Expressive Power of Infinitary Logic and Absolute co-Hopfianity

TL;DR

This paper investigates the limits of absolute co-Hopfianity in abelian groups and modules, demonstrating the absence of such groups above a certain cardinal using infinitary logic, contrasting with recent results on absolute Hopfian groups.

Contribution

It introduces a logical framework to show the non-existence of absolute co-Hopfian abelian groups beyond the first beautiful cardinal, extending the analysis to modules over rings.

Findings

No absolute co-Hopfian abelian groups above the first beautiful cardinal

Use of infinitary logic to establish non-existence results

Extension of results to modules over commutative rings

Abstract

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We use the infinitary logic to show that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.