Mathematics and Statistics 12(2): 204-210, 2024 DOI: 10.13189/ms.2024.120210 http://www.hrpub.org Variations of Rigidity for Abelian Groups Inessa I. Pavlyuk 1 , Sergey V. Sudoplatov 1,2 1 Department of Algebra and Mathematical Logic, Faculty of Applied Mathematics and Computer Science, Novosibirsk State Technical University, Russia 2 Laboratory of Logical Systems, Sobolev Institute of Mathematics, Novosibirsk 630090, Russia Received November 9, 2023; Revised March 5, 2024; Accepted March 19, 2024 Cite This Paper in the Following Citation Styles (a): [1] Inessa I. Pavlyuk, Sergey V. Sudoplatov, ”Variations of Rigidity for Abelian Groups,” Mathematics and Statistics, Vol.12, No.2, pp. 204-210, 2024. DOI: 10.13189/ms.2024.120210 (b): Inessa I. Pavlyuk, Sergey V. Sudoplatov (2024). Variations of Rigidity for Abelian Groups. Mathematics and Statistics, 12(2), 204-210. DOI: 10.13189/ms.2024.120210 Copyright ©2024 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract A series of basic characteristics of structures and of elementary theories reflect their complexity and richness. Among these characteristics, four kinds of degrees of rigidity and the index of rigidity are considered as measures of how far the given structure is situated from rigid one, both with respect to the automorphism group and to the definable closure, for some or any subset of the universe, which has the given finite cardinality. Thus, a natural question arises on a classification of model-theoretic objects with respect to rigidity characteristics. We apply a general approach of studying the rigidity values and related classification to abelian groups and their theories. We describe possibilities of degrees and indexes of rigidity for finite abelian groups and for standard infinite abelian groups. This description is based both on general consideration of rigidity, on its application for finite structures, and on their specificity for abelian groups including Szmielew invariants, combinatorial formulas for cardinalities of orbits, links with dimensions, and on their combinations. It shows how characteristics of infinite abelian groups are related to them with respect to finite ones. Some applications for non-standard abelian groups are discussed. Keywords Rigidity, Abelian Group, Semantic Degree of Rigidity, Syntactic Degree of Rigidity, Index of Rigidity Mathematics Subject Classification (2020) 03C50, 03C30, 20K21 1 Introduction The class of abelian groups is rich enough [1, 2] and admits a good elementary classification by Szmielew invariants [3, 4, 5] reducing abelian groups to standard ones which are represented by direct sums of a given collection of standard groups. It is broadly investigated both semantically, with respect to struc- tures of abelian groups [6, 7, 8] and their syntax [9, 10]. In the present paper we continue to study families of abelian groups and their theories, starting with possibilities of closures, ranks and approximations. We apply rigidity characteristics [11] for the class of standard abelian groups describing possi- bilities of both semantic and syntactic degrees of rigidity, and indexes of rigidity. The paper is organized as follows. In Section 2, we col- lect preliminary notions, notations and results on degrees and indexes of rigidity, and degrees of algebraicity. In Section 3, we define Szmielew invariants, standard groups, and links of Szmielew invariants. In Section 4, we describe indexes and degrees of rigidity for finite abelian groups (Corollary 4.2 and Theorem 4.11). Theorem 4.11 asserts a dichotomy for tuples of degrees of rigidity. In Section 5, degrees and indexes of rigid- ity for standard infinite abelian groups are found (Theorems 5.4 and 5.11). In Section 6, the considered approach is illus- trated for the group of integers and its variations. Described values are based on cardinalities of orbits, Szmielew invariants including dimensions of abelian groups, on Euler function, and their combinations. 2 Preliminaries Throughout we use standard model-theoretic and group- theoretic notions and notations [5, 16].