Bassian-Finite Abelian Groups

Peter V. Danchev; Patrick W. Keef

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arXiv · 2507.11388

Bassian-Finite Abelian Groups

Peter V. Danchev·Patrick W. Keef

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Abstract

We introduce a new class of Abelian groups which lies strictly between the classes of co-Hopfian groups and Dedekind-finite groups, calling these groups {\it Bassian-finite}. We prove the surprising fact that in the torsion case the Bassian-finite property coincides with the co-Hopficity, thus extending a recent result by Chekhlov-Danchev-Keef in Siber. Math. J. (2026), and we construct a torsion-free Bassian-finite group which is {\it not} co-Hopfian as well as a Dedekind-finite group which is {\it not} Bassian-finite. Some other closely relevant things are also established. E.g., we extend a construction of a countable Butler group that is {\it not} completely decomposable, due to Arnold-Rangaswamy in Boll. Un. Mat. Ital. (2007), to find a Butler group of countably infinite rank which is Bassian-finite, but {\it not} completely decomposable.

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Peter V. Danchev, Patrick W. Keef. 2025-07-15. Bassian-Finite Abelian Groups. https://arxiv.org/abs/2507.11388

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