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We extend the Ahlbrandt--Ziegler analysis of interpretability in aleph_0-categorical structures by showing that existential interpretation is controlled by the monoid of self--embeddings and positive existential interpretation of structures without constant endomorphisms is controlled by the monoid of endomorphisms in the same way as general interpretability is controlled by the automorphism group.

It is known that exactly eight varieties of Heyting algebras have a model-completion, but no concrete axiomatisation of these model-completions were known by now except for the trivial variety (reduced to the one-point algebra) and the variety of Boolean algebras. For each of the six remaining varieties we introduce two axioms and show that 1) these axioms are satisfied by all the algebras in the model-completion, and 2) all the algebras in this variety satisfying these two axioms have a certain embedding property. For four of these six varieties (those which are locally finite) this actually provides a new proof of the existence of a model-completion, this time with an explicit and finite axiomatisation.

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima's representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal sprectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. For example we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.

In this paper we introduce a notion of dimension and codimension for
every element of a distributive bounded lattice *L*. These notions
prove to have a good behavior when *L* is a co-Heyting algebra. In
this case the codimension gives rise to a pseudometric on *L* which
satisfies the ultrametric triangle inequality. We prove that the
Hausdorff completion of *L* with respect to this pseudometric is
precisely the projective limit of all its finite dimensional
quotients. This completion has some familiar metric properties, such
as the convergence of every monotonic sequence in a compact subset. It
coincides with the profinite completion of *L* if and only if it is
compact or equivalently if every finite dimensional quotient of *L* is
finite. In this case we say that *L* is precompact. If *L* is
precompact and Hausdorff, it inherits many of the remarkable
properties of its completion, specially those regarding the join/meet
irreducible elements. Since every finitely presented co-Heyting
algebra is precompact Hausdorff, all the results we prove on the
algebraic structure of the latter apply in particular to the
former. As an application, we obtain the existence for every positive
integers *n,d* of a term *t _{n,d}* such that in every co-Heyting
algebra generated by an

We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

This article aims to classify those reducts of expansions of
(**Q**,<) by unary predicates which eliminate quantifiers, and in
particular to show that, up to interdefinability, there are only
finitely many for a given language. Equivalently, we wish to classify
the closed subgroups of Sym(**Q**) containing the group of all
automorphisms of (**Q**,<) fixing setwise certain subsets. This
goal is achieved for expansions by convex predicates, yielding
expansions by constants as a special case, and for the expansion by a
dense, co-dense predicate. Partial results are obtained in the general
setting of several dense predicates.

This paper is a collection of results about topologies in stable groups. The first section defines and studies variants of the Srour topology. The second section examines the behaviour of terms in these topologies and particurlarly the question of generic equations. The last section is devoted to the notion of completeness and possible applications to the bad group problem.

Many complete first order theories admit nice independence relations,
*e.g.* simple theories and o-minimal theories.
In the classical examples, these independence relations are controlled
by rather simple families of formulae.
It has been observed by Srour (and others) that these formulae are
usually ``positive'' in some sense.
This yields the possibility to define locally noetherian topologies
in a model that are closely linked to the independence relation.

In this paper, we recall Srour's definition of equational independence (as in [S], [HS]), and we show that it is the same as local non-forking with respect to equations. Pushing further the development of [PS], we then give a comprising topological characterisation, generalising the definition of independence in algebraically closed fields via irreducibility and varieties.

Following [HS], a stable theory is said to be *almost
equational* if equational independence is non-forking. We generalise
this concept to arbitrary theories and suggest that it may be an
interesting setting to work in. Presently, no simple theory is known
which is not almost equational. We conclude with some criteria for
almost equationality and show many classical structures to be almost
equational.

The main purpose of this article is to shed new light on old things and to put different aspects together. It is based on the work of Hrushovski, Pillay, and Srour.

[HS] E. Hrushovski, G. Srour: *On Stable Non-Equational Theories*, manuscript.

[PS] A. Pillay, G. Srour: *Closed sets and chain conditions in stable
theories*, JSL **49** No. 4 (1984) pp.1350-1362.

[S] G. Srour: *The notion of independence in categories of
algebraic structures*,
part I: APAL **38** (1988) pp. 185-213;
part II: APAL **39** (1988) pp. 55-73;
part III: APAL **47** (1988) pp. 269-294.

