Variations of Rigidity for Abelian Groups

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.