# On $pp$-elimination and stability in a continuous setting

Nicolas Chavarria Gomez
University of Notre Dame  Anand Pillay
University of Notre Dame Supported by NSF grants DMS-1665035, DMS-1760212,
DMS-2054271, and a SDV at the Fields Institute, Toronto.

###### Abstract

We generalize “$pp$-elimination” for modules, or more generally, abelian
structures, to a continuous logic setting where the abelian structure is
equipped with a homomorphism to a compact (Hausdorff) group. We conclude that
the continuous logic theory of such a structure is stable.

##  1 Introduction

The model theory of modules, or more generally abelian structures, comprises
an important chapter in model theory and mathematical logic, feeding into
geometric stability theory (see Chapter 4 of [10]) as well as representation
theory [11].

Among the key facts in the context of left $R$-modules, for $R$ a unitary
ring, and the corresponding language $L_{R}$, is $pp$ elimination: for a given
module $M$, any formula $\phi(\bar{x})$ is equivalent in $M$ to a (finite)
Boolean combination of $pp$ (positive primitive) formulas.

In unpublished work, Fisher introduced the more general notion of an abelian
structure as a many sorted structure, where each sort has an abelian group
structure, and the distinguished relations are subgroups of Cartesian powers
of sorts, and where similar theorems hold. In particular one obtains stability
of the first order theories of abelian structures, in a very strong form which
has come to be known as $1$-basedness. Moreover any one-based group is
essentially an abelian structure.

It is natural to ask what, if anything, is the analogue of an abelian
structure in a continuous logic framework where formulas are real-valued
rather than Boolean valued; in particular, what is the continuous logic
analogue of a one-based group. Moreover, are there analogues of
$pp$-elimination? This is what we try to answer in the current paper.

In addition to the motivation described above, some other inspiration came
from Hrushovski’s recent work on the theory of finite fields with an additive
character, as a theory in continuous logic [8]. Hrushovski proved (among many
other things) simplicity of the theory, so it was natural to imagine that with
only the additive structure on the field we would have stability.

Our set-up will, for simplicity of presentation, be a one-sorted abelian
structure $A$, that is an abelian group $(A,+,-,0)$ equipped with a collection
$\mathcal{S}$ of subgroups of various Cartesian powers of $A$ (including $=$,
the diagonal in $A^{2}$), together with a homomorphism $f$ from $A$ to a
compact (Hausdorff) group $\mathbb{T}$ which we may and will assume to be also
commutative, and which we will write additively. Our results will generalize
without much trouble to that of a many sorted abelian structure equipped with
a compatible collection of homomorphisms from the sorts to compact Hausdorff
groups. We write this “structure” as
$M=(A,+,-,0,P,f,{\mathbb{T}})_{P\in{\mathcal{S}}}$. The precise formalism will
be described in detail in the next section, where we give a continuous logic
appropriate for the study of a first order structure equipped with a map to a
compact space, and will be closer to [7] than to [2], although ultimately
equivalent.

For now we will just give an informal description of the results and notions.

The first order logic (FOL) part of the structure is obtained by forgetting
about $\mathbb{T}$ and $f$, and is just the abelian structure
$M^{-}=(A,+,-,0,P)_{P\in{\mathcal{S}}}$, in the appropriate language $L^{-}$.
The FOL-atomic formulas are of the form $P({\bar{t}}({\bar{x}}))$, where
$P\in{\mathcal{S}}$ and ${\bar{t}}$ is a sequence of terms. The positive
primitive ($pp$) formulas are by definition formulas of the form
$\exists{\bar{y}}(\phi({\bar{x}},{\bar{y}}))$ where
$\phi({\bar{x}},{\bar{y}})$ is a (finite) conjunction of FOL-atomic formulas
(and we witness all free variables which occur). So we repeat that in so far
as this FOL-structure is concerned, the classical results say that every
formula is equivalent to a Boolean combination of positive primitive formulas
(see [6] for example). By a $pp^{*}$-formula for $M$ we will mean something of
the form $\exists{\bar{y}}(\phi({\bar{x}},{\bar{y}}))$ where now
$\phi({\bar{x}},{\bar{y}})$ is a finite conjunction of FOL-atomic formulas and
“expressions” of the form $f(x_{i})=c$, $f(y_{j})=c$ for $c\in{\mathbb{T}}$
and $x_{i},y_{j}$ variables (ranging over $A$) from ${\bar{x}}$, ${\bar{y}}$
respectively.

Note that a $pp$ formula $\phi({\bar{x}})$ defines a subgroup of the relevant
Cartesian power $A^{n}$ of $A$, which we call a $pp$-subgroup. If
$\phi({\bar{x}})$ is a $pp^{*}$-formula and all the parameters $c$ from
$\mathbb{T}$ which appear in the formula are $0$, then again $\phi({\bar{x}})$
defines a subgroup of the relevant $A^{n}$, which we call a $pp^{*}$-subgroup
of $A^{n}$. In general a $pp^{*}$-formula will define a coset of a
$pp^{*}$-subgroup.

We will be defining syntax and semantics, in particular the language $L$ of
the structure $M$, the notions of continuous logic (CL) $L$-formulas,
saturation, etc. All $FOL$ formulas in the language $L^{-}$ of $M^{-}$ will be
continuous logic $L$\- formulas, and $pp^{*}$-formulas will also be such
formulas.

We will say that the “continuous theory” $Th_{CL}(M)$ is stable if whenever
$({\bar{a}}_{i},{\bar{b}}_{i})$ is an indiscernible sequence in a saturated
model and $i<j$ then $({\bar{a}}_{i},{\bar{b}}_{j})$ has the same type as
$({\bar{a}}_{j},{\bar{b}}_{i})$.

Our results, expressed somewhat informally, are:

######  Theorem 1.1.

(informal version.) Assume $M$ to be saturated (in the appropriate sense).
Then two finite tuples with the same $pp^{*}$-type have the same type. We
conclude that $Th_{CL}(M)$ is stable, as well as every formula being
equivalent (in a suitable approximate sense) to a suitable combination of
$pp^{*}$-formulas and negated $pp$-formulas.

The precise statements, in light of the “theory” developed in the next
section, appear in Theorem 3.3, Corollary 3.8, and Remark 3.7.

Thanks to Ehud Hrushovski and Tommy Kucera for some comments when the second
author gave a preliminary talk on this material in Spring 2021. Kucera pointed
out the possible connection to his own work on the model theory of topological
modules in the Flum-Ziegler logic $L_{t}$ [9].

We would also like to thank Ward Henson for many discussions about continuous
logic in its various forms, as well as for encouraging us in the current
project.

##  2 Continuous logic

Here we will introduce a version of continuous logic suitable for our purposes
and for possible future work on related problems. The expression “continuous
logic” has come to be identified with the specific formalism introduced in [2]
and [3], and is basically the formalism used in [8]. Various earlier versions
include the theory developed by Chang and Keisler [4], and Henson’s
“approximate logic” or “positive bounded logic” [7]. There is also the
“positive logic and compact abstract theories” of Ben-Yaacov ([1]), which we
will not say much about, although it is relevant.

