String numbers of abelian groups

TL;DR

This paper investigates the global string number for abelian groups, providing calculations, characterizations of groups with zero string number, and exploring stability properties, thus advancing understanding of combinatorial entropy in algebraic contexts.

Contribution

It introduces the global string number for abelian groups, computes it in general, and characterizes groups with zero string number, including stability analysis.

Findings

Calculated string numbers for various abelian groups

Characterized abelian groups with zero string number

Analyzed stability properties of the string number

Abstract

The string number of self-maps arose in the context of algebraic entropy and it can be viewed as a kind of combinatorial entropy function. Later on its values for endomorphisms of abelian groups were calculated in full generality. We study its global version for abelian groups, providing several examples involving also Hopfian abelian groups. Moreover, we characterize the class of all abelian groups with string number zero in many cases and discuss its stability properties.