THE MODEL THEORY OF GROUPS
Notre Dame Mathematical Lectures
Number 11
THE MODEL THEORY OF GROUPS
Ali Nesin and Anand Pillay, editors
UNIVERSITY OF NOTRE DAME PRESS
NOTRE DAME, INDIANA
Copyright © 1989 by
University of Notre Dame Press
Notre Dame, Indiana 46556
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
The Model theory of groups.
(Notre Dame mathematical lectures; 11 )
1. Group theory. I. Nesin, AH. II. Pillay, Anand. III. Series:
Notre Dame mathematical lectures; no. 11.
QA1 .N87 no. 11 a [QA171 ]
510s
88-40322
ISBN 0-268-01371-3
[512'.22]
Manufactured in the United States of America
CONTENTS
MODEL THEORY, STABILITY THEORY AND STABLE GROUPS,
AnandPillay
AN INTRODUCTION TO ALGEBRAICALLY CLOSED FIELDS AND
VARIETIES,
Bruno Poizat
1
41
COUNTABLY CATEGORICAL EXPANSIONS OF PROJECTTVE SPACES,
Simon Thomas
68
GENERIC FORMULAS AND TYPES A LA HODGES,
Victor Harnik
88
SOME NOTES ON STABLE GROUPS,
John Baldwin
100
NON-ASSOCIATIVE RINGS OF FINITE MORLEY RANK,
AliNesin
117
MODULES WITH REGULAR GENERIC TYPES,
Ivo Herzog and Philipp Rothmaler
138
MODEL THEORETIC VERSIONS OF WEIL'S THEOREM ON PREGROUPS,
Elisabeth Bouscaren
177
ON SUPERSTABLE FIELDS WITH AUTOMORPHISMS,
Ehud Hrushovski
186
ON THE EXISTENCE OF 0-DEFINABLE NORMAL SUBGROUPS OF A
STABLE GROUP,
AnandPillay
THE GROUP CONFIGURATION - AFTER E. HRUSHOVSKI,
Elisabeth Bouscaren
v
192
199
Preface
This volume contains contributions by participants in the Stable Group
seminar at Notre Dame. This seminar began during the 1985-86 Notre Dame
Logic Year and continued up to December 1987.
Except for Simon Thomas' paper on expansions of projective spaces, all
the papers here are connected in one way or another with stable group theory.
This volume contains introductory, expository and research papers in the area
and we hope will be of interest to beginners as well as "experts". We cover
much of the material in Poizat's book Groupes Stables and go beyond it in
some respects (for example Bouscaren's paper on Hrushovski's important
group configuration theorem in which the presence of a definable group is
recognized from a certain forking configuration). So we hope our volume will
serve as a useful complement to Poizat's book.
We would like to thank the University of Notre Dame (Mathematics
Department and College of Science), the National Science Foundation (Grant
DMS 85 09920) and the Argonne Universities Association Trust Fund, for their
support of the 1985-86 Logic Year at Notre Dame, which in addition to funding
a conference in April 1986, allowed us to invite many of the contributors to this
volume.
We would also like to thank Joan Hoerstman, Karen Jacobs and Leona
Strickland for their excellent and patient work in preparing the manuscript.
AliNesin
Anand Pillay
Notre Dame, Indiana
October 1988
MODEL THEORY, STABILITY THEORY & STABLE GROUPS
Anand Pillay*
The aim of this chapter is to introduce the reader to the theory of stable
groups not to give a rigorous exposition of the general theory. Thus we tend to
proceed from the concrete to the abstract, with several examples and analyses of
special cases along the way. On the other hand, getting to grips with stable
groups presupposes some understanding of the point of view of model theory
in general and stability theory in particular, and the first few sections are
devoted to the latter.
1. MODEL THEORY
By a relational structure M we understand a set M (called the universe
or underlying set of M) equipped with relations Rj of arity ni < co say, for
n
lei.
i
Namely, for i e I, R± is a subset of the Cartesian product M .
Here I and <ni : i G I> depend on M and are called the signature of M. We
also insist that I always contains a distinguished element i= such that Ri- is
the diagonal {(a,a) : ae M} c, M2. Often the distinction between M and M
is blurred. The model theorist is interested in certain subsets of M and of Mn
(the definable sets) which are obtained in a simple fashion from the RI. So
J9(M) is a collection of subsets of Mn, n< co, which can be characterized as
follows:
(i) Every RIE J9(M).
(ii) If n < co, Xe J9(M) is a subset of Mn and n is a permutation of
{l,...,n} then 7c(X) = {(a^i) .....a^n)) : (ai,...,an) e X} e £)(M).
(iii) J9(M) is closed under Boolean combinations, i.e. if n < co and
* Partially supported by NSF Grant DMS 8601289
2
Model Theory, Stability Theory & Stable Groups
X,Y e JB(M) are subsets of M" then XU Y , X f l Y , M n - X areallin
JB(M).
