Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $\mu$ is in a certain class ${\mathcal T}$ ($\mu$ triples) we show that for …
Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $\mu$ is in a certain class ${\mathcal T}$ ($\mu$ triples) we show that for independent $I$ with $|I| >1$, ${\rm dcl}^*(I)= \emptyset$ (* means not in ${\rm dcl}$ of a proper subset). This implies the only definable truly $n$-ary function $f$ ($f$ `depends' on each argument), occur when $n=1$. We prove, indicating the dependence on $\mu$, for Hrushovski's original construction and including analogous results for the strongly minimal $k$-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, ${\rm sdcl}^*(I) =\emptyset$, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k = p^n$. The proofs depend on our introduction for appropriate $G \subseteq {\rm aut}(M)$ the notion of a $G$-normal substructure ${\mathcal A}$ of $M$ and of a $G$-decomposition of ${\mathcal A}$. These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit.
Boris Zil'ber conjectured that all strongly minimal theories are bi-interpretable with one of the “classical” sorts: theories of algebraically closed fields, theories of infinite vector spaces over division rings and …
Boris Zil'ber conjectured that all strongly minimal theories are bi-interpretable with one of the “classical” sorts: theories of algebraically closed fields, theories of infinite vector spaces over division rings and theories with trivial algebraic closure relations. Hrushovski produced the first two classes of counterexample to this conjecture in [10] and [9]. Subsequently, in [8], the author gave an explicit axiomatization of a special case of [9] from which model completeness could quickly be deduced. It was unclear at that writing whether the model completeness result was true in the general case or was due to peculiarities of the case under consideration. The main new result of this paper is model completeness, not only of the general case in [9], but also of the theories described in [10]. Specifically, we present a general framework in which producing a strongly minimal theory is reduced to finding an elementary class of theories satisfying certain requirements (see below). We present the theories of [10] and [9] as special instances of such theories, giving an explicit axiomatization from which model completeness immediately follows in each case. We hope by presenting these constructions in parallel, using common language and extracting common elements, to make easier both the exploitation of the ideas involved in their making and their comparison with other recent constructions of a similar flavor. For a selection of such constructions, see [6], [1], [2] and [3]. For more general background, see [2], [4] and [11].
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. …
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.
Let G be a group on which a measure m is defined. If $A,B \subset G$ we define $A \oplus B = C = \{ c|c = a + b,a …
Let G be a group on which a measure m is defined. If $A,B \subset G$ we define $A \oplus B = C = \{ c|c = a + b,a \in A,b \in B\}$. By ${A_k} \subset G$ we denote a subset of G consisting of k elements. Given ${A_k}$ we define $s({A_k}) = \inf m\{ B|B \subset G,{A_k} \oplus B = G\}$ and ${c_k} = {\sup _{{A_k} \subset G}}s({A_k})$. Theorems 1, 2, and 3 deal with the problem of determining ${c_k}$. In the dual problem we are given B, $m(B) > 0$, and required to find minimal A such that $A \oplus B = G$ or, sometimes, $m(A \oplus B) = m(G)$. Theorems 5 and 6 deal with this problem.
Abstract We construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T , generic automorphisms …
Abstract We construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T , generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.
The subject matter of this note is the notion of a dependence structure on an abstract set. There are a number of different approaches to this topic and it is …
The subject matter of this note is the notion of a dependence structure on an abstract set. There are a number of different approaches to this topic and it is known that many of these lead to precisely the same structure. Axioms are given here to specify the minimal dependent sets for such a structure. They are closely related to conditions introduced by Hassler Whitney in [1] and to a certain “elimination axiom” for the independent sets.
Abstract Soare [20] proved that the maximal sets form an orbit in ${\cal E}$ . We consider here ${\cal D}$ -maximal sets , generalizations of maximal sets introduced by Herrmann …
Abstract Soare [20] proved that the maximal sets form an orbit in ${\cal E}$ . We consider here ${\cal D}$ -maximal sets , generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of ${\cal D}$ -maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the ${\cal D}$ -maximal sets. Although these invariants help us to better understand the ${\cal D}$ -maximal sets, we use them to show that several classes of ${\cal D}$ -maximal sets break into infinitely many orbits.
Introduction.We show that every substitution on finitely many symbols, which replaces each symbol by a block of length 2 or more, effectively determines at least one almost periodic point under …
Introduction.We show that every substitution on finitely many symbols, which replaces each symbol by a block of length 2 or more, effectively determines at least one almost periodic point under the shift transformation of symbolic dynamics.The orbit-closures of these almost periodic points, called substitution minimal sets, are analyzed topologically to some extent and in particular it is proved that under certain conditions their structure groups are n-adic groups.Several of the known symbolic minimal sets, such as the Morse minimal set, are definable by the present method of construction.Minimal sets exhibiting new properties also appear.The first two sections contain general theorems on the trace relation of transformation groups and on i^-adic transformation groups, some of these results being used to study substitution minimal sets in the third section.A few particular examples of substitution minimal sets are described in the last section.As general references for notions, notation, and terminology occuring here, consult [3; 1].
