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\begin{document}
\title{\.Istanbul Model Theory Seminar Notes 2012}
\author{David Pierce}
\date{April 5, 2012; compiled, \today}
\maketitle

We are studying Hrushovski's `Stable group theory and approximate
subgroups' \cite{MR2833482}.  Secondary sources include 
\begin{compactitem}
\item
Notes by Lou van den Dries \cite{vdD-H},
\item
Terence Tao's blog \cite{TT-H}.
\end{compactitem}
These notes were first prepared for my talk on March 29 and then
revised afterwards.  I consulted my notes from earlier talks by Piotr
Kowalski (February 16) and G\"onen\c c Onay (February 23 and March 1;
the three sessions between then and March 29 were devoted to talks by
Bruno Poizat and C\'edric Milliet on other matters).

The appendix contains notes that I wrote soon after March 1, in
an attempt to justify the trouble van den Dries \cite{vdD-H} takes to
establish notation for many-sorted structures.

I expect to speak again on April 5 and then to add to these notes (and
perhaps edit what is already here).

\newpage

\tableofcontents

\newpage

\section{Setting}

We fix a complete theory $T$.  
The signature of $T$ may be \emph{many-sorted}; this means there are
variables for each sort, and function-symbols and predicates `know'
which sorts their arguments can come from. 

We let letters like $x$ and $y$ denote (finite) tuples of (distinct) variables.
In particular, $x$ is of the form $(x_i\colon i<n)$, where each $x_i$
belongs to a sort $s(i)$.

Say $\str M\models T$.  We denote by $M_x$ the set of instantiations of $x$
in $\str M$; that is,
\begin{equation*}
M_x=\prod_{i<n}M_{s(i)}.
\end{equation*}

Let $A$ be a set of parameters from $\str M$.  On the set of formulas
in $x$ over $A$ we have the interpretation map $\phi\mapsto\phi^{\str M}$,
where
\begin{equation*}
\phi^{\str M}=\{a\in M_x\colon\str M\models\phi(a)\};
\end{equation*}
the range of the map is called
\begin{equation*}
\Def{M_x}A,
\end{equation*}
the set of subsets of $M_x$ that are definable over $A$.
We identify formulas that have the same interpretation:
\begin{equation*}
\phi=\psi\iff\phi^{\str M}=\psi^{\str M}.
\end{equation*}
Since $T$ is complete, this identification is independent of choice of
$\str M$.  Considered under this identification, the set of formulas
in $x$ over $A$ is 
\begin{equation*}
\Lin xA,
\end{equation*}
and this is a \emph{Boolean algebra,} the \textbf{Lindenbaum algebra}
in $x$ over $A$ with  respect to $\str M$.  This algebra is isomorphic
to $\Def{M_x}A$ under $\phi\mapsto\phi^{\str M}$: 
\begin{align*}
	(\phi\lor\psi)^{\str M}&=\phi^{\str M}\cup\psi^{\str M},&
	\top^{\str M}&=M_x,\\	
	(\phi\land\psi)^{\str M}&=\phi^{\str M}\cap\psi^{\str M},&
	\bot^{\str M}&=\emptyset.
\end{align*}
Here $\top$ stands for $\bigwedge_{i<n}x_i=x_i$, and $\bot$ for its negation.\pagebreak

Like every Boolean algebra, $\Lin xA$ has a \textbf{Stone space,}
\begin{equation*}
\St xA,
\end{equation*}
the set of \emph{ultrafilters} of the algebra.  First, a \textbf{filter} of $\Lin xA$ is a nonempty subset $F$ such that
\begin{compactitem}
\item
$F$ is closed under $\land$;\footnote{We can formulate this condition as the closure of $F$ under $(\phi_i\colon i<n)\mapsto\bigwedge_{i<n}\phi_i$ for all $n$ in $\upomega$.  But in case $n=0$, this nullary operation can be understood as the formula $\top$ (since this formula is true if only if $\phi_i$ is true for each $i$ in $\emptyset$.  Thus this formulation implies that $F$ is nonempty.}
\item
if $\phi\in F$, then $\phi\lor\psi\in F$.
\end{compactitem}
Then an \textbf{ultrafilter} is a maximal proper filter, equivalently a filter $p$ such that
\begin{compactitem}
\item
$\phi\notin p$ if and only if $\lnot\phi\in p$.
\end{compactitem}
One way to prove this equivalence is to note that
$\Lin xA$ is also an \emph{associative ring} with addition
$(\phi,\psi)\mapsto\lnot(\phi\liff\psi)$ and multiplication $\land$;
it is in particular a \emph{Boolean ring}\footnote{Boolean rings are commutative and have characteristic $2$: if squaring is the identity, then $x+y=(x+y)^2=x+xy+yx+y$, so $0=xy+yx$.} because 
\begin{equation*}
\phi\land\phi=\phi.
\end{equation*}
Then filters are duals of ideals: $F$ is a filter if and
only if $\{\lnot\phi\colon\phi\in F\}$ is an ideal.  Quotients of commutative rings by maximal ideals are fields, and the only Boolean field is the two-element field.  This gives the characterization of ultrafilters.