(iv) If Xe JD(M) andYe J9(M) then Xx Ye J9(M).
(v) If X e J0(M) is a subset of Mn+m, then the projection of X on
n
M is in J9(M).
(vi) If Xe J9(M) is a subset of Mn+m and âe Mn then Xa =
{be Mm: (a,b) e X} is in J9(M).
(vii) Nothing else is in J9(M).
We call J9(M) the class of definable sets of M.
These definable sets can be defined (and usually are) syntactically.
Associated to the relational structure M (in fact to its signature) is a language
L(M) consisting of symbols: Pi for each ie I, "variables" Xj for each j<co,
and logical symbols A (and), v (or), -i (not), V (for all) and 3 (there exists).
L(M) -formulas are constructed from these symbols as follows: if xj are
variables then PiXi...xn. is an (atomic) formula. If 9,\|/ are formulas and x
is a variable then (pA\|/, cpvy, -i 9, (3x)cp, (Vx)(p are all formulas. A variable
x is said to be free in the formula 9 if some occurrence of x in 9 is not in
the scope of any quantifier. We write 9(xi,...,xn) to mean that xi,...,xn are
the free variables in the formula 9. We then define "9(xi,...,xn) is true of
(ai,...,an) in M" (where ai,...,ane M) as follows:
If 9 is atomic, say Pyi... yn, and for some permutation n of
{l,...,n}, xi = y^i) then 9(xi,...,xn) is true of (ai,...,an) in M if
(ao(l)>-,ao(n)) e p where a =7C"1If \\f is (3xn+i) 9 and xn+i is a free variable of 9, then \|/(xi,...,xn)
is true of (ai,...,an) in M if there is an+i e M such that 9(xi,...,xn,xn+i)
is true of (ai,...,an+i) in M. Similarly for y = (Vxn+i) 9.
The clauses for A,v, -i are obvious. We abbreviate "9(xi,...,xn) is
true of (ai,...,an) in M" by the notation M 1= 9(ai,...,an). (Note this
notation depends on our having listed the free variables in 9 in a certain order).
By abuse of everything, we can and will think of M N= 9(ai,...,an) as
saying that 9 is true when we substitute ai for xj.
AnandPillay
3
It is now routine to check that
Fact 1.1. If X c Mn, then X e J9(M) if and only if there are an L(M)
formula <p(xi,...,xn,yi,...,ym) and bi,...,bm e M such that
X ={âe Mn: M (=cp(â,b)}.
The syntactic approach to defining definable sets appears at first to be
preferable as one can make the following definition.
Definition 1.2. Xe J9(M), a subset of Mn, is said to be A-definable or
defined over A (forACiM) if in Fact 1.1 we can choose (p with b c: A.
Example 1.3. Let K be an algebraically closed field. We can consider K
as a relational structure in the above sense by choosing {0}, {1}, and the
graphs of addition and multiplication as the "distinguished" relations. Note that
if PI,... J?r are polynomials in n-variables over K, then the subset V of Kn
consisting of the simultaneous zero set of these polynomials is a definable set.
These are called the affine algebraic sets. Finite Boolean combinations of such
sets are called constructible sets of K, and either Tarski's "quantifier
elimination theorem" (quantifier elimination in a language with function
symbols for addition and multiplication) or Chevalley's theorem states
Fact: The constructible sets of K are precisely the definable sets of K.
For an affine algebraic set X c: Kn there is an algebraic-geometrical
notion of X being defined over k (k a subfield of K) which may have some
discrepancy with the model theoretic notion (Definition 2.2). Namely: let I(X)
CL K[xi,...,Xn] be the ideal of polynomials which vanish on X. According to
the algebraic geometer X is defined over k d K if I(X) can be generated as
an ideal by polynomials in k[xi,...,xn].
We do have (for X£.Kn affine algebraic and k subfield of K)
Fact: X is defined over k in the model theoretic sense iff X is defined
over k
in the sense of algebraic geometry (where p = char K).
So if k is perfect, or char k = 0, the notions agree.
Model Theory, Stability Theory & Stable Groups
1.4. The usual procedure in model theory is to start with a language L and to
consider various subclasses of L-structures. So L will essentially be a
signature as above, Le. will consist of a set of relation symbols of specified
arity and an L-structure will be a relational structure equipped with corresponding relations of the right arity. This enables us to compare L-structures in
various respects. For instance, by an L-sentence we mean an L-formula
which has no free variables. An L-structure M is said to be a model of a set
F of L-sentences if for every a e F, M h a, i.e. every a e F is true in M.