In Chang and Keisler’s monograph, [4], formulas take values in compact
Hausdorff spaces $C$. The set up (both syntax and semantics) is rather hard to
parse. I am unaware of any work by others which makes use of this set-up.
However, the monograph built on earlier work by Chang, which as far as we
understand was influenced by Lukasiewicz’s work on infinite valued
propositional logic.

Henson’s positive bounded logic is a logic for expansions of normed vector
spaces. There are two sorts, the vector space sort $V$ and the reals
$\mathbb{R}$. Formulas are either true or false. As far as we understand one
of the motivations was to find the right logic for formulating a Los theorem
appropriate for “metric ultraproducts”.

In the Ben-Yaacov, Berenstein, Henson, and Usvyatsov version [2], [3],
relations and formulas have values in the real unit interval $[0,1]$, and
structures are also equipped with a metric $d$ (with values in $[0,1]$) which
replaces the equality relation and coincides with it when the metric is
discrete. On the other hand they also talk about “conditions”, which are
typically of the form that a given formula has a given value, and these
conditions are true or false.

In all the set-ups the relevant (locally) compact spaces ($C$, $[0,1]$ or
$\mathbb{R}$) are fixed (i.e. not allowed to move when passing to other
models), whether they are treated as separate sorts or repositories of truth
values.

We will consider “continuous logic” structures $M$ of the form $(M^{-},f,C)$
where $M^{-}$ is a classical first order structure (with equality) and $f$ is
a map from the universe of $M^{-}$ to a compact Hausdorff space $C$. (This
easily extends to the context where $M^{-}$ is many sorted and we have a
collection of maps from Cartesian powers of various sorts of $M^{-}$ to
various compact spaces.) In particular the metric on $M$ is discrete. We could
use the tools of [2], as the map $f$ to $C$ can be understood in terms of maps
to $[0,1]$, and the classical first order structure on $M^{-}$ can be
understood in terms of $\\{0,1\\}$ valued formulas. But because our structure
is mixed (a classical first order part and a continuous part) it will be
conceptually convenient to consider $(M^{-},f,C)$ as two-sorted, where
moreover formulas are true or false. In fact our set-up will be close to that
of Henson and Iovino [7] in various ways, although our underlying structure
$M^{-}$ is not a normed vector space. The proofs of the basic model theory
results in this section are completely routine, but we may give some hints,
sketches, or references to [7] for similar things. The acronym FOL stands for
(classical) first order logic, and CL will stand for continuous (first order)
logic (denoted by CFOL in other places).

I guess that our point of view, especially regarding our context, is to see
continuous logic as the logic of the standard part map: what exactly is
preserved when passing to an elementary extension and applying the standard
part map to the “continuous part” of the structure.

We will start with the “structure” $M$, produce a language and then define
what other CL structures for the logic are, and then define CL-formulas and
the semantics.

So let us fix the structure $M=(M^{-},f,C)$.

Let $L^{-}$ be the language or vocabulary of the one sorted structure $M^{-}$
(including equality) as a FOL structure.

$L$ will be a $2$-sorted language, a sort $P$ for $M^{-}$ and a sort $Q$ for
$C$. $L^{-}$ will be a sublanguage of $L$, with the proviso that all the
symbols are restricted to the $P$-sort. We will have a function symbol
(without loss of generality $f$) for the function $f:M^{-}\to C$. We also have
predicate symbols $D$ for every closed subset $D$ of $C^{n}$ (as $n$ varies),
in particular a predicate for equality on $C$. Note that in particular we have
essentially constant symbols for each element $c$ of $C$ (represented as the
singleton $\\{c\\}$).

In an arbitrary FOL $L$-structure, the $P$ sort will be an $L^{-}$-structure,
and the $Q$ sort will be equipped with the various relations $D\subset Q^{n}$.
But:

######  Definition 2.1.

By a CL $L$-structure we mean an FOL $L$-structure such that the
interpretation of the $Q$-sort is precisely $C$ with the tautological
interpretations of predicate symbols $D$. In particular $M$ itself is a CL
$L$-structure.

In our context we may suppress $L$ and just talk about CL-structures.

Let us note in passing that the cardinality of $L$ will be greater than the
cardinality of $C$. We could economize a bit by working with a smaller
collection of closed subsets of $C^{n}$ (satisfying suitable denseness
properties), but in the interests of a simple presentation we will ignore this
for now.

Note that all the $L$-terms have domain in some $P^{n}$, and those that are
$Q$ valued consist of the composition of an $L^{-}$-term with $f$. We now
define the CL $L$-formulas, a subset of the collection of FOL $L$-formulas.
The only variables appearing in the CL formulas, whether free or quantified,
will be of sort $P$.

######  Definition 2.2.

(i) Any FOL $L^{-}$-formula is a CL formula.
(ii) Suppose that $t_{1},..,t_{n}$ are $Q$-valued terms and $D$ is (a
predicate symbol for) a closed subset of $C^{n}$, then $D(t_{1},..,t_{n})$
(which we may also write as $(t_{1},..,t_{n})\in D$) is a CL-formula. If $D$
is a singleton $\\{(c_{1},..,c_{n})\\}\in C^{n}$, we may write this formula as
${\bar{t}}={\bar{c}}$.
(iii) If $\phi$, $\psi$ are CL-formulas, so are $(\phi\wedge\psi)$ and
$(\phi\vee\psi)$.
(iv) If $\phi$ is a CL-formula and $x$ is a variable of sort $P$ then $\exists
x\phi$ and $\forall x\phi$ are CL-formulas.
(v) Nothing else is a CL-formula.
(vi) a CL sentence is a CL formula without free variables.

So there are no negations in CL-formulas other than inside
$L^{-}$-subformulas.

######  Remark 2.3.

As a CL-structure is also a (FOL) $L$-structure and a CL-formula is an FOL
$L$-formula, the usual satisfaction relation $N\models\phi({\bar{a}})$ where
$N$ is an $L$-structure, $\phi({\bar{x}})$ an $L$-formula, and ${\bar{a}}$ a
suitable tuple from $N$, specializes to the case where $N$ is a CL-structure
and $\phi({\bar{x}})$ a CL-formula, and ${\bar{a}}$ a tuple (from the $P$-sort
of $N$). So there is no need for a separate definition of truth for CL-
formulas and CL-structures.

In usual model theory, especially stability, we are often concerned just with
behaviour and definability in a saturated model. For the CL-version, the above
definitions suffice for working in “saturated models”, but to work with
“arbitrary models” (or even to define what we mean by saturation) we need the
notion of approximate truth and satisfaction, as in [7]. This is really about
the quantifiers. In “official” continuous logic [2] where formulas have values
in $[0,1]$, approximate truth is directly built into the set-up via the
quantifiers inf and sup in place of $\exists$ and $\forall$.

