)}_{i\leq 2n},$$

where $\operatorname{med}_{n}\colon[0,1]^{2n+1}\to[0,1]$ is the median value combination:

$$\displaystyle\operatorname{med}_{n}(t_{\leq 2n})=\bigwedge_{w\in[2n+1]^{n+1}}\,\bigvee_{i\in w}\,t_{i}=\bigvee_{w\in[2n+1]^{n+1}}\,\bigwedge_{i\in w}\,t_{i}.$$

Thus in particular $\tilde{\varphi}^{0}=\tilde{\varphi}$ and every instance of $\tilde{\varphi}^{n}$ is a $\tilde{\varphi}$-predicate.

By abuse of notation we may sometimes write $d_{p}\varphi(\bar{y})=d\varphi(\bar{y},c)$, where $c\in M^{eq}$ is the canonical parameter for the definition. This canonical parameter is called the canonical base of $p$, denoted $\mathrm{Cb}(p)$.

We recall that for $A\subseteq B\subseteq\mathcal{M}$, $p\in\operatorname{S}_{\varphi}(B)$ does not fork over $A$ if it admits an extension $p_{1}\in\operatorname{S}_{\varphi}(M)$ which is definable over $\operatorname{acl}^{eq}(A)$. In this case $p_{1}$ itself does not fork over $A$ or $B$. A type over a model clearly admits a unique non forking extension to any larger model (and therefore set), so this definition does not depend on the choice of $\mathcal{M}$.

We proved in [BU, Proposition 7.15] that every $\varphi$-type over a set $A$ admits a non forking extension to every model (and therefore every set) containing $A$. A minor enhancement of that result will be quite useful.

It follows that if the theory is stable then every complete type over a set admits non forking extensions. The same fact was proved in [BU] using a somewhat longer “gluing” argument.

Proposition 2.6.(iii) is the analogue of the finite equivalence relation theorem in continuous logic. It has already appeared as [BU, Proposition 7.17]. In case $p\in\operatorname{S}_{\varphi}(A)$ is stationary, the unique non forking extension to $B\supseteq A$ will be denoted $p{\restriction}^{B}$. Similarly, we write $d_{p}\varphi$ for the definition of $p{\restriction}^{M}$ where $\mathcal{M}\supseteq A$ is any model (and this does not depend on the choice of $\mathcal{M}$). Thus, in hindsight, in the statement of Proposition 2.6, the definitions $\chi_{\bar{b}}$ are uniquely determined, $\chi_{\bar{b}}=d_{\bar{b}/\operatorname{acl}^{eq}(A)}\tilde{\varphi}$.

We pass to forking of single conditions.

We may define the non forking degree of $\varphi(\bar{x},\bar{b})$ over $A$ to be

$$\displaystyle\operatorname{nf}\bigl{(}\varphi(\bar{x},\bar{b})/A\bigr{)}=\inf\bigl{\{}r\colon\varphi(\bar{x},\bar{b})\leq r\text{ does not fork over }A\bigr{\}}.$$

An easy compactness argument shows that the infimum is attained and the condition $\varphi(\bar{x},\bar{b})\leq\operatorname{nf}\bigl{(}\varphi(\bar{x},\bar{b})/A\bigr{)}$ does not fork over $A$. In addition, by the existence of non-forking types we have $\operatorname{nf}\bigl{(}\varphi(\bar{x},\bar{b})/A\bigr{)}+\operatorname{nf}\bigl{(}\neg\varphi(\bar{x},\bar{b})/A\bigr{)}\leq 1$.

Since a continuous function to $[0,1]$ can only take into account countably many arguments, we may always assume that $\alpha\leq\omega$. Notice that any connective constructed using $\vee$ and $\wedge$ alone is faithful (so in particular the median value connective $\operatorname{med}_{n}\colon[0,1]^{2n+1}\to[0,1]$ is). Similarly, any uniform limit of faithful combinations is faithful.

Notice that a similar proof yields that if $\varphi(\bar{x},\bar{y})$ is stable then the restriction mapping $\rho_{A,\varphi}\colon X_{\varphi}\to\operatorname{S}_{\varphi}(A)$ is open, where $X_{\varphi}\subseteq\operatorname{S}_{\varphi}(B)$ denotes the set of $\varphi$-types which do not fork over $A$.

It follows from Lemma 2.14 that a $\tilde{\varphi}$-type (and therefore a $\varphi$-type) over an arbitrary set $A$ is definable over $A$, but of course the same definition applied to a larger set need not give a consistent complete type. This yields the following (adaptation of a) classical result:

It follows that if $X$ is an $A$-type-definable set and $Y\subseteq X$ is a type-definable subset, then $Y$ is type-definable over $AB$ for some countable $B\subseteq X$. If $Y$ is a definable set then it is definable over $AB$ (by Lemma 1.3, since the definable predicate $d(\bar{x},Y)$ is $AB$-invariant).