A set of L-sentences F is said to be consistent if it has a model. A consistent
set of sentences F is said to be a complete theory if for every L-sentence a
either ae F or -icie F, equivalenûy for some M, F= {G: M l=a}; in the
latter case F being called the theory of M. Two L-structures M and N are
called elementarily equivalent if they have the same theory, equivalently they
satisfy the same L-sentences. As an example, any two algebraically closed
fields of the same characteristic, say p, are elementarily equivalent; in other
words the set TACF_ of sentences (in the language in Example 1.3 for
example) true in all algebraically closed fields of characteristic p is a complete
theory.
A crucial tool in model theory is the compactness theorem: a set of
sentences F is consistent iff every finite subset of F is consistent. This gives
substance to the following important notion: Let M, N be L-structures with
M a substructure of N (M ÇL N, with the obvious meaning). M is said to be
an elementary substructure of N, M < N, if for every formula cp(x) of L
and à" G M, we have M 1= cp(â) iff N 1= q>(a).
Let us remark that if M < N then any definable set X jc Mn in M
has a canonical extension to a definable set X' Cj, Nn in N. Namely, let
cp(x,y), âÇ^M be such that cp(x,a) defines X in M. Then let X/ =
{x e Nn : N 1= cp(x,a)}. Note that X c= X' and X' does not depend on the
particular choice of (p and a.
The compactness theorem yields for any infinite M, elementary
extensions N of M of arbitrarily large cardinality. Another consequence of
Anand Hllay
5
Tarski's quantifier elimination is that if KI c: K2 are algebraically closed
fields then KI < K2, noting the following characterisation : let MI c:
then MI < M2 iff for any non-empty Mi-definable subset X ofM2,
1.5. Saturated models.
Let K be an infinite cardinal. The structure N is said to be Kr
saturated if for any A C N with I AI < K and any collection Xi, i e I of
A-definable subsets of N with the finite intersection property ( . fl X. ?* 0 for
all finite J ^ I), we have .0 X. ^ 0. Again the compactness theorem gives for
any M and K some K-saturated N > M.
It is worth noting that the definition above of K-saturation would be
equivalent if we allowed the Xi to be A-definable subsets of Nn for any
n > 1 This apparently stronger fact follows by use of the existential quantifier.
One can think of the property of K-saturation of N as meaning that for
any M < N with IMI < K, any situation that can happen in some elementary
extension of M already happens in N. (In this sense N is like a universal
domain. In fact, what Weil calls a universal domain - an algebraically closed
field of infinite transendence degree K over the prime field- is K-saturated).
Moreover if M = N and IMI < K then there is an elementary embedding
(obvious meaning) of M into N. It will be convenient to assume that any
complete theory has models which are K-saturated and of cardinality K. for
arbitrarily large K. Such a model, N say, will have homogeneity properties
in addition to saturation properties, which are pointed out subsequently. (For
stable theories the existence of such models is guaranteed. Otherwise, it
depends on set theory).
Let us now fix such a model N (K-saturated of cardinality K for some
large K). A, A', B etc. will denote subsets of N of cardinality <K, and M,
M', MI,... elementary substructures of N of cardinality < K (often called
models). We now introduce the important notion of a type.
6
Model Theory, Stability Theory & Stable Groups
Let A CN. By a complete n-tvpe over A we mean a maximal
consistent collection of A-definable subsets of Nn (where consistent means
having the finite intersection property). Alternatively, with some abuse of
earlier notation, a complete n-type over A is a maximal set F of formulas of
the form cp(xi,...,xn,a) where if d A and for (pi,—,<pme F, N 1=
Let bi,...,bne N. By the type of b over A (in N if you wish),
tp(b /A) is meant the collection of A-definable subsets of Nn containing b.
tp(b /A) is clearly a complete n-type over A. Conversely, saturation of N
implies that every complete n-type F over A is the form tp(b/A), for some b
e Nn. b is said to realize F. The set of complete n-typesover A is denoted
Sn(A), and types themselves are usually denoted by p, q etc.
The fact that N is saturated in its own cardinality gives us a nice
characterization: if t>i e Nn, t>2e Nn then tp(bi/A) = tp(t)2/A) iff there is
an automorphism f of N such that f(bi) = b2 and f fixes A pointwise.
(Similarly for types of infinite tuples of cardinality < K).
Saturation of N also enables us to give the notion "definable over N" a
"Galois theoretic" interpretation. Firstly, the compactness theorem yields: Let
X ci Nn be definable, let A <^ N and suppose that whether or not some b e
Nn is in X depends only on tp(b/A). Then X is A-definable. In conjunction
with the previous observation this shows that for definable X ci Nn, X is Adefinable iff for every automorphism f of N which fixes A pointwise, f(X)
= X.
N e <l
It will be sometimes convenient (especially when dealing with groups)
to work in a structure which is "closed under definable quotients". We can
1.6.
construct from N such a universe, Neci, which is "essentially" the same as N.
Informally, N^i is the disjoint union of a collection of universes, one of which
is N, and each being picked out by a new predicate. Each new universe is
identified, by means of a new function symbol, with the set of classes of a