There is an embedding $\phi\mapsto[\phi]$ of the Boolean algebra $\Lin xA$ in the algebra $\pow{\St xA}$, where
\begin{equation*}
[\phi]=\{p\colon\phi\in p\}.
\end{equation*}
In particular
\begin{align*}
[\phi\land\psi]&=[\phi]\cap[\psi],&
[\top]&=\St xA,&
[\lnot\phi]&=\St xA\setminus[\phi].
\end{align*}
Thus $\{[\phi]\colon\phi\in\Lin xA\}$ is a basis of open sets for a
topology on $\St xA$, and these basic open sets are also closed.  The
topology is compact.\footnote{One proof of this is as follows.
  Suppose $\Gamma$ is a subset of $\Lin xA$ such that, for every
  finite subset $\Delta$ of $\Gamma$, 
\begin{equation*}
\bigcap_{\phi\in\Delta}[\phi]\neq\emptyset.
\end{equation*}
Since $\Delta$ is finite, this means
$[\bigwedge_{\phi\in\Delta}\phi]\neq\emptyset$, that is,
$\bigwedge_{\phi\in\Delta}\phi$ generates a proper filter of $\Lin xA$
(namely the set of formulas implied by
$\bigwedge_{\phi\in\Delta}\phi$).  This being so for \emph{every}
finite subset $\Delta$ of $\Gamma$, the set $\Gamma$ itself generates
a proper filter (namely the set of formulas implied by
$\bigwedge_{\phi\in\Delta}\phi$ for \emph{some} finite subset $\Delta$
of $\Gamma$).  This filter embeds in an ultrafilter $p$, for the same
reason that proper ideals embed in maximal ideals.  Thus 
\begin{equation*}
p\in\bigcap_{\phi\in\Gamma}[\phi].
\end{equation*}
Therefore $\St xA$ is compact.  For the compactness of first-order logic, see note~\ref{1com} below.}

Similarly, $\{\phi^{\str M}\colon\phi\in\Lin xA\}$ is a basis for the
\textbf{$A$-topology} on $M_x$.  Closed sets in this topology can be
called \textbf{$A$-closed;} open sets, \textbf{$A$-open.}  Thus, 
\begin{compactitem}
\item
the $A$-closed sets are are the \textbf{$\bigwedge$-definable,} or
  \textbf{type-definable,} sets over $A$; 
\item
the $A$-open sets are the \textbf{$\bigvee$-definable} sets over $A$.
\end{compactitem}
From $M_x$ to $\St xA$ there is a map $a\mapsto\tp aA$, where
\begin{equation*}
\tp aA=\{\phi\colon a\in\phi^{\str M}\}.
\end{equation*}
This map is continuous with respect to the $A$-topology, because under
the map the inverse image of $[\phi]$ is $\phi^{\str M}$. 
If the image of $\phi^{\str M}$ is $[\phi]$---that is, if the map is
surjective---, then the map is also closed and open. 

We can make the map surjective, and we can ensure that the inverse
image of a singleton is one orbit under $\Aut{\str M}A$.  We do this
by replacing $\str M$ with a \emph{monster model} or \textbf{universal
  domain,} $\U$. 
\begin{compactitem}
\item
$\U$ is $\card{\str M}^+$-saturated (for all $\str M$ that we shall
  consider), so we may assume 
\begin{equation*}
\str M\prec\U.
\end{equation*}
\item
$\U$ is $\card{\str M}^+$-homogeneous: for all $A$ from $\str M$, if
  $\tp aA=\tp bA$ then $a\mapsto b$ extends to an automorphism of
  $\U$. 
\end{compactitem}



\section{Keisler measures}

Now $\str M$ is just some structure, and $A$ is a set of parameters
from $\str M$. 
A function $\mu_x$ from $\Lin xA$ to the closed interval $[0,\infty]$
of $\R\cup\{\infty\}$ is a 
\textbf{Keisler measure} if
\begin{equation*}
\mu_x(\phi\lor\psi)=\mu_x(\phi)+\mu_x(\psi)
\end{equation*}
whenever $\phi$ and $\psi$ are mutually contradictory, that is,
$\phi\land\psi=\bot$; in a word, $\mu_x$ is \emph{additive.} 
Usually also
\begin{equation*}
\mu_x(\top)=1,
\end{equation*}
in which case $\mu_x$ is a \emph{probability measure} and
\begin{equation*}
\mu_x(\lnot\phi)=1-\mu_x(\phi).
\end{equation*}
We may consider $\mu_x$ also as having domain $\Def{M_x}A$ (in which
case the term \emph{measure} is more suggestive). 
We may also consider Keisler measures on $\Def XA$ for some $A$-open
subset $X$ of $\str M_x$.  

\begin{example}\label{ex:tp}
If $p\in\St xA$, we can define $\mu_x$ on $\Lin xA$ by
\begin{equation*}
\mu_x(\phi)=\begin{cases}
	1,&\text{ if }\phi\in p,\\
	0,&\text{ otherwise,}
\end{cases}
\end{equation*}
or on $\Def{M_x}A$ by
\begin{equation*}
\mu_x(X)=\begin{cases}
	1,&\text{ if $p=\tp aA$ for some $a$ in $X$},\\
	0,&\text{ otherwise.}
\end{cases}
\end{equation*}
\end{example}