The following is an adaptation of some of the definitions in Chapters 5 and 6
of [7] to our context. It is a bit long-winded, but straightforward. It is
very similar to the situation with hyperimaginaries in FOL where we only
really see them in saturated models, and one can ask what exactly is discerned
about them in arbitrary models.

######  Definition 2.4.

(i) Let $D$ be a closed subset of $C^{n}$. By an approximation to $D$, we mean
a closed neighbourhood $D^{\prime}$ of $D$, namely a closed subset of $C^{n}$
such that for some open subset $U$ of $C^{n}$, we have that $D\subseteq
U\subseteq D^{\prime}$.
(ii) Let $\phi$ be a CL-formula. By an approximation to $\phi$ we mean a CL-
formula obtained from $\phi$ by replacing every appearance of any closed set
$D$ in $\phi$ by an approximation to $D$. The formal inductive definition is
similar to the definition at the bottom of p. 24 and top of p. 25 in [7].
(iii) For $\Gamma$ a set of CL-formulas, closed under finite conjunctions, let
$\Gamma^{+}$ be the set of approximations of formulas in $\Gamma$ (so with the
same free variables).
(iv) Let $\Gamma({\bar{x}})$ be a collection of CL-formulas with free
variables $\bar{x}$, let $N$ be a CL-structure, and ${\bar{a}}$ a tuple from
$N^{-}$ of the relevant size. We say that ${\bar{a}}$ approximately satisfies
(or realizes) $\Gamma({\bar{x}})$ in $N$,
$N\models_{approx}\Gamma({\bar{a}})$, if $N\models\phi({\bar{a}})$ for all
$\phi({\bar{x}})\in\Gamma^{+}$.
(v) Two CL-structures $N_{1}$, $N_{2}$ are said to be “approximately CL-
elementarily equivalent” if they approximately satisfy the same CL sentences.
(vi) Finally we say that (for CL-structures $N_{1}$, and $N_{2}$) that $N_{1}$
is an approximate elementary substructure of $N_{2}$, written
$N_{1}\prec_{approx}N_{2}$, if for every CL-formula $\phi({\bar{x}})$ and
${\bar{a}}$ a tuple from $N_{1}$, $N_{1}\models_{approx}\phi({\bar{a}})$ iff
$N_{2}\models_{approx}\phi({\bar{a}})$.

Note that if $\phi$ is a FOL $L^{-}$ formula then (by definition or
convention) the only approximations of $\phi$ are itself. Also if $D$ is a
clopen subset of $C^{n}$, then $D$ is an approximation of itself.

######  Remark 2.5.

Let $M$ be a CL-structure, $\bar{a}$ a tuple from $M$ and $\phi({\bar{x}})$ a
CL-formula.
(i) If $M\models\phi({\bar{a}})$ then $M\models_{approx}\phi({\bar{a}})$.
(ii) If $\phi$ is quantifier-free then $M\models\phi({\bar{a}})$ if and only
if $M\models_{approx}\phi({\bar{a}})$.

Given a CL-structure $N$ we can introduce constant symbols for some set
$A\subseteq N^{-}$ and talk about CL-formulas over $A$.

######  Proposition 2.6.

Let $N$ be a CL-structure, and $\Gamma({\bar{x}})$ be a collection of CL-
formulas over $N^{-}$ which is finitely approximately (or approximately
finitely) satisfiable in $N$, namely for every finite subset $\Gamma^{\prime}$
of $\Gamma$, we have
$N\models_{approx}\exists{\bar{x}}(\bigwedge\Gamma^{\prime}({\bar{x}}))$. Then
there is an approximate elementary extension $N^{\prime}$ of $N$ and some
tuple ${\bar{a}}$ from $N^{\prime}$ which realizes $\Gamma({\bar{x}})$ in
$N^{\prime}$.

###### Proof.

Let $\Gamma^{+}$ be the collection of approximations to the collection of
finite subsets of $\Gamma$. Consider $N$ as a FOL $L$-structure. Consider
$\Gamma^{+}$ as a set of $FOL$ $L$-formulas over $N$. By FOL compactness we
find a sufficiently saturated FOL elementary extension $N^{*}$ of $N$ and
${\bar{a}}$ in $N^{*}$ which realizes $\Gamma^{+}$. Let $C^{*}$ be the
interpretation of the $Q$ sort in $N^{*}$ and $f^{*}$ the interpretation in
$N^{*}$ of $f$ (from the $P$ sort to the $Q$ sort). As $C^{*}$ is equipped
with all the $L$-structure coming from $C$ we have the standard part map $\pi$
from $C^{*}$ onto $C$. Let $f^{\prime}$ be the composition of $f^{*}$ with
$\pi$ from $(N^{*})^{-}$ to $C$. We let $N^{\prime}=((N^{*})^{-},f,C)$ as a
CL-structure. We have to show that
(i) $N^{\prime}$ is an approximate elementary extension of $N$ and
(ii) ${\bar{a}}$ (which is in $N^{\prime-}$) realizes $\Gamma({\bar{x}})$ in
$N^{\prime}$.

(i) is left to the reader and only uses that $N^{*}$ is an FOL elementary
extension of $N$ (not the saturation of $N^{*}$).
We now consider (ii) which does use saturation of $N^{*}$ and we will do just
a special case, namely that of a single formula of the form $\phi({\bar{x}})$:
$\exists{\bar{y}}(f({\bar{x}},{\bar{y}})\in D)$ where ${\bar{x}}$ is an
$n$-tuple, ${\bar{y}}$ a $k$-tuple, and $D$ a (predicate symbol for) a closed
subset of $C^{n+k}$. The approximations to $\phi({\bar{x}})$ have the form
$\phi^{\prime}({\bar{x}})$: $\exists{\bar{y}}(f({\bar{x}},{\bar{y}})\in
D^{\prime})$ where $D^{\prime}$ is an approximation to $D$. Note that a finite
intersection of approximations to $D$ is also an approximation to $D$. By the
assumption that ${\bar{a}}$ realizes $\Gamma^{+}$ in $N^{*}$ and saturation of
$N^{*}$ there is ${\bar{b}}$ in $(N^{*})^{-}$ such that
$f^{*}({\bar{a}},{\bar{b}})\in(D^{\prime})^{*}$ for all approximations
$D^{\prime}$ to $D$. Hence (as $C$ was compact Hausdorff),
$f^{*}({\bar{a}},{\bar{b}})\in(D^{*})^{0}$ where $(D^{*})^{0}$ is the
intersection of all $U^{*}$ where $U$ is an open neighbourhood of $D$ in
$C^{n+k}$, and thus after composing with the standard part map $\pi$, we see
that $N^{\prime}\models f({\bar{a}},{\bar{b}})\in D$, so
$N^{\prime}\models\phi({\bar{a}})$. The case of general $\phi\in\Gamma$ is
proved by induction, but this existential case is the key step, and the
“positive” nature of the logic is precisely set up to allow the above argument
to go through, explaining why we referred to continuous logic as the logic of
the standard part map. ∎

######  Definition 2.7.

The CL-structure $M$ is said to be $\kappa$-saturated, if whenever
$\Gamma({\bar{x}})$ is a collection of CL-formulas over a subset $A$ of
$M^{-}$ of cardinality $<\kappa$ and $\Gamma$ is finitely approximately
satisfiable in $M$, then $\Gamma$ is satisfiable (or realized) in $M$.

