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 mono Dedekind-finite. We prove the surprising fact that in the torsion case the mono Dedekind-finite property coincides with the co-Hopficity, thus extending a recent result by Chekhlov–Danchev–Keef [Sib. Math. J. (2026), to appear], and we construct a torsion-free mono Dedekind-finite group which is not co-Hopfian as well as a Dedekind-finite group which is not mono Dedekind-finite. Some other closely relevant things are also established. For example, we extend the construction due to Arnold–Rangaswamy [Boll. Unione Mat. Ital. Sez. B, Artic. Ric. Mat. (8) 10 (3), 605–611 (2007)] of a countable Butler group that is not completely decomposable to find a Butler group of countably infinite rank which is mono Dedekind-finite, but not completely decomposable.