Let (X, T) and (Y, T) be transformation groups with the same phase group T. A homomorphism f: X-Y is a continuous mapping such that (xt)f= (xf)t (xCX, tE T). …
Let (X, T) and (Y, T) be transformation groups with the same phase group T. A homomorphism f: X-Y is a continuous mapping such that (xt)f= (xf)t (xCX, tE T). A transformation group (X, T) is called a minimal if (xT)= cls [xt/tC T] = X(x CX). In this note it will be proved that given any abstract group T there exists a minimal (M, T) with compact phase space M such that any minimal (X, T) with compact X is a homomorphic image of (M, T). Furthermore this universal minimal set is unique up to an isomorphism, and given xCM, tCT with t$e then xt$x. For a more complete discussion of several notions involved above see [21 and [3 ]. DEFINITION 1. The fl-compactification as a transformation group. Let T be a discrete group, let f,T be the ,B-compactification of T, and let tG T. Then the map s-*st of T into f,T is continuous and so may be extended to a map of fBT into f,T. Thus each element of T may be identified with a homeomorphism of ,BT onto f,T. Under this identification (fiT, T) becomes a transformation group. Henceforth all transformation groups (X, T) will be assumed to have compact phase spaces, X, and discrete phase group T.
Abstract Suppose D ⊂ M is a strongly minimal set definable in M with parameters from C . We say D is locally modular if for all X, Y ⊂ …
Abstract Suppose D ⊂ M is a strongly minimal set definable in M with parameters from C . We say D is locally modular if for all X, Y ⊂ D , with X = acl( X ∪ C )∩ D , Y = acl( Y ∪ C ) ∩ D and X ∩ Y ≠ ∅, We prove the following theorems. Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then in M eq there is a definable pseudoplane . (For a discussion of M eq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3]. Theorem 2. Suppose M is stable and D , D ′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular .
Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our …
Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family. Problems encountered in this research include the minimum number of steps required to arrive to the last family as well as a characterization of that last family; we solve all those problems. We also define a class of simple families ($n$-minimal constructible) and we analyze the relationships between partitions and separability (our new concept) that lead to interesting results such as finding families based on partitions that generate finite algebras. We prove a number of new results about $n$-minimal constructible families; one major result is that every finite algebra of sets has a generating family which is $n$-minimal constructible for all natural $n$ and we compute the minimum number of steps required to generate an algebra. Another interesting result is a connection between this construction process and Baire's Theorem. This work has a number of possible applications, particularly in the fields of economics and computer science.
Le resultat principal de cette these est l'etude de l'ampleur dans des expansions des structures geometriques et de SU-rang omega par un predicat dense/codense independant. De plus, nous etudions le …
Le resultat principal de cette these est l'etude de l'ampleur dans des expansions des structures geometriques et de SU-rang omega par un predicat dense/codense independant. De plus, nous etudions le rapport entre l'ampleur et l'equationalite, donnant une preuve directe de l'equationalite de certaines theories CM-triviales. Enfin, nous considerons la topologie indiscernable et son lien avec l'equationalite et calculons la complexite indiscernable du pseudoplan libre
Recall that a complete first-order theory with infinite models is strongly minimal if in any of its models, every parameter-definable subset of the model is finite or cofinite. Classical examples …
Recall that a complete first-order theory with infinite models is strongly minimal if in any of its models, every parameter-definable subset of the model is finite or cofinite. Classical examples are theories of vector spaces and algebraically closed fields; also the degenerate example of the theory of infinite ‘pure’ sets where the only structure comes from equality. Algebraic closure in a strongly minimal structure satisfies the exchange condition, so gives rise to notions of dimension and independence (corresponding to linear dimension/ independence and transcendence degree/ algebraic independence in the two classical examples).
We construct an infinite projective plane with Lenz-Barlotti class I. Moreover, the plane is almost strongly minimal in a very strong sense: each automorphism of each line extends uniquely to …
We construct an infinite projective plane with Lenz-Barlotti class I. Moreover, the plane is almost strongly minimal in a very strong sense: each automorphism of each line extends uniquely to an automorphism of the plane.
We show that the groups $AGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 2$) and $PGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 3$), seen as closed subgroups of $S_{\omega }$, are maximal-closed.
We show that the groups $AGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 2$) and $PGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 3$), seen as closed subgroups of $S_{\omega }$, are maximal-closed.
