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

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.

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.

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 order indiscernibles of divisible ordered abelian groups

1984 ◽  
Vol 49 (1) ◽  
pp. 151-160
Author(s):  
David Rosenthal
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Simple Condition ◽  
Algebraic Invariants ◽  
Image Position ◽  
And Algebra ◽  
Simplified Form

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in CNote that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.

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