\begin{example}\label{ex:counting}
If $\str M$ is finite and one-sorted, and $x=(x_i\colon i<n)$, we can
define\footnote{This example, given by Hrushovski \cite[\S2.6,
    p.~197]{MR2833482}, is mentioned on the last of Pillay's slides
  \cite{Pillay-first-order}.} 
\begin{equation*}
\mu_x^{\str M}(\phi)=\frac{\card{\phi^{\str M}}}{\card M^n}.
\end{equation*}
Thus $\mu_x^{\str M}$ is the \textbf{counting measure} on $\Lin xM$.
Given an infinite family $(\str M^i\colon i\in I)$ of finite
structures (of the same signature), we can form an ultraproduct $\str
N$ of the family, and then on $\Lin xN$ 
we can define $\mu_x(\phi)$ as the standard part of the image of
$(\mu_x^i(\phi)\colon i\in I)$ in ${}^*[0,1]$ or just ${}^*\R$.
Indeed, in the Ravello volume, Hrushovski \cite[Addendum,
  p.~209]{MR2159717} says: 
\begin{quote}
In an ultraproduct $k$ of finite fields, one has the nonstandard
counting measure; and one can let $\mu(V)=\st(\card V/\card
k^{\dim(V)})$ (the standard part)\dots This recovers the
generalization of Lang--Weil in Kieffe and
\cite{MR1162433}\footnote{\label{CDM}Cited by Hrushovski as
Chatzidakis, van den Dries, Macintyre, `Definable sets over finite
fields', Paris 7 Logique prepublication \textbf{23}; but I have not
yet been able to obtain either version.}\dots
\end{quote}
What is going on is the following.  We select a non-principal
ultrafilter of the Boolean algebra of subsets of $I$; elements of this
ultrafilter will be considered \emph{large.}  Then $\str N$ is the
Cartesian product of the structures $\str M^i$, but with two elements
identified if their entries agree on a large set of indices.  By the
result called \emph{\L o\' s's Theorem,} a sentence is true in $\str
N$ if and only if it is true in $\str M_i$ for each $i$ in a large set
of indices.\footnote{\label{1com}Now we can prove compactness of first-order logic.  Say $\Gamma$ is a set of sentences, and every finite subset $\Delta$ of $\Gamma$ has a model, $\str M_{\Delta}$.  A certain ultraproduct of these $\str M_{\Delta}$ will be a model of $\Gamma$.  Indeed, writing $\powf{\Gamma}$ for the set of these $\Delta$, we let $(\Delta)$ be the set of all elements of $\powf{\Gamma}$ that include $\Delta$.  Then $(\Delta)\cap(\Delta')=(\Delta\cup\Delta')$.  Thus the sets $(\Delta)$ generate a proper filter $F$ of $\pow{\powf{\Gamma}}$.  Now we take the ultraproduct $\str N$ of the $\str M_{\Delta}$ with respect to an ultrafilter that includes $F$.  For each $\Delta$ in $\powf{\Gamma}$, the set $(\Delta)$ of indices is large, and $\Delta$ is true in $\str M_{\Theta}$ for each $\Theta$ in $(\Delta)$; thus $\Delta$ is true in $\str N$.  This is the proof of Bell and Slomson \cite[Thm 5.4.1]{MR0269486}, who trace it to a 1958 article by Morel, Scott, and Tarski.}
In particular, ${}^*\R$ is the \emph{ultrapower} that results when
each $\str M^i$ is $\R$.  In this case, ${}^*\R$ can be understood as
the quotient of $\R^I$ by a non-principal maximal ideal $P$.  By \L
o\'s's Theorem, ${}^*\R$ is an ordered field, and it is
non-Archimedean.  If $S$ is the ring of its finite elements, and
$\mathfrak m$ is the ring of its infinitesimal elements, then
$\mathfrak m$ is a maximal ideal of $S$, and the quotient map
$x\mapsto x+\mathfrak m$ from $S$ to $S/\mathfrak m$ is an isomorphism
when restricted to the image of $\R$ in ${}^*\R$ under
$x\mapsto(x\colon i\in I)+P$.  Thus the standard part map from $S$ to
$\R$ is induced; this is a ring homomorphism, and in particular
$\mu_x$ is additive. 

To make $\mu_x$ \emph{definable} in $\str N$, for each formula
$\phi(x,y)$, for each $\alpha$ in $\Q$, we introduce a new atomic
formula, denoted by 
\begin{equation*}
\Qq x\phi(x,y),
\end{equation*}
and we expand each $\str M^i$ so that
\begin{equation*}
\Qq x\phi(x,y)^{\str M^i}=\{b\in M_y^i\colon\mu_x^i(\phi(x,b))\leq\alpha\}.
\end{equation*}
Repeat $\upomega$ times (so we have a formula $\Qq x\phi(x,y)$ for
every formula $\phi$).  
For every formula $\phi$ and every $\alpha$ in $\Q$, the following are
equivalent: 
\begin{gather*}
	\mu_x(\phi(x,b))\leq\alpha,\\
	\{i\colon\mu_x^i(\phi(x,b))\leq\alpha\}\text{ is large,}\\
	\str N\models\Qq x\phi(x,b).
\end{gather*}
Thus
\begin{equation*}
\mu_x(\phi(x,b))=\inf\{\alpha\colon\str N\models\Qq x\phi(x,b)\}.
\end{equation*}
We can take this as the \emph{definition} of $\mu_x$, and then we can
establish additivity as in the Dedekind construction of $\R$: 
\begin{compactitem}
\item
If $\gamma,\delta\in\R$, then
\begin{equation*}
\inf\{x\in\Q\colon\gamma\leq x\}+\inf\{y\in\Q\colon\delta\leq
y\}=\inf\{z\in\Q\colon\gamma+\delta\leq z\}. 
\end{equation*}
\item
If $\phi(x,b)\land\psi(x,b)=\bot$, then the sentences
\begin{gather*}
\Qq x\phi(x,b)\land\Qq[\beta]x\psi(x,b)
\lto\Qq[\alpha+\beta]x(\phi(x,b)\lor\psi(x,b),\\ 
\lnot\Qq
x\phi(x,b)\land\lnot\Qq[\beta]x\psi(x,b)
\lto\lnot\Qq[\alpha+\beta]x(\phi(x,b)\lor\psi(x,b))  
\end{gather*}
are true in each $\str M^i$ and therefore in $\str N$.
\end{compactitem}
A curiosity is that, while
\begin{align*}
\mu_x(\phi(x,b))<\alpha
&\implies\str N\models\Qq x\phi(x,b)\\
&\implies \mu_x(\phi(x,b))\leq\alpha,
\end{align*}
we need not have the converse of the second implication.  That is,
possibly $\mu_x(\phi(x,b))=\alpha$, although $\str N\models\lnot\Qq
x\phi(x,b)$, because $\mu_x^i(\phi(x,b))>\alpha$ for a large set of
$i$. 
\end{example}