The topics of reference, realism, and structure have been discussed extensively in the philosophy of mathematics of the last decades. There have been some parallel discussions in certain parts of …
The topics of reference, realism, and structure have been discussed extensively in the philosophy of mathematics of the last decades. There have been some parallel discussions in certain parts of philosophy of science and metaphysics. The technical apparatus which serves as the formal backdrop to these discussions is model theory, a branch of mathematical logic, and the key results in model theory which have been thought to be of primary philosophical significance are the famous categoricity theorems of Dedekind and Zermelo. The aim of this paper is to survey this recent literature, using basic ideas and results from model theory to aid in organizing the discussion. Some of the specific topics covered are: Putnam's permutation argument and the just more theory manoeuvre; the Newman objection and notions of conservation; the import of the L\owenheim-Skolem theorems; the invocation of second-order logic in debates about realism and reference; the distinctive challenges posed by quasi-categoricity results in set theory, the structuralism of Shapiro and Parsons, and the aims and goals of recent work on uncountable categoricity.
I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property.A …
I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property.A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.
Abstract This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion …
Abstract This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for L ω 1 ,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.
We show that the generic automorphism is axiomatizable in the green field of Poizat (once Morleyized) as well as in the bad fields that are obtained by collapsing this green …
We show that the generic automorphism is axiomatizable in the green field of Poizat (once Morleyized) as well as in the bad fields that are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain ‘bad pseudofinite fields’ in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatization. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, for example, the free fusion of two strongly minimal theories having the definable multiplicity property. We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.
We study model-theoretic and stability-theoretic properies of the nonabelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn. We point out …
We study model-theoretic and stability-theoretic properies of the nonabelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn. We point out analogies between the free group and so-called bad groups of finite Morley rank and we prove non CM-triviality of the free group.
We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies (one topology on each M n ). The structure is almost determined by …
We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies (one topology on each M n ). The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.
Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the …
Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the set $U:=\{ y:E(t,y) \wedge y' \neq 0 \}$ where $t$ is an element with $t'=1$. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U$. Moreover, the induced structure on Cartesian powers of $U$ is given by special subvarieties. If $E$ has some special form then all fibres $U_s:=\{ y:E(s,y) \wedge y' \neq 0 \}$ (with $s$ non-constant) have the same properties. In particular, since the $j$-function satisfies an Ax-Schanuel theorem of the required form (due to Pila and Tsimerman), our results will give another proof for a theorem of Freitag and Scanlon stating that the differential equation of $j$ defines a strongly minimal set with trivial geometry (which is not $\aleph_0$-categorical though).
We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using …
We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an ω-categorical sparse graph has no ω-categorical expansion with extremely amenable automorphism group.
Abstract. Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e. , ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely …
Abstract. Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e. , ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module R U to have an indecomposable local dual
Abstract We will study some aspects of the local structure of models of certain C -minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C -minimal structure …
Abstract We will study some aspects of the local structure of models of certain C -minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C -minimal structure in which the algebraic closure has the exchange property and which is locally modular, we can construct an infinite type-definable group around any non trivial point (a term to be defined later).
Abstract It is possible to define a combinatorial closure on alternating bilinear maps with few relations similar to that in [2]. For the ℵ 0 -categorical case we show that …
Abstract It is possible to define a combinatorial closure on alternating bilinear maps with few relations similar to that in [2]. For the ℵ 0 -categorical case we show that this closure is part of the algebraic closure.
The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the …
The paper is an extended version of the talk in the Logic Colloquium-2000 at Paris. We discuss a series of results and problems around Hrushovski's construction of counter-examples to the Trichotomy conjecture.
Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem …
Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>is any non-locally modular …
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>is any non-locally modular strongly minimal structure interpreted in an algebraically closed field<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>of characteristic zero, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>itself interprets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>; in particular, any non-1-based structure interpreted in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>is mutually interpretable with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.
Abstract The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that …
Abstract The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that the complex exponential field is quasiminimal.
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both …
§1. Introduction . By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ (or any real closed field) where algebra alone determines the ordering and hence the topology of the field: In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but this will be too coarse to give a diferentiable structure. A celebrated example of how partial algebraic and topological data ( G a locally euclidean group) determines a differentiable structure ( G is a Lie group) is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason. The main result which we discuss here (see [13] for the full version) is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨ M , <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of M n , n = 1, 2, …, in some first order expansion ℳ of ⟨ M , <⟩.
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We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of …
We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of green elements. A definable subgroup of an algebraic group in the green or red field is an extension of the coloured points of a multiplicative or additive algebraic group by an algebraic group. In particular, a simple group in a coloured field is algebraic.