######  Proposition 2.8.

(i) Every CL-structure has a $\kappa$-saturated approximate elementary
extension.
(ii) Suppose $N^{*}$ is a $\kappa$-saturated FOL $L$-structure, and
$N^{\prime}$ is the CL-structure obtained from $N^{*}$ by applying the
standard part map to the $Q$ sort (as in the proof of Proposition 2.6). Then
$N^{\prime}$ is $\kappa$-saturated as a CL-structure.
(iii) If $M$ is an $\omega$-saturated CL-structure and $\phi({\bar{x}})$ is a
CL-formula and ${\bar{a}}$ a tuple from $M^{-}$ then
$M\models_{approx}\phi({\bar{a}})$ iff $M\models\phi(\bar{a})$. In particular
for $\sigma$ an $L$-sentence $M\models_{approx}\sigma$ iff $M\models\sigma$

###### Proof.

(i) and (ii) follow from Proposition 2.6 and its proof. The proof of (iii) is
routine and by induction on the complexity of $\phi({\bar{x}})$.

∎

We will be mainly interested in saturated CL-structures, but in the background
are “complete” CL-theories. So we define, as in [7].

######  Definition 2.9.

(i) Let $M$ be a CL-structure. By the CL-theory of $M$, $Th_{CL}(M)$ we mean
the set of CL-sentences $\sigma$ such that $M\models_{approx}\sigma$.
(ii) By a complete CL-theory we mean $Th_{CL}(M)$ for some CL-structure $M$.
(iii) Given a complete CL-theory $T$, by a model of $T$ we mean a CL-structure
$N$ such that $N\models\sigma$ for all CL-sentences $\sigma\in T$.
(iv) Likewise by an approximate model of the complete CL-theory $T$ we mean a
CL-structure $N$ such that $N\models_{approx}\sigma$ for all $\sigma\in T$.

Maybe it would be better to call $Th_{CL}(M)$ as defined above the approximate
theory of $M$, but never mind. Anyway, ith our definitions, note that an
arbitrary $CL$-structure is only an approximate models of its CL-theory.

######  Remark 2.10.

(i) Let $M,N$ be CL-structures, then $Th_{CL}(M)=Th_{CL}(N)$ iff
$M\equiv_{approx}N$, namely $M$ and $N$ approximately satisfy the same CL-
sentences.
(ii) If $T$ is a complete CL-theory and $N$ is an $\omega$-saturated
approximate model of $T$ then $N$ is a model of $T$.
(iii) If $T$ is a complete CL-theory, then there is a closed (so compact)
subspace $C_{T}$ of $C$ such that for every approximate model $M=(M^{-},f,C)$
of $T$, $f(M^{-})$ is dense in $C_{T}$, and moreover if $M$ is
$\omega$-saturated then $f(M^{-})=C_{T}$.

###### Proof.

(i) and (ii) are obvious from the definitions and Proposition 2.8.
For (iii), fix an approximate model $M=(M^{-},f,C)$ of $T$, and let
$C_{0}=cl(f(M^{-}))$. So for every $c\in C_{0}$ and closed neighbourhood $D$
of $\\{c\\}$, $M\models\exists x(f(x)\in D)$. So the sentence $\exists
x(f(x)=c)$ is approximately true in $M$ so is in $T=Th_{CL}(M)$, so
approximately true in all models of $T$, and actually true in any
$\omega$-saturated model of $T$, by 2.8. ∎

So given the complete CL-theory $T$, and $C_{T}$ as in Remark 2.10(iii), we
may assume that in the underlying language $L$, $C=C_{T}$, namely that in a
saturated model $f$ is surjective.

There is no harm in assuming that every complete CL-theory $T$ has
$\kappa$-saturated models of cardinality $\kappa$ for suitable $\kappa$. And
two such models of $T$ will be isomorphic. As usual given a complete CL-theory
$T$ we feel free to work in such a $\kappa$-saturated model ${\bar{M}}$ of
size $\kappa$ for some very big $\kappa$. But we should realize that arbitrary
models of $T$ (of cardinality $\leq\kappa$) will in general be only
approximately elementarily embeddable in $\bar{M}$.

We now discuss types and type spaces. For $A\subset\bar{M}$ and ${\bar{b}}$ a
tuple from $M$, we will compute $tp({\bar{b}}/A)$ in ${\bar{M}}$ and it is
just the collection of CL-formulas $\phi({\bar{x}})$ over $A$ such that
$\bar{M}\models\phi({\bar{b}})$.

We now fix a complete CL theory $T$ and saturated model ${\bar{M}}$ as above.
${\bar{x}}=(x_{1},..,x_{n})$ is a tuple of variables ranging over the $P$
sort).

######  Definition 2.11.

$S_{\bar{x}}(A)$ (or $S_{n}(A)$) is the set of $tp({\bar{b}}/A)$ for
${\bar{b}}$ in $\bar{M}$, equipped with the topology: $X\subseteq
S_{\bar{x}}(A)$ is a basic closed set iff there is some CL-formula
$\phi({\bar{x}})$ over $A$ such that
$X=\\{tp({\bar{b}}/A):{\bar{M}}\models\phi({\bar{b}})\\}$.

Then by the compactness theorem $S_{\bar{x}}(A)$ is a compact Hausdorff space.
If $A$ is omitted we mean types over $\emptyset$.

If $\Phi$ is a family of CL-formulas, then we can talk about
$tp_{\Phi}({\bar{b}}/A)$ (or sometimes $\Phi tp({\bar{b}})$): it is the
collection of $CL$-formulas with parameters $\phi({\bar{x}},{\bar{a}})$ such
that $\phi({\bar{x}},{\bar{y}})$ is in $\Phi$, ${\bar{a}}$ is a tuple from $A$
and ${\bar{M}}\models\phi({\bar{b}},{\bar{a}})$.

######  Definition 2.12.

We will say that $T$ has “quantifier elimination” down to formulas in $\Phi$,
if whenever ${\bar{b}}$ and ${\bar{c}}$ are finite tuples of the same length,
and $tp_{\Phi}({\bar{b}})=tp_{\Phi}({\bar{c}})$ then
$tp({\bar{b}})=tp({\bar{c}})$.

This notion is rather more meaningful when the collection $\Phi$ has some
closure properties. We will call the collection $\Phi$ of CL-formulas closed
if $\phi$ contains the quantifier-free formulas, and is closed under $\wedge$,
$\vee$, and $\neg$ (i.e. if $\phi$ is a FOL $L^{-}$-formula which is in
$\Phi$, then so is $\neg\phi$), as well as approximations. Fix the tuple
${\bar{x}}$ of variables. Then $S_{n,\Phi}$ denotes the set of
$tp_{\Phi}({\bar{b}})$ for ${\bar{b}}$ an $n$-tuple from ${\bar{M}}$. Assuming
that $\Phi$ is closed, $S_{n,\Phi}(T)$ is a compact Hausdorff space when
equipped with the topology where a basic closed set is the collection of
$\Phi$-types containing a given formula $\phi({\bar{x}})\in\Phi$.