Important structures like algebraically closed fields or vector spaces
carry a natural topology, namely the Zariski topology in the first
case or the topology generated by the affine subspaces as closed sets
in the second case. In both cases, the definable sets are exactly the
constructible sets, i.e. boolean combinations of closed sets. It has
been noticed several times (e.g. in [PS]) that such a topology
is necessarily noetherian and can only exist in aleph_{0}-stable
structures. However, from a model theoretic point of view, the
important point about the Zariski topology is not the noetherianity,
but the possibility to distinguish closed sets among the definable
ones.

Roughly speaking, geometric stability theory aims to classify structures up to bi-interpretability by well-known structures. With Hrushovski's and Zil'ber's work on Zariski geometries ([HZ1], [HZ2]) -in contrast with Hrushovski's counter-examples to Zil'ber's conjecture in [H]- the importance of an abstract Zariski topology for this program has become evident. We are therefore interested in topologies which are closely related to the definable sets in a structure M and which could replace this abstract Zariski topology.

More precisely, we will consider notions of *closedness* in
arbitrary structures M. Each such notion provides a
family of definable closed sets, which should be stable under positive
boolean combinations. As in the case of the Zariski topology, we get a
family of generated topologies, one on each power M^{n}. Of course,
these topologies should be linked in various ways, for example by
continuous projections. This is best expressed by Hrushovski's notion
of an *f-space*, i.e. a contravariant functor from finite sets
to topological spaces. We want these topologies to be controlled by
the definable closed sets, and we want a certain uniformity in the
definition. The topology should depend on a first order theory rather
than on the single model, and apply to a large class of theories. In
particular, elementary maps are required to be continuous.

A first attempt to find such a topology was done by Srour (for our purposes best in [PS]). Here we give a new approach to this topology implicitly introduced by Srour. We also examine related topologies and start to study them systematically as candidates to replace an abstract Zariski topology.

[H] E. Hrushovski: *A New Strongly Minimal Set*, APAL **62** (1993) pp. 147- 166.

[HZ1] E. Hrushovski, B. Zil'ber: *Zariski Geometries*,
Journal AMS **9** No. 1 (1996) pp. 1-56.

[HZ2] E. Hrushovski, B. Zil'ber, *Zariski Geometries*,
Bull.AMS **28** No. 2 (1993) pp. 315-323.

[PS] A. Pillay, G. Srour: *Closed sets and chain conditions in stable
theories*, JSL **49** No. 4 (1984) pp.1350-1362.

Several attempts have been done to distinguish ``positive'' information in an arbitrary first order theory, i.e. to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yield a class of theories, namely those where all definable sets are constructible, i.e. boolean combinations of closed sets. Here are some examples, ordered by strength:

CLOSED SETS | DEFINITION | CLASS OF THEORIES |

normal | conjugates are equal or disjoint | normal theories |

weakly normal | infinitely many pairwise distinct conjugates are inconsistent | one-based theories |

Srour closed | no infinite descending chains of intersections of conjugates | equational theories |

normalized | distinct conjugates have smaller (e.g. in the sense of Morley rank) symmetric difference | (all omega-stable theories) |

Weak normality describes a rather small class of theories which are well understood by now (see, for example, [P]). On the other hand, normalization is so weak that all theories, in a suitable context, are normalizable (see [HH]). Thus equational theories form an interesting intermediate class of theories. Few work has been done so far. The original work of Srour [S] adopts a point of view that is closer to universal algebra than to stability theory. The fundamental definitions and model theoretic properties can be found in [PS], though some easy observations are missing there. Hrushovski's example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS]. In fact, it is an expansion of the free pseudospace constructed independently by Baudisch and Pillay in [BP] as an example of a strictly 2-ample theory. Strong equationality, defined in [H], is also investigated in [HS].

The definitions of equationalilty are not the same in all these articles. The main purpose of this paper is to clarify definitions and to show that equationality is a rather robust concept.

[BP] A. Baudisch, A. Pillay: *A free pseudospace*, preprint 1998.

[HH] V. Harnik, L. Harrington: *Fundamentals of forking*, APAL
**26** (1984) pp. 245-286.

