Some model theory of abelian groups

1972 ◽  
Vol 37 (2) ◽  
pp. 335-342 ◽  
Author(s):  
Paul C. Eklof
Keyword(s):  
Model Theory ◽  
Abelian Groups ◽  
Existential Sentence ◽  
Complete Theories

AbstractWe study the relations between abelian groups B and C that every universal (resp. universal-existential) sentence true in B is also true in C, and give algebraic criteria for these relations to hold. As a consequence we characterize the inductive complete theories of abelian groups and prove that they are exactly the model-complete theories.

The model theory of finitely generated finite-by-abelian groups

1984 ◽  
Vol 49 (4) ◽  
pp. 1115-1124
Author(s):  
Francis Oger
Keyword(s):  
Model Theory ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Elementary Embeddings

AbstractIn [O1], we gave algebraic characterizations of elementary equivalence for finitely generated finite-by-abelian groups, i.e. finitely generated FC-groups. We also provided several examples of finitely generated finite-by-abelian groups which are elementarily equivalent without being isomorphic.In this paper, we shall use our previous results to describe precisely the models of the theories of finitely generated finite-by-abelian groups and the elementary embeddings between these models.

A result concerning cardinalities of ultraproducts

1974 ◽  
Vol 39 (1) ◽  
pp. 43-48 ◽  
Author(s):  
H. Jerome Keisler ◽  
Karel Prikry
Keyword(s):  
Model Theory ◽  
Abelian Groups ◽  
Image Position

The cardinality problem for ultraproducts is as follows: Given an ultrafilter over a set I and cardinals αi, i ∈ I, what is the cardinality of the ultraproduct ? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter over a set I of power κ (uniform means that all elements of have power κ). It is open whether every countably incomplete has the property that, for all infinite α, the ultra-power has power ακ. However, it is shown in [4] that certain countably incomplete , namely the κ-regular , have this property.This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the κ-good ultrafilters, do have this property. The κ-good ultrafilters are important in model theory because they are exactly the ultrafilters such that every ultraproduct modulo is κ-saturated (see [5]).Let be an ultrafilter on a set I. Let αi, n, i ∈ I, n ∈ ω, be cardinals and αi, n, ≥ αi, m if n < m. Let.Then ρn are nonincreasing and therefore there is some m and ρ such that ρn = ρ if n ≥ m. We call ρ the eventual value (abbreviated ev val) of ρn.

From “Metabelian ℚ-Vector Spaces” to New ω-Stable Groups

1996 ◽  
Vol 2 (1) ◽  
pp. 84-93 ◽  
Author(s):  
Olivier Chapuis
Keyword(s):  
Group Theory ◽  
Model Theory ◽  
Stability Theory ◽  
Abelian Groups ◽  
Vector Spaces ◽  
Universal Theory ◽  
Direct Power ◽  
Metabelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

The aim of this paper is to describe (without proofs) an analogue of the theory of nontrivial torsion-free divisible abelian groups for metabelian groups. We obtain illustrations for “old-fashioned” model theoretic algebra and “new” examples in the theory of stable groups. We begin this paper with general considerations about model theory. In the second section we present our results and we give the structure of the rest of the paper. Most parts of this paper use only basic concepts from model theory and group theory (see [14] and especially Chapters IV, V, VI and VIII for model theory, and see for example [23] and especially Chapters II and V for group theory). However, in Section 5, we need some somewhat elaborate notions from stability theory. One can find the beginnings of this theory in [14], and we refer the reader to [16] or [21] for stability theory and to [22] for stable groups.§1. Some model theoretic considerations. Denote by the theory of torsion-free abelian groups in the language of groups ℒgp. A finitely generated group G satisfies iff G is isomorphic to a finite direct power of ℤ. It follows that axiomatizes the universal theory of free abelian groups and that the theory of nontrivial torsion-free abelian groups is complete for the universal sentences. Denote by the theory of nontrivial divisible torsion-free abelian groups.

Infinitary model theory of abelian groups

1976 ◽  
Vol 25 (1-2) ◽  
pp. 97-107 ◽  
Author(s):  
Paul C. Eklof
Keyword(s):  
Model Theory ◽  
Abelian Groups

scholarly journals Truncations of ordered abelian groups

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Paola D’Aquino ◽  
Jamshid Derakhshan ◽  
Angus Macintyre
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Abelian Groups ◽  
Forthcoming Paper ◽  
Peano Arithmetic ◽  
Local Rings ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Nonstandard Models ◽  
Quotient Rings

AbstractWe give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

scholarly journals COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM

2014 ◽  
Vol 20 (1) ◽  
pp. 39-79 ◽  
Author(s):  
JOHN T. BALDWIN
Keyword(s):  
Formal Methods ◽  
Model Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Specific Content ◽  
First Order ◽  
Complete Theories ◽  
The Stability ◽  
Second Order Logic

AbstractWe propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.

scholarly journals Method of the rheostat for studying properties of fragments of theoretical sets

2020 ◽  
Vol 100 (4) ◽  
pp. 152-159
Author(s):  
Aibat Yeshkeyev ◽  
Keyword(s):  
Model Theory ◽  
Complete Theory ◽  
Existentially Closed ◽  
Complete Theories ◽  
Syntactic Similarity ◽  
Definition Of

In this article discusses the model-theoretical properties of fragments of theoretical sets and the rheostat method. These two concepts: theoretical set and rheostat are new. The study of this topic in the framework of the study of Jonsson theories, the Jonsson spectrum, classes of existentially closed models of such fragments is a new promising class of problems and their solution is closely related to many problems that once defined the classical problems of model theory. The purpose of this article is to determine the rheostat of the transition from complete theory to Jonsson theory, which will be consistent with the corresponding concepts for any α and any α-Jonsson theory. For this we define a theoretical set. On the basis of research by the author formulated a model-theoretical definition of the concept of a rheostat in the transition from complete theories to ϕ(x)-theoretically convex Jonsson sets. Also was formulated an application of h-syntactic similarity to α-Jonsson theories.

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