We define a generalized version of CM-trivialityq and show that in the presence of enough regular types! or solubility. a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable …
We define a generalized version of CM-trivialityq and show that in the presence of enough regular types! or solubility. a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups. Introductioll. Recently, Hrushovski [8] has introduced a new geometric property, CM-triviality (CM standing for Cohen-Macaulay), which holds for the stable relational structures obtained via his amalgamation process. Furthermore, Baudisch has shown in [2] that his new 81-categorical nilpotent group is CM-trivial. For a partial converse, Pillay proved in [11] that any CM-trivial group of finite Morley rank must be nilpotent, using the Borovik-Cherlin-Corredor-Nesin analysis of a bad group of finite Morley rank. In section one, we shall reformulate the definition of CM-triviality in a local version more suitable to theories of infinite rank, and indeed to general stable theories, and deduce its basic properties. We shall then generalize Pillay's result in section two to CM-trivial stable groups with additional properties: either solubility, or torsion-freeness plus smallness, or the existence of enough regular types. The last condition means that every type should be non-orthogonal to a regular type; in particular this case includes superstable and hyperstable (see [17]) groups. Its proof uses a generalization of the analysis of bad groups of finite Morley rank, which will be dealt with in the last section. All theories in this paper are assumed to be stable, and groups type-definable, unless otherwise stated. §1. CM-triviality. Hrushovski defines CM-triviality in [8] as follows: DEFINITION 1.1. A stable theory is CM-trivial if the following equivalent conditions hold in Teq: ( 1 ) If B1, B2, and E are algebraically closed, B1 and B2 are independent over E, acl(sis2) n acl(EBi) = Bi for i = 1, 2, and Bi n E = A for i = 1, 2, then B1 and B2 are independent over A. (2) If B1 and B2 are independent over some E, then they are independent over acl(slB2) n acl(E) Received February 8, 1996; revised May 23, 1997. 1991 Mathematics Subject Classificatio7?. primary 03C45, secondary 20F99. The author would like to thank Andreas Baudisch and the Freie Universitat Berlin for their hospitality. (O 1998 ASSOCiatiOII fOr SYmbOIiC LOgiC 0022-48 1 2/98/6304-00 1 6/$3 .30
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. …
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3. As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.
We prove that if T is a stable theory with only a finite number (>1) of countable models, then T contains a type-definable pseudoplane. We also show that for any …
We prove that if T is a stable theory with only a finite number (>1) of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal (in the sense of [9]).
An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as …
An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between “positive” and “negative” information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a “topological” significance, for example with the Zariski topology on (the set of n -tuples of) an algebraically closed field, the “closed” sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain “special” definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an “equation”. Given a complete theory T in a language L , an L -formula φ ( x̄, ȳ ) is said to be an equation (in x̄ ) if any collection Φ of instances of φ (i.e. of formulae φ ( x̄, ā )) is equivalent to a finite subset Φ′ ⊂ Φ .
Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one type of models of power K. This notion was …
Introduction. A theory, 1, (formalized in the first order predicate calculus) is categorical in power K if it has exactly one type of models of power K. This notion was introduced by Los [ 9] and Vaught [ 16] in 1954. At that time they pointed out that a theory (e.g., the theory of dense linearly ordered sets without end points) may be categorical in power N0 and fail to be categorical in any higher power. Conversely, a theory may be categorical in every uncountable power and fail to be categorical in power N0 (e.g., the theory of algebraically closed fields of characteristic 0). Los' then raised the following question. Is a theory categorical in one uncountable power necessarily categorical in every uncountable power? The principal result of this paper is an affirmative answer to that question. We actually prove a stronger result, namely: If a theory is categorical in some uncountable power then every uncountable model of that theory is saturated. (Terminology used in the Introduction will be defined in the body of the paper; roughly speaking, a model is saturated, or universalhomogeneous, if it contains an element of every possible elementary type relative to its subsystems of strictly smaller power.) It is known(2) that a theory can have (up to isomorphism) at most one saturated model in each power. It is interesting to note that our results depend essentially on an analogue of the usual analysis of topological spaces in terms of their derived spaces and the Cantor-Bendixson theorem. The paper is divided into five sections. In ?1 terminology and some meta-mathematical results are summarized. In particular, for each theory, 1, there is described a theory, *, which has essentially the same models as z but is neater to work with. In ?2 is defined a topological space, S(A), corresponding to each subsystem, A, of a model of a theory, 1; the points of S(A) being the isomorphism types of elements with respect to A. With each monomorphism (= isomorphic imbedding), f: A -+ B, is associated a dual continuous map, f*: S(B) -+ S(A). Then there is defined for each S(A) a decreasing sequence ISa(A) I of subspaces which is analogous to (but different from)