######  Lemma 2.13.

Assume that $\Phi$ is closed. Then $T$ has quantifier elimination down to
formulas in $\Phi$ if for every CL-formula $\phi({\bar{x}})$ and approximation
to it, $\phi^{\prime}({\bar{x}})$ there is a formula
$\theta({\bar{x}})\in\Phi$ such that in any approximate model $M$ of $T$ for
any tuple ${\bar{a}}$ from $M$, we have: if $M\models_{approx}\phi({\bar{a}})$
implies $M\models_{approx}\theta({\bar{a}})$, and if
$M\models_{approx}\theta({\bar{a}})$ then
$M\models_{approx}\phi^{\prime}({\bar{a}})$.

###### Proof.

We will fix $n$ and consider CL-formulas with $n$ variables and work in the
type spaces $S_{n}(T)$, and $S_{n,\Phi}(T)$. We identify formulas with the
sets they define. The “quantifier elimination” assumption implies that the
restriction map from $S_{n}(T)$ to $S_{n,\Phi}(T)$ is a homeomorphism. Let
$\phi({\bar{x}})$ be a CL-formula, so $\phi$ defines a closed set in
$S_{n,\Phi}(T)$, namely given by a possibly infinite conjunction
$\bigwedge_{i\in I}\psi_{i}({\bar{x}})$ of $\Phi$-formulas. If this was a
finite conjunction we would be finished, but it need not be, so we require
some approximations. Choose an arbitrary approximation $\phi^{\prime}$ to
$\phi$. As we can always make $\phi^{\prime}$ smaller, we may assume that
$\phi^{\prime}$ is the closure of an open neighbourhood $U$ of $\phi$ (in the
type space). The complement of $U$ is closed, so compact and covered by the
union of the complements of the $\psi_{i}$’s, so by a finite subunion. It
follows that for some $n$ we have that
$\phi({\bar{x}})\subseteq\bigwedge\psi_{i}({\bar{x}})\subseteq\phi^{\prime}({\bar{x}})$.
Let $\theta({\bar{x}})=\wedge_{i=1,..,n}\psi_{i}({\bar{x}})$. So now working
with truth of formulas in the saturated model ${\bar{M}}$ of $T$ we have that
$\phi({\bar{x}})$ then $\theta({\bar{x}})$ implies $\phi^{\prime}({\bar{x}})$.
This suffices, using Proposition 2.8 (iii).

∎

Finally we discuss stability. We have already discussed types, and thus we
obtain the notion of an indiscernible sequence.

######  Definition 2.14.

$T$ is stable if whenever $({\bar{a}}_{i},{\bar{b}}_{i}):i<\omega)$ is an
indiscerible sequence then
$tp(({\bar{a}}_{i},{\bar{b}}_{j}))=tp(({\bar{a}}_{j},{\bar{b}}_{i}))$ whenever
$i<j$.

One can adapt this definition to give stability of any given CL-formula
$\phi({\bar{x}},{\bar{y}})$. In any case, the definition makes sense in the
classical FOL context where it is correct.

We finish this expository section with some general remarks, questions and
problems:

######  Problem 2.15.

One possible point of view is that the formalism developed in this section
could be a mechanism for adjoining new KP-strong types to a first order
theory. So one should formulate a version of the set-up where any
homeomorphism of $C$ with itself lifts to an automorphism of a saturated model
$M$. It can probaby be done by working, not with all closed subsets of $C^{n}$
but with all closed subsets invariant under homeomorphisms of $C$.

The second question concerns under what situations one of our CL-theories is
“essentially first order”. For example:

######  Remark 2.16.

Suppose that $M=(M^{-},f,C)$ is a CL-structure, where $C$ is profinite (i.e.
has a basis of clopens). Let $M^{--}$ be the (first order) expansion of
$M^{-}$ by adding predicates for preimages of all clopen subsets of $C^{n}$
under $f$. Then the first order theory $Th(M^{--})$ of $M^{--}$ is the “same
as” $Th_{CL}(M)$, for example has the same models/approximate models.

######  Problem 2.17.

Suppose that $M^{-}$ is a saturated FOL structure in language $L^{-}$. Let $E$
be a bounded $\emptyset$-type-definable equivalence relation on $M^{-}$ (or on
some sort in $M^{-}$). Formalise the idea that adjoining $X/E$ as a new sort
and working as in Problem 2.15 “adds no new structure to $M^{-}$”.

It will be routine to define the notion of $1$-basedness for a stable CL-
theory, The content of the main result of this paper is that the CL-theory of
an abelian structure equipped with a homomorphism to a compact Hausdorff group
is not only stable but also $1$-based. It would be nice to have a converse,
extending the theorems about $1$-based groups in FOL to the current CL
context.

######  Problem 2.18.

Suppose that $G$ is an FOL expansion of a group, and $f:G\to C$ is a map to a
compact space $C$. Suppose that the CL-theory of $M=(G,f,C)$ is stable and
$1$-based. Show that $G$ is essentially an abelian structure and up to a
suitable sense of piecewise translation $C$ has a group structure compatible
with its topology and $f$ is a homomorphism.

##  3 Abelian structures with a homomorphism to a compact group

Having introduced our formalism in Section 2, we can now be more precise about
the results of this paper.

We will take an abelian structure $A$ to be an abelian group $(A,+,-,0)$
equipped with a family ${\mathcal{S}}$ of subgroups of various Cartesian
powers of $A$. We treat $A$ as a structure in a language $L^{-}$ (with
equality) which has function symbols $+$, $-$, constant symbol $0$ and
predicate (or relation) symbols $S$ for each $S\in{\mathcal{S}}$. One can
identify equality with a distinguished subgroup $S_{=}$ of $A^{2}$, if one
wishes.

An atomic $L^{-}$ formula is by definition a formula of the form
$P(t_{1},..,t_{n})$ where $t_{1},..,t_{n}$ are terms and $P\in{\mathcal{S}}$
is $n$-ary. And by definition a positive primitive $L^{-}$-formula is
something of the form $\exists{\bar{y}}(\phi({\bar{x}},{\bar{y}}))$ where
$\phi$ is a finite conjunction of atomic formulas, and we exhibit all the free
variables of $\phi$ by the tuple ${\bar{x}},{\bar{y}}$.

On the other hand given a ring $R$ (with a $1$), the language $L_{R}$ of
$R$-modules has again symbols $+,-,0,1$ as well as unary function symbols
$\lambda_{r}$ for each $r\in R$. The only relation symbol is equality. One
again has the notion of a positive primitive formula Any (left) $R$ module can
be considered as an abelian structure by replacing each $\lambda_{r}$ by its
graph. Then the positive primitive formulas of $L_{R}$ coincide with those in
the corresponding abelian structure language, insofar as their interpretations
are concerned. So in this sense $R$-modules are a special case of abelian
structures, and from now on we will only talk about (enriched) abelian
structures.