[H] E. Hrushovski: *A New Strongly Minimal Set*, APAL **62** (1993) pp. 147- 166.

[HS] E. Hrushovski, G. Srour: *On Stable Non-Equational Theories*, manuscript.

[P] A. Pillay, *Geometric Stability Theory*, Oxford University Press, 1996.

[PS] A. Pillay, G. Srour: *Closed sets and chain conditions in stable
theories*, JSL **49** No. 4 (1984) pp.1350-1362.

[S] G. Srour: *The notion of independence in categories of
algebraic structures*,
part I: APAL **38** (1988) pp. 185-213;
part II: APAL **39** (1988) pp. 55-73;
part III: APAL **47** (1988) pp. 269-294.

*Short abstract:*

Zariski groups are aleph_{0}-stable groups with an
axiomatically given Zariski topology and thus abstract generalizations
of algebraic groups. A large part of algebraic geometry can be
developed for Zariski groups. As a main result, any simple smooth
Zariski group interprets an algebraically closed field, hence is
almost an algebraic group over an algebraically closed field.

*Introduction:*

Model theory came naturally across the notion of aleph_{0}-stable groups of finite
Morley rank. These are groups where a finite dimension is assigned to all definable
sets that behaves much as dimension in algebraic geometry. In fact, these groups
share many properties with algebraic groups. It was even conjectured that they were
essentially algebraic groups. Removing trivial obstacles, Cherlin's conjecture states
that a simple aleph_{0}-stable group (of finite rank) is an algebraic group over an
algebraically closed field. The theory now is far developed: see [BN] for an
algebraic and [P] for a more model theoretic introduction to the subject.

The main results about aleph_{0}-stable groups of finite Morley rank are the
following: If an aleph_{0}-stable group of finite Morley rank has a definable,
connected, solvable, non-nilpotent subgroup, then an algebraically closed field is
interpretable (Zil'ber). If the group is simple, it is interpretable in the field
possibly equipped with some extra structure coming from the group (Hrushovski). Hence
the main problems are to eliminate the so-called bad groups, where no connected,
solvable, non-nilpotent subgroup are definable, and to show that the group is
algebraic over the field it interprets.

As long as the general Cherlin conjecture is still unsolved, it is natural to consider weaker forms. One possibility is motivated by Hrushovski's and Zil'ber's work on strongly minimal sets and Zil'ber's conjecture, which is a similar problem. The main part of Zil'ber's conjecture asserted that a non-locally modular strongly minimal set was an algebraic curve. Hrushovski constructed counter-examples to this, but Hrushovski and Zil'ber succeeded in proving the conjecture for special strongly minimal sets, so-called Zariski geometries. These are structures equipped with Noetherian topologies as abstract Zariski topologies. More precisely, their result characterizes the Zariski topology of smooth algebraic curves over algebraically closed fields.

In the light of their work, it seems natural to consider ``Zariski groups'' defined
in this article: aleph_{0}-stable groups with an axiomatically given Zariski
topology.

The main interest of Cherlin's conjecture, at least from an algebraic point of view, is the hope to get an abstract characterization of algebraic groups, not mentioning fields and varieties. While a positive solution of the general conjecture would characterize the constructible structure of algebraic groups, a solution for Zariski groups would provide a characterization of the Zariski topology of algebraic groups.

This article gives an approach to Cherlin's conjecture for Zariski groups. In particular I show that there are no bad smooth Zariski groups, hence that any simple smooth Zariski group interprets an algebraically closed field. Most probably, this result follows also from Hrushovski's and Zil'ber's work [HZ1],[HZ2]. The problem is to show that there is a strongly minimal subset of a smooth Zariski group satisfying the dimension formula. Anyhow, I hope my proof is of interest because my methods are more elementary and might be more easily understood.

[BN] A. Borovik, A. Nesin, *Groups of Finite Morley Rank*, Clarendon Press, Oxford 1994.

[P] B. Poizat: *Groupes stables*, Nur al-Mantiq wal Ma'rifah, Lyon 1987.

[HZ1] E. Hrushovski, B. Zil'ber: *Zariski Geometries*,
Journal AMS **9** No. 1 (1996) pp. 1-56.