Suppose now $\mu_x$ is a Keisler measure on $\Lin x{\U}$.
If for all $y$ and all formulas $\phi(x,y)$ in no
parameters,\footnote{The qualification that $\phi$ must have no
  parameters is not made explicit by Hrushovski \cite[\S2.6,
    p.~197]{MR2833482}.} the value of $\mu_x(\phi(x,b))$ depends only
on $\tp bA$, then $\mu_x$ is \textbf{$A$-invariant.}  In this case,
the function 
\begin{equation*}
b\mapsto\mu_x(\phi(x,b))
\end{equation*}
on $\U_y$ has the factor $b\mapsto\tp bA$; the other factor can be
called $\mu_{\phi}$,\footnote{Van den Dries \cite[p.~11]{vdD-H} uses
  $\mu_{\phi}$ for the whole function $b\mapsto\mu_x(\phi(x,b))$.  I
  use it here for the factor mainly to have a label for the
  commutative diagram.  Hrushovski just calls the factor $g$.} 
  so that we have the following commutative diagram
  (where we rely on the $\card A^+$-saturation of $\U$ to be able to define $\mu_{\phi}$ on all of $\St yA$):
\begin{equation*}
\xymatrix@!{
\U_y\ar[r]^{\phi} \ar[d]_{\mathrm{tp}} \ar[dr] & \Lin x{\U}\ar[d]^{\mu_x}\\
\St yA \ar[r]_{\mu_{\phi}}&[0,\infty]
}
\end{equation*}

\begin{theorem}
For a Keisler measure $\mu_x$ on $\Lin x{\U}$, the following are equivalent:
\begin{compactenum}
\item
For each $y$ disjoint from $x$, for each $\phi(x,y)$ in $\Lin{x,y}{\emptyset}$, the function $b\mapsto\mu_x(\phi(x,b))$ from $\U_x$ to $[0,\infty]$ is continuous with respect to the $A$-topology on $\U_x$.
\item
$\mu_x$ is $A$-invariant,
and for each $y$ disjoint from $x$, for each $\phi(x,y)$ in $\Lin{x,y}{\emptyset}$, the induced function $\mu_{\phi}$ (from $\St yA$ to $[0,\infty]$ such that $\mu_x(\phi(x,b))=\mu_{\phi}(\tp bA)$) is continuous.
 \end{compactenum}
\end{theorem}

\begin{proof}
In the presence of $A$-invariance, since the map $b\mapsto\tp bA$ is continuous and open, continuity of the `diagonal' maps
$b\mapsto\mu_x(\phi(x,b))$ with respect to the
$A$-topology is equivalent to continuity of the $\mu_{\phi}$.

If $A$-invari\-ance fails, that is, $\tp cA=\tp{c'}A$ for some $c$ and
$c'$ in $\U_x$, but 
\begin{equation*}
\mu_x(\phi(x,c))\leq\alpha<\mu_x(\phi(x,c')),
\end{equation*}
then $\{b\colon\mu_x(\phi(x,b))\leq\alpha\}$ is not $A$-closed, since
every $A$-closed (or $A$-open) set contains both $c$ and $c'$, or
neither. 
\end{proof}

Under the equivalent conditions of the theorem,
$\mu_x$ is called \textbf{$A$-defin\-able.}\footnote{Hrushovski just says $\mu_x$
  is $A$-definable if it is $A$-invariant and \emph{in addition} the
  maps $\mu_{\phi}$---his $g$---are continuous.} 
  If we work with an arbitrary structure $\str M$ instead of $\U$, we may take the first condition as $A$-definability.
This condition means just that for each $\phi$ and each
$\alpha$ in $\Q$ the sets 
\begin{align*}
&\{b\colon\mu_x(\phi(x,b))\leq\alpha\},&
&\{b\colon\mu_x(\phi(x,b))\geq\alpha\}
\end{align*}
are $A$-closed.  So we have this continuity in Example \ref{ex:counting}.

In Example \ref{ex:tp}, where
\begin{equation*}
\mu_x(\phi)=\begin{cases}
	1,&\text{ if }\phi\in p,\\
	0,&\text{ otherwise,}
\end{cases}
\end{equation*}
then we have $A$-definability of $\mu_x$ if and only if the sets
\begin{equation*}
\{b\colon\phi(x,b)\in p\}
\end{equation*}
are $A$-clopen, that is, $A$-definable.  This condition is that $p$
itself is \emph{definable} over $A$. 

More generally, suppose $B$ is another parameter set.  If $\U_x=\prod_{i<n}\U_{s(i)}$, we can let
\begin{equation*}
B_x=\prod_{i<n}(\U_{s(i)}\cap B).
\end{equation*}
This set has the $A$-topology induced from $\U_x$, and then an \textbf{$A$-definable} subset of $B_x$ can be understood as an $A$-clopen subset.

Now suppose $A\included B$.
An element $p$ of $\St xB$ is called \textbf{definable} over $A$ \cite[Defn 1.1]{MR719195} if for each $y$ disjoint from $x$, for each $\phi(x,y)$ in $\Lin{x,y}{\emptyset}$, there is a formula $\dee\phi(x,y)$ in $\Lin yA$ such that for all $b$ in $B_y$
\begin{equation*}
\phi(x,b)\in p\iff\U\models\dee\phi(x,b).
\end{equation*}
We can define $\mu_x$ on $\Lin xB$ as before; then this measure is $A$-definable as before if and only if $p$ is $A$-definable.

\begin{theorem}
Suppose $A\included B$.  All types in $x$ over $B$ are $A$-definable if and only if all $\U$-definable subsets of $B_x$ are $A$-definable.
\end{theorem}

\begin{proof}
Exercise, or see \cite[\S6.7, `Definability of types']{MR94e:03002}.
\end{proof}

The present situation can be depicted in one big commutative diagram:
\begin{equation*}
\xymatrix@!C{
\Lin{x,y}{\emptyset}\times B_y\ar@{.>}[dd]\ar[dr]\ar[rr]&&\Lin xB\ar[dd]\\
&\pow{B_y}\times B_y\ar[dr]&\\
\Lin yA\times B_y\ar@{.>}[ur]\ar[rr]&&\Lin{}B
}
\end{equation*}
Here $\Lin{}B$ is the set of sentences over $B$ \emph{modulo} $T$, so
it is the $2$-element Boolean algebra $\{\bot,\top\}$.  The first
component of the main diagonal map takes $(\phi(x,y),b)$ to $(C,b)$,
where $C=\{c\in B_y\colon\phi(x,c)\in p\}$; the next component takes
this pair to $\top$ if and only if $b\in C$. 