It is well-known and immediate to check that if $\phi({\bar{x}},{\bar{y}})$ is
a $pp$ formula of $L^{-}$, then $\phi({\bar{x}},{\bar{0}})$ defines a subgroup
of $A^{n}$ (a so-called $pp$-subgroup) and that for any tuple ${\bar{b}}$ from
$A$ (of length that of ${\bar{y}}$), the formula $\phi({\bar{x}},{\bar{b}})$
defines a coset (translate) of the subgroup.

We now add a homomorphism $f$ from $A$ to a compact Hausdorff group which we
call $\mathbb{T}$, and we are interested in $M=(A,f,{\mathbb{T}})$ as a CL-
structure and its theory $T=Th_{CL}(M)$ as a CL-theory. (So when we write $A$
we have implicitly all its abelian structure.) So $T$ is a complete CL-theory
in a language $L$, which is fixed from now on. From Remark 2.10 (iii), we may
assume that $f(A)$ is dense in ${\mathbb{T}}$.

######  Remark 3.1.

Let $(B,g,{\mathbb{T}})$ be an approximate model of $T$. Then $(B,+,-,0)$ is
an abelian group, the interpretations of the distinguished predicate symbols
of $L^{-}$ are subgroups (or the appropriate $B^{n}$), and $g$ is a
homomorphism (with dense image).

###### Proof.

As the first order $L^{-}$ theory of $B$ is part of $T$, the first two parts
are immediate. For the last part, note that the expression $\sigma$: $\forall
x\forall y(f(x+y)-f(x)-f(y)=0)$ is a CL-sentence which is in $T$. If
$(B,g,{\mathbb{T}})$ is an approximate model of $\sigma$, then it is also a
model of $\sigma$, so $g$ is a homomorphism. $g(B)$ being dense follows from
2.10 (iii). ∎

######  Definition 3.2.

By a $pp^{*}$-formula we mean informally a $pp$-$L^{-}$-formula,
$\exists{\bar{x}}(\phi({\bar{x}},{\bar{y}}))$ where $\phi$ is a conjunction of
atomic $L^{-}$ formulas, and where in addition we specify the values under $f$
of none, some, or all, of the variables in ${\bar{x}}$, ${\bar{y}}$. More
formally, a $pp^{*}$-formula, is a CL-formula of the following form:

$\exists{\bar{y}}(\phi({\bar{x}},{\bar{y}})\wedge\bigwedge_{i\in
I}(f(x_{i})=c_{i})\wedge\bigwedge_{j\in J}(f(y_{j})=d_{j}))$

where $\phi({\bar{x}},{\bar{y}})$ is a finite conjunction of atomic
$L^{-}$-formulas, $I$ is a subset of the indices of ${\bar{x}}$, and $J$ a
subset of the indices of ${\bar{y}}$.

We now pass to a saturated model of $T$, which we again call
$M=(A,f,{\mathbb{T}}$ and now $f$ is surjective. By a $pp$-formula we mean a
$pp$ $L^{-}$ formula. Such a formula defines a subgroup of the relevant
$A^{n}$ called a $pp$-subgroup. By a $pp^{-}$ formula we mean the negation of
a $pp$-formula which, note is a CL-formula. Given a $pp^{*}$-formula
$\psi({\bar{x}})$ as in Definition 3.2 where any $c_{i}$, $d_{j}$ which appear
are $0$, then $\psi({\bar{x}})$ also defines a subgroup of $A^{n}$, which we
may call a $pp^{*}$-subgroup. Of course a $pp$-subgroup is a special case of a
$pp^{*}$-subgroup. As in the proof of 2.10 (iii) the image of a $pp^{*}$
subgroup of $A^{n}$ is a closed subgroup of ${\mathbb{T}}^{n}$.

We often identify formulas with the sets the define. We will sometimes let $K$
denote $Ker(f)$, a subgroup of $A$. The notions $pp^{*}$-type and $pp$-type
are the obvious ones (as with $\Phi$-type in Section 2).

Our main theorem, “quantifier elimination down to $pp^{*}$-formulas” is:

######  Theorem 3.3.

Suppose $\bar{a}$ and $\bar{b}$ are finite tuples from $M$ (namely from the
sort $A$) such that $pp^{*}tp({\bar{a}})=pp^{*}tp({\bar{b}})$ then
$tp({\bar{a}})=tp({\bar{b}})$.

Here $tp$ refers to CL-type in the structure $M$.

Theorem 3.3 will be proved by a standard back-and-forth argument. We will need
the following lemma, the proof of which is a simple version of the proof of
the Theorem.

######  Lemma 3.4.

Let ${\bar{a}}$, ${\bar{b}}$ be tuples of the same length in $M$. Suppose that
$pp^{*}tp({\bar{a}})\subseteq pp^{*}tp({\bar{b}})$ and
$pp^{-}tp({\bar{a}})\subseteq pp^{-}tp({\bar{b}})$. Then
$pp^{*}tp({\bar{a}})=pp^{*}tp({\bar{b}})$.

###### Proof.

Given the hypotheses we have to prove that $pp^{*}tp({\bar{b}})\subseteq
pp^{*}tp({\bar{a}})$. Let $\psi({\bar{x}})$ be a $pp^{*}$-formula:
$\exists{\bar{y}}(\phi({\bar{x}},{\bar{y}})\wedge{\bar{f}}({\bar{x}})={\bar{r}}\wedge{\bar{f}}({\bar{y}})={\bar{s}})$,
where notation has the obvious interpretation (in particular we are specifying
the value under $f$ of all variables). And suppose that
$M\models\psi({\bar{b}})$. First note that the value of ${\bar{a}}$ under
${\bar{f}}$ is part of $pp^{*}tp({\bar{a}})$, so holds of ${\bar{b}}$ too,
whereby
(*) ${\bar{f}}({\bar{a}})={\bar{r}}$.