[HZ2] E. Hrushovski, B. Zil'ber, *Zariski Geometries*,
Bull.AMS **28** No. 2 (1993) pp. 315-323.

Zariski geometries were introduced by E. Hrushovski and B. Zil'ber as abstract models of algebraic curves to get a class of structures where the Zil'ber trichotomy holds. In my thesis, I define a higher dimensional generalization of the notion of Zariski geometry and I start the systematic study of these structures.

A Zariski geometry is given by a family of Noetherian topologies such
that the topological dimension shares several properties of the
concept of dimension in algebraic geometry (chapter 1), and such that
the essential properties are preserved under elementary extensions. It
follows that Zariski geometries are special aleph_{0}-stable
structures where topological and model theoretical properties are
closely linked (chapter 2).

Besides trivial and linear structures, algebraic varieties over algebraically closed fields form the most important class of examples (chapter 4).

A structural description of Zariski geometries requires a deep knowledge of the interpretable structures. Chapter 3 of this thesis shows that the topological behaviour of imaginary sets is quite complex. This analysis leads to the definition of a variety over a Zariski geometry and to the appropriate notion of morphisms. Special important types of varieties are smooth and complete varieties. They are defined and studied separately.

As an application of the techniques developed so far, I prove a structure theorem for Zariski groups (defined over Zariski geometries as algebraic groups over fields), namely that there are no bad smooth Zariski groups. Combinig this result with a theorem of Zil'ber, one concludes that every simple smooth Zariski group interprets an algebraically closed field. This proves one half of Cherlin's conjecture for smooth Zariski groups and almost provides an abstract characterization of simple algebraic groups over algebraically closed fields.

Zariski-Geometrien wurden von E. Hrushovski und B. Zil'ber eingeführt als abstrakte Modelle algebraischer Kurven, um über eine Klasse von Strukturen zu verfügen, in denen die Zil'bersche Trichotomie gilt. In meiner Doktorarbeit definiere ich eine beliebig endlich-dimensionale Verallgemeinerung jener Zariski-Geometrien und beginne eine systematische Untersuchung dieser Strukturen.

Eine Zariski-Geometrie ist eine durch mehrere noethersche Topologien
bestimmte Struktur, in denen der topologische Dimensionsbegriff
Axiomen genügt, welche Eigenschaften der Dimension in
algebraischen Mannigfaltigkeiten sind (Kapitel 1). Außerdem wird
gefordert, daß die wichtigsten Eigenschaften unter elementaren
Erweiterungen erhalten bleiben. Zariski-Geometrien sind dann
aleph_{0}-stabile Strukturen, in denen die topologischen
Eigenschaften eng mit den modelltheoretischen verwoben sind (Kapitel
2).

Neben trivialen und linearen Strukturen bilden algebraische Mannigfaltigkeiten über algebraisch abgeschlossenen Körpern die wichtigste Beispielklasse (Kapitel 4).

Eine strukturelle Beschreibung der Zariski-Geometrien verlangt eine genaue Kenntnis der darin interpretierbaren Strukturen. Das Verhalten der Topologien auf definierbaren Mengen imaginärer Elemente erweist sich als sehr komplex. Aus diesen Untersuchungen ergibt sich eine Definition von Mannigfaltigkeiten über Zariski-Geometrien und der zugehörigen Morphismen. Als wichtige Typen werden glatte und vollständige Mannigfaltigkeiten definiert und gesondert betrachtet (Kapitel 3).

Als Anwendung der bislang entwickelten Techniken beweise ich einen Struktursatz für Zariski-Gruppen. Dabei sind Zariski-Gruppen in gleicher Weise über Zariski-Geometrien definiert wie algebraische Gruppen über Körpern. Ich zeige, daß keine sogenannten schlechten glatten Zariski-Gruppen existieren. Mit Hilfe eines Satzes von Zil'ber besagt dies, daß jede einfache glatte Zariski-Gruppe einen algebraisch abgeschlossenen Körper interpretiert. Damit ist Cherlins Vermutung für glatte Zariski-Gruppen zur Hälfte bewiesen, und fast eine abstrakte Charakterisierung einfacher algebraischer Gruppen über algebraisch abgeschlossenen Körpern gewonnen.