A type over $A$ (that is, an element of some $\St xA$) is called
simply \emph{definable} if it is $A$-definable.  Then all types over
$\emptyset$ are definable.  The following are equivalent:
\begin{compactenum}
\item 
The theory $T$ is \emph{stable} (that is, $\kappa$-stable for some
$\kappa$).
\item
All types over \emph{models} of $T$ are
definable  \cite[Cor.~1.21]{MR719195}.
\item
All types over all parameter sets are definable \cite[Cor 6.7.11]{MR94e:03002}.
\end{compactenum}
  So here is a hint that
definable Keisler measures are a generalization of types for unstable
theories. 

\section{Ideals}

if $\mu_x$ is a Keisler measure on $\Lin xA$, then the set
$\{\phi\colon \mu_x(\phi)=0\}$ is an \emph{ideal} of the Boolean ring
$\Lin xA$. 

\subsection*{The forking ideal}

Another example is the \emph{forking ideal.}\footnote{G\"onen\c c
  talked about this on March 1.}  A formula $\phi$ \textbf{forks} over $A$ if
$\phi\neq\bot$ and there are finitely many formulas $\theta_i$ such
that 
\begin{compactitem}
\item
$\phi\lto\bigvee_i\theta_i=\top$, that is, $T\vdash\phi\lto\bigvee_i\theta_i$;
\item
each $\theta_i$ \emph{divides} over $A$.
\end{compactitem}
For present purposes, one might say that $\bot$ also forks, since as
things are the \emph{forking ideal} in $x$ over $A$ will be the set 
\begin{equation*}
\{\phi\in\Lin x{\U}\colon\phi\text{ forks over }A\}\cup\{\bot\}.
\end{equation*}

If $\phi(x,y)$ is a formula\footnote{Probably $\phi$ has no parameters.} and $b\in\U_y$, the formula
$\phi(x,b)$ \textbf{divides} over $A$ if $b$ belongs to an
\emph{indiscernible} sequence $(b_i\colon i<\upomega)$ over $A$ such
that $\{\phi(x,b_i)\colon i<\upomega\}$ is inconsistent.  Recall that
the indiscernibility means that, for all $m$ in $\upomega$, if 
\begin{equation}\label{eqn:n}
n(0)<\dots<n(m-1)<\upomega,
\end{equation}
then for all formulas $\psi$ over $A$
\begin{equation*}
T\vdash\psi(b_0,\dots,b_{m-1})\liff\psi(b_{n(0)},\dots,b_{n(m-1)}).
\end{equation*}
So in $(\Q,<)$, every increasing sequence of elements is indiscernible
over $\emptyset$.  In a vector space, a basis is indiscernible as a
\emph{set} (under any ordering it is an indiscernible sequence). 

If $\{\phi(x,b_i)\colon i<\upomega\}$ is inconsistent, then by
compactness the subset $\{\phi(x,b_i)\colon i<m\}$ is inconsistent for some $m$
in $\upomega$; that is, the formula $\bigwedge_{i<m}\phi(x,b_i)$ is (`identically') false (in $T$).  By indiscernibility, if \eqref{eqn:n} holds, then $\bigwedge_{i<m}\phi(x,b_{n(i)})$ is false.
In short,
$\{\phi(x,b_i)\colon i<\upomega\}$ is \textbf{$m$-inconsistent.}  

Easily, if $\phi$ divides, so does $\phi\land\psi$.
However, if $(b_i\colon i<\upomega)$ and $(c_i\colon i<\upomega)$ are
indiscernible over $A$, it does not follow that $(b_ic_i\colon
i<\upomega)$ is indiscernible; so it is not immediate that if $\phi$
and $\psi$ divide, so does $\phi\lor\psi$. 

However, the definition of \emph{forking} ensures that the forking
formulas, along with $\bot$, do compose an ideal.  Also, in
\emph{stable} theories, forking and dividing are the same
\cite[ch.~6]{MR719195}. 

\subsection*{Possible properties}

Now say $X$ is an $A$-definable set.\footnote{Van den Dries
  \cite[p.~12]{vdD-H} allows $X$ to be $A$-open.  Hrushovski, and
  therefore we, do this presently.}  We may form the Boolean ring
$\Def X{\U}$ of definable subsets of $X$.  If $X=\theta^{\U}$ for
some $\theta$ in $\Lin xA$, we can define
\begin{equation*}
\Lin X{\U}=\{\theta\land\phi\colon\phi\in\Lin x{\U}\}
\end{equation*}
and identify this with $\Def X{\U}$.
An ideal of the Boolean ring $\Lin X{\U}$ is \textbf{$A$-invariant} if for all
formulas $\phi(x,y)$ and all $b$ in $\U_y$, the answer to the question of whether
$\phi(x,b)$ is in the ideal depends only on $\tp bA$.   

The forking ideal is invariant.
If $\mu_x$ is $A$-invariant, then so is the ideal $\{P\colon\mu_x(P)=0\}$ mentioned above.

Let $I$ be an ideal of $\Def X{\U}$.  A subset $\Phi$ of $\Lin X{\U}$ (that is, a partial type in  $x$ defining a subset of $X$) is \textbf{$I$-wide}
if $\Phi$ implies no formula
in $I$, that is, the \emph{filter} generated by $\Phi$ does not
intersect $I$.  If $I$ is the zero ideal of a measure, then $I$-wideness of $\Phi$ means no element of $\Phi$ has measure $0$.
The filter
\begin{equation*}
\{\theta\land\lnot\phi\colon\phi\in I\}
\end{equation*}
is the maximal $I$-wide partial type, unless $I$ is the improper ideal.

