2025/07/15 by Peter Danchev, Patrick W. Keef, Danchev, Peter V. +1
Mathematics · #20K10 #20K20 #20K21 #20K30 #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2507.11388
openalex publication_date 2025/07/15 · openalex created_date 2025/10/08 · openalex updated_date 2026/07/28
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.