Mathematics and Statistics Vol. 8(2), pp. 220 - 224
DOI: 10.13189/ms.2020.080218
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Approximations for Theories of Abelian Groups

Inessa I. Pavlyuk ^1,2, Sergey V. Sudoplatov ^1,3,*
1 Novosibirsk State Technical University, K.Marx avenue, 20, 630073, Novosibirsk, Russia
² Novosibirsk State Pedagogical University, Vilyuiskaya street, 28, 630126, Novosibirsk, Russia
³ Sobolev Institute of Mathematics, Academician Koptyug avenue, 4, 630090, Novosibirsk, Russia

ABSTRACT

Approximations of syntactic and semantic objects play an important role in various fields of mathematics. They can create theories and structures in one given class by means of others, usually simpler. For instance, in certain situations, infinite objects can be approximated by finite or strongly minimal ones. Thus, complicated objects can be collected using simplified ones. Among these objects, Abelian groups, their first order theories, connections and dynamics are of interests. Theories of Abelian groups are characterized by Szmielew invariants leading to the study and descriptions of approximations in terms of these invariants. In the paper we apply a general approach for approximating theories to the class of theories of Abelian groups which characterizes the approximability of a theory of Abelian groups by a given family of theories of Abelian groups in terms of Szmielew invariants and their limits. We describe some forms of approximations for theories of Abelian groups. In particular, approximations of theories of Abelian groups by theories of finite ones are characterized. In addition, we describe approximations by quasi-cyclic and torsion-free Abelian groups and their combinations with respect to given families of prime numbers. Approximations and closures of families of theories with respect to standard Abelian groups for various sets of prime numbers are also described.

KEYWORDS
Approximation, Theory, Abelian Group, Approximable Theory

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Inessa I. Pavlyuk , Sergey V. Sudoplatov , "Approximations for Theories of Abelian Groups," Mathematics and Statistics, Vol. 8, No. 2, pp. 220 - 224, 2020. DOI: 10.13189/ms.2020.080218.

(b). APA Format:
Inessa I. Pavlyuk , Sergey V. Sudoplatov (2020). Approximations for Theories of Abelian Groups. Mathematics and Statistics, 8(2), 220 - 224. DOI: 10.13189/ms.2020.080218.