We generalize to the case where $X$ is merely $A$-open.  If
$X=\bigcup_iX_i$, the $X_i$ being definable over $A$, we let 
\begin{equation*}
\Lin X{\U}=\bigcup_i\Lin{X_i}{\U}.
\end{equation*}
Then a subset $I$ is an ideal if each $I\cap\Lin{X_i}{\U}$ is an
ideal.  This is Hrushovski's definition.  Alternatively, it seems we
could just define 
\begin{equation*}
\Lin X{\U}=\{\phi\in\Lin x{\U}\colon\phi^{\U}\included X\}.
\end{equation*}
This may not be a Boolean \emph{algebra.}  It is still a Boolean
\emph{ring,} possibly without a unit; so it has ideals and filters.  If $I$ is a proper ideal, then the maximal $I$-wide partial type is
\begin{equation*}
\{\psi\land\lnot\phi\colon\psi\in\Lin X{\U}\text{ and }\phi\in I\}.
\end{equation*}

\section{$S_1$ rank}

In the Ravello volume, Hrushovski \cite[Defn~4.1, p.~176]{MR2159717} defines
$\Sone(\theta)$ for formulas $\theta$ in $\Lin x{\U}$ (here $\U$ need
only be $\upomega$-saturated): 
\begin{compactenum}
\item
$\Sone(\theta)>0$ if $\card{\theta^{\U}}\geq\upomega$.
\item
$\Sone(\theta)\geq n+1$ if for some set $A$ of parameters,
  $\theta\in\Lin xA$ and there is an indiscernible sequence
  $(b_i\colon i\in I)$ over $A$ and a formula
  $\phi(x,y)$\footnote{Presumably with no parameters?} such that 
\begin{align*}
	\Sone(\phi(x,b_1)\land\phi(x,b_2))&< n,&
	\Sone(\theta\land\phi(x,b_i))&\geq n
\end{align*}
for each $i$ in $I$.
\end{compactenum}
(Hrushovski has $>$ for $\geq$ and $\leq$ for $<$ in the second part
of this definition.  He notes that $\Stwo(\theta)$ can be defined the
same way, but with $\Stwo(\theta)>0$ if $\Sone(\theta)>n$ for all $n$
in $\upomega$.) 

Compare with Morley rank:
\begin{compactenum}
\item
$\RM(\theta)\geq0$ if $\card{\theta^{\U}}>0$.
\item
$\RM(\theta)\geq\alpha+1$ if there is a sequence $(\psi_i\colon
  i<\upomega)$ of formulas (with parameters) such that 
\begin{align*}
	\psi_i\land\psi_j{}^{\U}&=\emptyset,&
	\RM(\theta\land\psi_i)&\geq\alpha
\end{align*}
whenever $i<j<\upomega$.
\end{compactenum}

Then $\Sone(\theta)$ is the least $n$ such that $\Sone(\theta)\not>n+1$.  

(For a subset $\Gamma$ of $\Lin x{\U}$, $\Sone(\Gamma)$ is the least
of the $\Sone(\theta)$ such that $\theta$ is in the filter generated
by $\Gamma$.  Then 
\begin{math}
\Sone(a/B)=\Sone(\tp aB)
\end{math}.  Similarly for $\RM$.)

Suppose $\theta$ in $\Lin x{\U}$ has Morley rank $1$ and Morley
\emph{degree} $1$ (that is, no two disjoint definable subsets have
lower rank).  Then $\theta$ or $\theta^{\U}$ is called
\textbf{strongly minimal.}  Every definable subset of $\theta^{\U}$ is
finite or cofinite; moreover, by compactness, for every $\phi(x,y)$,
there is $n$ in $\vnn$ such that $\theta\land\phi(x,a)^{\U}$ is
smaller than $n$ or infinite.  Hrushovski asserts the same in case
$\theta$ has $\Sone$ rank $1$: 

\begin{theorem}[\protect{\cite[Lem.~4.1, p.~176]{MR2159717}}]
Suppose $\theta$ in $\Lin x{\U}$ has $\Sone$ rank $1$.  Then for every
$\phi(x,y)$, there is $n$ in $\vnn$ such that
$\theta\land\phi(x,a)^{\U}$ is smaller than $n$ or infinite.  
\end{theorem}

\begin{proof}
Suppose not, so that there are $a_m$ for infinitely many $m$ such that
$\theta\land\phi(x,a_m)^{\U}$ has size $m$.  Write
$\theta\land\phi(x,a_m)^{\U}$ as $D_m$.  For every $m$, there is a
least $m'$ such that $m<m'$ and, for infinitely many $n$ 
\begin{equation}\label{eqn:mm'}
D_m\cap D_{m'}=D_m\cap D_n.
\end{equation}
 Hence there is an infinite set of indices such that \eqref{eqn:mm'} holds whenever $m<m'\leq n$.\footnote{Hrushovski appeals to Ramsey's theorem for this.}  In this case
 \begin{equation*}
D_m\setminus D_{m'}=D_m\setminus D_n.
\end{equation*}
Thus the sets $D_m\setminus D_{m'}$ are disjoint.  Therefore their sizes are bounded, since $\Sone(\theta)=1$.\footnote{Apparently if the sizes were unbounded, by compactness we could assume that the parameters composed an indiscernible sequence.}  This means that the sets $D_m\cap D_{m'}$ are unbounded in size.  But by \eqref{eqn:mm'} they form a chain:
\begin{equation*}
D_m\cap D_{m'}=D_m\cap D_{m'}\cap D_{m''}\included D_{m'}\cap D_{m''}.
\end{equation*}
Perhaps by restricting the index set again\footnote{Hrushovski does not say this, but it seems to be needed.}, we may assume that the differences
\begin{equation*}
(D_{m'}\cap D_{m''})\setminus(D_m\cap D_{m'})
\end{equation*}
are strictly increasing in size.  Since they are disjoint, this contradicts $\Sone(\theta)=1$.
\end{proof}

