Abstract: The rank for families of theories is similar to Morley rank and can be considered as a measure for complexity or richness of these families. Increasing the rank by extensions of families we produce more rich families and obtaining families with the infinite rank that can be considered as “rich enough”. In the paper, we realize ranks for families of theories of abelian groups. In particular, we study ranks and closures for families of theories of finite abelian groups observing that the set of theories of finite abelian groups in not totally transcendental, i.e., its rank equals infinity. We characterize pseudofinite abelian groups in terms of Szmielew invariants. Besides we characterize e-minimal families of theories of abelian groups both in terms of dimension, i.e., the number of independent limits for Szmielew invariants, and in terms of inequalities for Szmielew invariants. These characterizations are obtained both for finite abelian groups and in general case. Furthermore we give characterizations for approximability of theories of abelian groups and show the possibility to count Szmielew invariants via these parameters for approximations. We describe possibilities to form d-definable families of theories of abelian groups having given countable rank and degree.

Cite: Павлюк И.И. , Sudoplatov S.V.
Ranks for families of theories of abelian groups
Известия Иркутского государственного университета. Серия: Математика (Bulletin of Irkutsk State University. Series Mathematics). 2019. V.28. P.95-112. DOI: 10.26516/1997-7670.2019.28.95 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Apr 25, 2019
Published print:	Jun 21, 2019
Published online:	Jun 21, 2019

Identifiers:

Web of science:	WOS:000476659000007
Scopus:	2-s2.0-85068622741
Elibrary:	38228682
OpenAlex:	W2950895861

Citing:

DB	Citing
Web of science	11
Scopus	12
Elibrary	18
OpenAlex	8

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