Now $\exists{\bar{y}}(\phi({\bar{x}},{\bar{y}}))$ is a $pp$-formula true of
${\bar{b}}$. Our assumptions imply that this $pp$-formula is true of
${\bar{a}}$. So the $pp$-formula with parameters $\phi({\bar{a}},{\bar{y}})$
is consistent, hence defines a coset (in suitable $A^{m}$) of the subgroup of
$A^{m}$ defined by $\phi({\bar{0}},{\bar{y}})$. By remarks above
${\bar{f}}(\phi({\bar{0}},{\bar{y}}))$ is a closed subgroup $H$ of
${\mathbb{T}}^{m}$, and ${\bar{f}}(\phi({\bar{a}},{\bar{y}}))$ defines a coset
(translate) $C$ of $H$ in ${\mathbb{T}}^{m}$. But $\phi({\bar{b}},{\bar{y}})$
is nonempty, so also ${\bar{f}}(\phi({\bar{b}},{\bar{y}}))$ is a coset $D$ say
of $H$ in ${\mathbb{T}}^{m}$. Pick any ${\bar{s}}^{\prime}$ in $C$. So
$M\models\exists{\bar{y}}(\phi({\bar{a}},{\bar{y}})\wedge{\bar{f}}({\bar{y}})={\bar{s}}^{\prime})$.
By our assumptions
$M\models\exists{\bar{y}}(\phi({\bar{b}},{\bar{y}})\wedge{\bar{f}}({\bar{y}})={\bar{s}}^{\prime})$
too. This means that $C\cap D\neq\emptyset$, hence $C=D$, as both are cosets
of the same subgroup $H$ of ${\mathbb{T}}^{m}$. Hence, as ${\bar{s}}\in D$,
${\bar{s}}\in C$ too, hence
$M\models\exists{\bar{y}}(\phi({\bar{a}},{\bar{y}})\wedge{\bar{f}}({\bar{y}})={\bar{s}})$.
Together with (*) this shows that $M\models\psi({\bar{a}})$, which completes
the proof of the lemma.

∎

Now we can give:
Proof of Theorem 3.3.
This will be closely related to a standard proof of $pp$ elimination in
modules (see [12], [6]) and recovers that result.

As the $pp^{*}$-type of a tuple contains the quantifier-free type of that
tuple, it will be enough to prove the following “back-and-forth” statement:

######  Lemma 3.5.

Suppose that ${\bar{a}}$ and ${\bar{b}}$ are finite tuples from $M$ (or rather
$A$) of the same length and $c\in A$. Then there is $d\in A$ such that
$({\bar{a}},c)$ and $({\bar{b}},d)$ have the same $pp^{*}$-type.

###### Proof.

Let $q({\bar{x}},y)=pp^{*}tp({\bar{a}},c)$. and let $q_{0}=pptp({\bar{a}},c)$.
If we can find $d$ satisfying
$q({\bar{b}},y)\cup\\{\neg\psi({\bar{b}},y):\psi({\bar{x}},y)\notin q_{0}\\}$,
then by Lemma 3.4 we will have that
$pp^{*}tp({\bar{a}},c)=pp^{*}tp({\bar{b}},d)$.

So we will prove:
CLAIM. $q({\bar{b}},y)\cup\\{\neg\psi({\bar{b}},y):\psi({\bar{x}},y)\notin
q_{0}\\}$ is consistent (i.e. realized in $M$).
Proof of CLAIM. By compactness it is enough to show that any finite subset of
the above set of CL-formulas is consistent. As $q$ is closed under finite
conjunctions we must show that for any $pp^{*}$-formula $\chi({\bar{x}},y)$ in
$q$ and $pp$-formulas $\psi_{1}({\bar{x}},y)$,..,$\psi_{n}({\bar{x}},y)$, not
in $q_{0}$, that:

(*)
$\chi({\bar{b}},y)\wedge\neg\psi_{1}({\bar{b}},y)\wedge...\wedge\neg\psi_{n}({\bar{b}},y)$
has a solution in $M$.

Identifying formulas with their solutions in $M$ this means we have to show
that $\chi({\bar{b}},y)$ is not covered by the union of the
$\psi_{i}({\bar{b}},y)$ ($i=1,..,n$).

Let us write explicitly $\chi({\bar{x}},y)$ as
$\exists{\bar{z}}(\phi({\bar{x}},y,{\bar{z}})\wedge({\bar{f}}({\bar{x}})={\bar{r}})\wedge(f(y)=s)\wedge({\bar{f}}({\bar{z}})={\bar{t}}))$.
(Note we can assume that all values of the variables $x_{i},y,z_{i}$ under $f$
are specified in $\chi$.)

Then $\phi({\bar{b}},y)$ defines a coset $X$ of the subgroup $H$ of $K$
defined by
$\exists{\bar{z}}(\phi({\bar{0}},y,{\bar{z}})\wedge(f(y)=0)\wedge({\bar{f}}({\bar{z}})={\bar{0}}))$.
On the other hand $\psi_{i}({\bar{b}},y)$ defines a coset of the subgroup
$H_{i}$ of $A$ defined by $\psi_{i}({\bar{0}},y)$.

Now the Neumann lemma (see Lemma 1.4 of [12]) says that $X$ is covered by the
$X_{i}$’s iff $X$ is covered by those $X_{i}$’s such that $H\cap H_{i}$ has
finite index in $H$. So we may assume that for each $i$, $H\cap H_{i}$ has
finite index in $H$.

On the other hand we have the following elementary inclusion-exclusion
principle (Fact 3.6 below).
Let us denote $H\cap\bigcap_{i=1,..,n}H_{i}$ by $K_{0}$, a finite index
subgroup of $H$. So for each $\Delta\subseteq\\{1,..,n\\}$,
$\bigcap_{i\in\Delta}X\cap X_{i}$ is a finite (disjoint) union of cosets of
$K_{0}$ and by $|\bigcap_{i\in\Delta}X\cap X_{i}|$ we will temporarily mean
the number of these cosets (namely the “index” of $K_{0}$ in
$\bigcap_{i\in{\Delta}}X\cap X_{i}$). We can use the same notation for $H$ and
$H_{i}$ in place of the $X$ and $X_{i}$. And let us note for now that

(**) $|\bigcap_{i\in\Delta}X\cap X_{i}|=|\bigcap_{i\in\Delta}H\cap H_{i}|$ if
and only if $\bigcap_{i\in\Delta}X\cap X_{i}\neq\emptyset$.

Then we have:

######  Fact 3.6.

$X$ is contained in $\bigcup_{i\in\Delta}X_{i}$ iff
$\sum_{\Delta\subseteq\\{1,..,n\\}}(-1)^{|\Delta|}|X\cap\bigcap_{i\in\Delta}X_{i}|=0$

Now replacing ${\bar{b}}$ by ${\bar{a}}$ we obtain $\chi({\bar{a}},y)$ (which
we call $Y$) and $\psi_{i}({\bar{a}},y)$ which we call $Y_{i}$, cosets of $H$
and $H_{i}$ respectively. Our assumptions apply to give that $Y$ is NOT
covered by the union of the $Y_{i}$. On the other hand Fact 3.6 also applies
to the $Y$ and $Y_{i}$. Now (*) says exactly that $X$ is not covered by the
$X_{i}$. Hence to prove this, it is enough, using (**) to prove:

Subclaim. For each $\Delta\subseteq\\{1,..,n\\}$,
$X\cap\bigcap_{i\in\Delta}\neq\emptyset$ iff
$Y\cap\bigcap_{i\in\Delta}Y_{i}\neq\emptyset$.
Proof of subclaim. The right hand side is expressed by a $pp^{*}$-formula true
of ${\bar{a}}$, so true of ${\bar{b}}$ by our hypotheses, yielding the left
hand side.