According to Hrushovski \cite[p.~177]{MR2159717},
\begin{quote}
The fact that  $\Sone(F)=1$ in the case of pseudo-finite fields was
shown in \cite{MR1162433}\footnote{See note~\ref{CDM}.}, using
an extension of the Lang--Weil estimates. 
\end{quote}

If $X$ has ordinal\footnote{Piotr gave this example (with `finite' for
  `ordinal') on February 16.} Morley rank $\alpha$ and has
Morley \emph{degree} $1$ (that is, has no two disjoint definable
subsets of its rank), then there is a Keisler measure on $\Lin X{\U}$
given by 
\begin{equation*}
\mu(\phi)=
\begin{cases}
	1,&\text{ if }\RM(\phi)=\alpha,\\
	0,&\text{ otherwise,}
\end{cases}
\end{equation*}
and this determines the ideal $\{\phi\colon\mu(\phi)=0\}$.  Even
without the assumption that $\dM(X)=1$, the set 
\begin{equation*}
\{\phi\colon\RM(\phi)<\alpha\}
\end{equation*}
is an ideal.  Similarly
\begin{equation*}
\{\phi\colon\Sone(\phi)<n+1\}
\end{equation*}
is an ideal.  
If $\Sone(\theta)=n+1$, then for all $A$ such that $\theta\in\Lin xA$,
for all indiscernible $(b_i\colon i<\upomega)$ over $A$ and all
$\phi(x,y)$ [over $A$], if 
\begin{equation*}
\Sone(\phi(x,b_0)\land\phi(x,b_1))<n,
\end{equation*}
then for some $i$ in $\upomega$,
\begin{equation*}
\Sone(\theta\land\phi(x,b_i))<n.
\end{equation*}
An arbitrary ideal $I$ that is invariant over $A$ is called an
\textbf{$\Sone$ ideal} (or $S1$ ideal) over $A$ if it has the
foregoing property, that is, for any $\phi(x,y)$ over $A$ and any
indiscernible $(a_i\colon i<\upomega)$ over $A$,
\begin{equation}\label{eqn:S1}
\text{if $\phi(x,a_0)\land\phi(x,a_1)\in I$,\qquad then
$\phi(x,a_0)\in I$} 
\end{equation}
---equivalently, $\phi(x,a_i)\in I$ for some and therefore all $i$ in $\upomega$,
by invariance of $I$ and indiscernibility of the
sequence.\footnote{Hrushovski \cite[Def.~2.8]{MR2833482} just as the
  conclusion as $\phi(x,a_i)\in I$ for some $i$ in $\upomega$; van den
  Dries \cite[p.~12]{vdD-H} observes that it then holds for all $i$.}  

We can replace \eqref{eqn:S1} with the condition that for some or all
$n$ in $\upomega$ 
\begin{equation*}
\text{if $\bigwedge_{i<2^n}\phi(x,a_i)\in I$,\qquad then
$\phi(x,a_0)\in I$.} 
\end{equation*}
Thus every $\Sone$ ideal includes the forking
ideal.\footnote{G\"onen\c c showed this on March 1 by the method of
  \cite[Lem.~1.18, p.~12]{vdD-H}.}  

\section{The stabilizer theorem}

We are now ready to state what appears to be the central result
(Theorem 3.5) of Hrushovski's paper.  (It is van den Dries's \cite[Thm
  2.6]{vdD-H}.) 

We let $G$ be a group definable over a model $\str M_0$.  

We let $X$ be an arbitrary subset of $G$, and we define
$\tilde G=\langle X\rangle$
(van den Dries calls this $\hat X$ \cite[p.~17]{vdD-H}).  Then
\begin{equation*}
\tilde G=\bigcup_{n\in\upomega}(X\cup X\inv)^{\leq n},
\end{equation*}
where $(X\cup X\inv)^{\leq n}$ comprises the elements of $\tilde G$ that, as words in $X$, have length $n$ or less.

We can form the isomorphic Boolean rings $\Def{\tilde G}{\U}$ and $\Lin{\tilde G}{\U}$.
But (apparently) these are too big.  Let
\begin{equation*}
\Def{\tilde G}{\U}^*=\bigcup_{n\in\upomega}(\Def{\tilde G}{\U}\cap\pow{(X\cup X\inv)^{\leq n}}).
\end{equation*}
(The notation is mine, although van den Dries uses the star for restrictions to $XX\inv X$.  Hrushovski refers to elements of $\Def{\tilde G}{\U}$ as definable, though to avoid confusion one might call them `star-definable'.)  We can let $\Lin{\tilde G}{\U}^*$ be the image of $\Def{\tilde G}{\U}^*$ in $\Lin{\tilde G}{\U}$.

Now let $\str M$ be another model (probably $\str M_0\included\str M$).  

Suppose $I$ is an ideal of $\Lin{\tilde G}{\U}^*$ that is
\begin{compactitem}
\item
$\str M$-invariant and
\item
$\Sone$ over $\str M$.
\end{compactitem}
Suppose also that $I$ is invariant (or closed) under left and right
translation by elements of $\tilde G$, that is, for all $g$ in $\tilde
G$, 
\begin{equation*}
\text{if $\phi(x)\in I$, \qquad then $\phi(g\inv x)$ and
  $\phi(xg\inv)$ are in $I$.}
\end{equation*}

Let $q=\tp a{\str M}$ for some $a$ in $\tilde G$, and suppose $q$ is $I$-wide.  That is, suppose $a\notin\phi^{\U}$ for any $\phi$ in $I$, and let $q=\tp a{\str M}$.

Suppose further that for some other realization $b$ of $q$, both $\tp a{b\str M}$ and $\tp b{a\str M}$ do not fork over $\str M$.

