M. I. Kargapolov. Ob eléméntarnoj téorii abélévyh grupp (On the elementary theory of Abelian groups). Algébra i logika, Séminar, vol. 1 no. 6 (1963), pp. 26–36.

1968 ◽  
Vol 32 (4) ◽  
pp. 535-536 ◽  
Author(s):  
J. Mennicke
Keyword(s):  
Abelian Groups ◽  
Elementary Theory ◽  
Algebra I

Model-Completeness and Elementary Properties of Torsion Free Abelian Groups

1974 ◽  
Vol 26 (4) ◽  
pp. 829-840
Author(s):  
Elias Zakon
Keyword(s):  
High Power ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Complete Classification ◽  
Formal Logic ◽  
Predicate Calculus ◽  
Symbolic Logic ◽  
Model Completeness ◽  
Free Abelian Groups ◽  
Torsion Free

The decidability of the elementary theory of abelian groups, and their complete classification by elementary properties (i.e. those formalizable in the lower predicate calculus (LPC) of formal logic), were established by W. Szmielew [13]. More general results were proved by Eklof and Fischer [2], and G. Sabbagh [12]. The rather formidable "high-power" techniques used in obtaining these remarkable results, and the length of the proofs (W. Szmielew's proof takes about 70 pages) triggered off several attempts at simplification. M. I. Kargapolov's proof [3] unfortunately turned out to be erroneous (cf. J. Mennicke's review in the Journal of Symbolic Logic, vol. 32, p. 535).

scholarly journals On the elementary theory of pairs of real closed fields. II

1982 ◽  
Vol 47 (3) ◽  
pp. 669-679 ◽  
Author(s):  
Walter Baur
Keyword(s):  
Power Series ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Predicate Symbol ◽  
Axiom System ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Real Closed Fields ◽  
Order Language ◽  
Closed Fields

Let ℒ be the first order language of field theory with an additional one place predicate symbol. In [B2] it was shown that the elementary theory T of the class of all pairs of real closed fields, i.e., ℒ-structures ‹K, L›, K a real closed field, L a real closed subfield of K, is undecidable.The aim of this paper is to show that the elementary theory Ts of a nontrivial subclass of containing many naturally occurring pairs of real closed fields is decidable (Theorem 3, §5). This result was announced in [B2]. An explicit axiom system for Ts will be given later. At this point let us just mention that any model of Ts, is elementarily equivalent to a pair of power series fields ‹R0((TA)), R1((TB))› where R0 is the field of real numbers, R1 = R0 or the field of real algebraic numbers, and B ⊆ A are ordered divisible abelian groups. Conversely, all these pairs of power series fields are models of Ts.Theorem 3 together with the undecidability result in [B2] answers some of the questions asked in Macintyre [M]. The proof of Theorem 3 uses the model theoretic techniques for valued fields introduced by Ax and Kochen [A-K] and Ershov [E] (see also [C-K]). The two main ingredients are(i) the completeness of the elementary theory of real closed fields with a distinguished dense proper real closed subfield (due to Robinson [R]),(ii) the decidability of the elementary theory of pairs of ordered divisible abelian groups (proved in §§1-4).I would like to thank Angus Macintyre for fruitful discussions concerning the subject. The valuation theoretic method of classifying theories of pairs of real closed fields is taken from [M].

scholarly journals The elementary theory of abelian groups

1972 ◽  
Vol 4 (2) ◽  
pp. 115-171 ◽  
Author(s):  
Paul C. Eklof ◽  
Edward R. Fischer
Keyword(s):  
Abelian Groups ◽  
Elementary Theory
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