Conversely, suppose the right hand side fails.
Case (i). $M\models\neg(\exists
y)(\exists{\bar{z}})(\phi({\bar{a}},y,{\bar{z}})\wedge\bigwedge_{i\in\Delta}\psi({\bar{a}},y))$.
Then as this means that a certain negated $pp$-formula is true of ${\bar{a}}$,
by our hypotheses it is also true of ${\bar{b}}$, so a fortiori, the left hand
side of the Subclaim fails.
Case (ii). $M\models(\exists
y)(\exists{\bar{z}})(\phi({\bar{a}},y,{\bar{z}})\wedge\bigwedge_{i\in\Delta}\psi({\bar{a}},y))$.
Then the set of $(y,{\bar{z}})\in A^{1+k}$ such that
$\phi({\bar{a}},y,{\bar{z}})\wedge\bigwedge_{i\in\Delta}\psi({\bar{a}},y)$
holds in ${\bar{M}}$, is a coset (translate) $Z$ of the subgroup $S$ of
$A^{1+k}$ defined by
$\phi({\bar{0}},y,{\bar{z}})\wedge\bigwedge_{i\in\Delta}\psi_{i}({\bar{0}},y)$.
Then ${\bar{f}}(Z)\subseteq{\mathbb{T}}^{1+k}$ is a coset of the (closed)
subgroup ${\bar{f}}(S)$ of ${\mathbb{T}}^{1+k}$. Now our assumption that the
right hand side in the subclaim fails means that $(s,{\bar{t}})$ (from the
formula $\chi$) is not in ${\bar{f}}(Z)$. Let
$(s^{\prime},{\bar{t}}^{\prime})$ be any point in ${\bar{f}}(Z)$. Then the
$pp^{*}$\- formula $\exists
y\exists{\bar{z}}(\phi({\bar{x}},y,{\bar{z}})\wedge\bigwedge_{i\in\Delta}\psi_{i}({\bar{x}},y)\wedge
f(y)=s^{\prime}\wedge{\bar{f}}({\bar{z}})={\bar{t}}^{\prime})$ is true of
${\bar{a}}$, so by assumption true of ${\bar{b}}$. But the set of
$(y,{\bar{z}})\in A^{1+k}$ such that
$\phi({\bar{b}},y,{\bar{z}})\wedge\bigwedge_{i\in\Delta}\psi({\bar{b}},y)$
holds, is a coset $Z^{\prime}$ of $S$, whereby ${\bar{f}}(Z^{\prime})$ is also
a coset of the subgroup ${\bar{f}}(S)$. We have just seen that
$(r^{\prime},{\bar{t}}^{\prime})\in{\bar{f}}(Z)\cap{\bar{f}}(Z^{\prime})$.
Hence ${\bar{f}}(Z)={\bar{f}}(Z^{\prime})$, and so
$(r,{\bar{t}})\notin{\bar{f}}(Z^{\prime})$ which proves that the left hand
side of the statement of the subclaim fails.

This proves the subclaim, the claim, as well as the lemma. ∎

As remarked above, this back and forth property for equality of $pp^{*}$-types
suffices, by a routine argument to prove Theorem 3.3

∎(of proof of Theorem 3.3)

Let $\Phi$ be the collection of CL-formulas obtained from the $pp^{*}$
formulas and negated $pp$-formulas by closing under $\wedge$, $\vee$, and
approximations.

######  Remark 3.7.

From Lemma 2.13 we obtain for $T$: for any formula $\phi({\bar{x}})$ and
approximation to it $\phi^{\prime}({\bar{x}})$ there is a formula
$\theta({\bar{x}})\in\Phi$ such that in a saturated model, $\phi({\bar{x}})$
implies $\theta({\bar{x}})$ implies $\theta^{\prime}({\bar{x}})$ (and likewise
with approximate satusfaction in arbitrary models of $T$).

######  Corollary 3.8.

$T$ is stable.

###### Proof.

This is very standard, but we do it anyway because of the somewhat different
set-up. Working in the saturated model, let
$(({\bar{a}}_{i},{\bar{b}}_{i}):i<\omega)$ be an indiscernible sequence of
pairs of finite tuples (from $A$). We have to show that if $i<j$ then
$({\bar{a}}_{i},{\bar{b}}_{j})$ and $({\bar{a}}_{j},{\bar{b}}_{i})$ have the
same type

By Theorem 3.3 we have to show that they satisfy the same $pp^{*}$-formulas.
Let $\phi^{*}({\bar{x}},{\bar{y}})$ be a $pp^{*}$-formula
$\exists{\bar{z}}(\phi({\bar{x}},{\bar{y}},{\bar{z}})\wedge({\bar{f}}({\bar{x}})={\bar{r}})\wedge({\bar{f}}({\bar{y}})={\bar{s}})\wedge({\bar{f}}({\bar{z}})={\bar{t}}))$.
Then for any ${\bar{b}}$, $\phi^{*}({\bar{x}},{\bar{b}})$, if consistent,
defines a coset of the group defined by
$\exists{\bar{z}}(\phi({\bar{x}},{\bar{0}},{\bar{z}})\wedge{\bar{f}}({\bar{x}})={\bar{0}}\wedge{\bar{f}}({\bar{z}})={\bar{0}})$.
And two such cosets are equal or disjoint. Let us assume that
$M\models\phi^{*}({\bar{a}}_{j},{\bar{b}}_{i})$ and show
$M\models\phi^{*}({\bar{a}}_{i},{\bar{b}}_{j})$ We may assume that there is
$k$ with $i<k<j$. Let $j^{\prime}>j$. By indiscernibility
$M\models\phi^{*}({\bar{a}}_{j^{\prime}},{\bar{b}}_{j})$. So
$\phi^{*}({\bar{x}},{\bar{b}}_{i})$ and $\phi^{*}({\bar{x}},{\bar{b}}_{j})$
are equivalent (define the same set). We also have, by indiscernibility, that
$M\models\phi^{*}({\bar{a}}_{k},{\bar{b}}_{i})$. Hence also
$M\models\phi^{*}({\bar{a}}_{k},{\bar{b}}_{j})$, and so again by
indiscernibility, we have $M\models\phi^{*}({\bar{a}}_{i},{\bar{b}}_{j})$.
Good.

By a symmetric argument we have
$M\models\phi^{*}({\bar{a}}_{i},{\bar{b}}_{j})$ implies
$M\models\phi^{*}({\bar{a}}_{j},{\bar{b}}_{i})$.

∎

######  Remark 3.9.

It might be natural to ask the question whether, if $G$ is a FOL group (with
additional structure) and $C$ is a compact group and $f$ a homomorphism $G$ to
$C$ and $(G,f,C)$ is stable (in the CL) sense, then this is explained by the
existence of a stable expansion of $G$ where $C=G/G^{00}$ after passing to a
saturated model. However then $C$ (or rather the image of $f$) will be
profinite. Thus Corollary 3.8 gives a negative answer to the question, as we
can of course find examples of our structures $(A,f,{\mathbb{T}})$ where
${\mathbb{T}}$ is not profinite.

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