Then there will be a certain normal subgroup $S$ of $\tilde G$ that is $\bigwedge$-definable over $\str M$.

$S$ (or rather its defining partial type) will be $I$-wide.

Hrushovski says  $S=(q\inv q)^2$, which we may write as
\begin{equation*}
S=q\inv qq\inv q;
\end{equation*}
and he says $qq\inv q$ is a coset of $S$.  He says moreover that, as a
consequence of the theorem, $S\included XX\inv XX\inv$, that is, 
\begin{equation*}
S\included X\inv XX\inv X.
\end{equation*}
But where Hrushovski has $q$, van den Dries has $q(X)$, which would
seem to mean the realizations of $q$ that belong to $X$.  (He does not
write $q(\hat X)$.) 

\appendix

\section{Sorted structures}

In the beginning, a \textbf{structure} is a set with distinguished
\emph{elements, operations,} and \emph{relations.}  The set is the
\textbf{universe} of the structure.  If this universe is $A$,
then a \textbf{relation} is a subset of some Cartesian power
$A^n$, where $n\in\upomega$; and an \textbf{operation} is a
function from 
some $A^n$ to $A$.  If $R$ is a relation on $A$, and $\vec b\in R$, we
write the \textbf{atomic sentence} 
\begin{equation*}
R\vec b.
\end{equation*}
In the binary case we usually write $b\mathrel Rc$ when $(b,c)\in R$.
If $f$ is an operation on $A$, then it is a certain kind of relation
on $A$, namely a relation for which it is meaningful to write another
\textbf{atomic sentence,} 
\begin{equation*}
f\vec b=c,
\end{equation*}
when $(\vec b,c)\in f$.  In the ternary case we may write $b\mathbin
fc=d$ when $(b,c,d)\in f$.  We may allow $f$ to be a nullary
operation, in which case $f$ by itself denotes an element of $A$. 
We combine atomic formulas and quantify their variables in the usual
way, for the sake of describing structures. 

This definition of structure turns out to be needlessly limiting.

On a \emph{field} $K$, division is not an operation; it is only a
`partial' operation, a function $(x,y)\mapsto x/y$ from
$K\times(K\setminus\{0\})$ to $K$.  We normally want a binary
operation $*$ to be `total', so that $x*y$ is always meaningful.  This
is a notational convenience, which we can achieve in the present case
by defining $x/0$ as $0$.  Alternatively, we can treat a field as
\emph{two} structures, abelian groups $(K,+)$ and
$(K^{\times},{}\cdot{})$, with the appropriate interactions, including
the function $(x,y)\mapsto x/y$ from $K\times K^{\times}$ to $K$, and
the relation $\{(x,x)\colon x\in K\setminus\{0\}\}$ from $K^{\times}$
to $K$. 

A vector space is also a pair of structures, namely an abelian group $V$ of
vectors and a field $K$ of scalars; and there is an action of the latter on
the former, that is, a certain function from $K\times V$ to $V$.  We
could treat this pair as a structure whose universe is the disjoint
union of $K$ and $V$; then each of these components would be a
singulary relation on the universe.  It may however be considered ugly
to introduce symbols for these relations.  Convention already supplies
a way to keep the two sets apart: we let boldface letters like $\mathbf v$
denote vectors, and plainface letters like $a$ denote scalars. 

Given a group $G$, we may want to consider it together with all of its
quotient groups.  If $M<N$, both being normal subgroups of $G$, then
there is a function $xM\mapsto xN$ from $G/M$ onto $G/N$.  Here we
probably do not want to treat the (disjoint) union of all of these
quotients as the universe of a structure, because, if there are
infinitely many quotients, then the Compactness Theorem would give us
an elementary extension with elements not in any of the original
quotients. 

We might restrict the last example so that the quotients of $G$ are
all \emph{definable} in $G$; then we might generalize so that,
starting from a structure $\str A$, for every definable relation $R$
on $A$, for every definable equivalence relation $E$ on $R$, we
consider a new structure whose universe is the quotient $R/E$. 

Now we can generalize the original definition.
On an indexed family $(A_s\colon s\in S)$ of sets, a \textbf{relation}
is a subset of $\prod_{i\in I}A_i$ for some finite subset $I$ of $S$,
and an \textbf{operation} is a function from $\prod_{i\in I}A_i$ to
$A_s$ for some finite subset of $I$ of $S$ and some $s$ in $S$.  A
\textbf{structure} then is an \emph{indexed} family of sets with some
relations and operations.  The sets in the family are \textbf{sorts.}
In a formula, the arguments of a relation symbol or a function symbol
carry the information that they belong to the appropriate sort.  Thus
there are no variables simply; there are \textbf{$s$-variables} for
the various indices $s$ of sorts.  There is also no requirement that
the sorts be disjoint, since our symbolism refers to an element of a
sort only through its index. 

%\bibliographystyle{amsplain}
%\bibliography{../references}

\def\rasp{\leavevmode\raise.45ex\hbox{$\rhook$}} \def\cprime{$'$}
  \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{1}

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\bibitem{MR1162433}
Zo{\'e} Chatzidakis, Lou van~den Dries, and Angus Macintyre, \emph{Definable
  sets over finite fields}, J. Reine Angew. Math. \textbf{427} (1992),
  107--135. \MR{1162433 (94c:03049)}

\bibitem{MR94e:03002}
Wilfrid Hodges, \emph{Model theory}, Encyclopedia of Mathematics and its
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\bibitem{MR2159717}
Ehud Hrushovski, \emph{Pseudo-finite fields and related structures}, Model
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\bibitem{MR2833482}
\bysame, \emph{Stable group theory and approximate subgroups}, J. Amer. Math.
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\bibitem{Pillay-first-order}
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\bibitem{MR719195}
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\bibitem{TT-H}
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\end{thebibliography}


\end{document}
