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On Strongly Hopfian Mixed Abelian Groups
Peter V. Danchev, Patrick W. Keef
The classes of abelian groups that are (uniformly) strongly Hopfian abelian groups, and dually, (uniformly) strongly co-Hopfian abelian groups have been studied by several authors,…
On Some Versions of Hopficity for Abelian Groups
Peter V. Danchev, Patrick W. Keef
We completely describe in certain important cases the class of commutative co-finitely Hopfian groups as defined by Bridson-Groves-Hillman- Martin in the journal Groups, Geometry,…
Strongly and Uniformly Strongly co-Hopfian Abelian Groups
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
We consider the so-called {\it strongly co-Hopfian} and {\it uniformly strongly co-Hopfian} Abelian groups, significantly generalizing some important results due to Abdelalim in th…
Bassian-Finite Abelian Groups
Peter V. Danchev, Patrick W. Keef
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 pr…
Hereditarily and Super Generalized co-Bassian Abelian Groups
Peter V. Danchev, Brendan Goldsmith, Patrick W. Keef
We completely characterize by finding necessary and sufficient conditions those co-Bassian and generalized co-Bassian Abelian groups having, respectively, the hereditary or the sup…
Two Generalizations of co-Hopfian Abelian Groups
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
By defining the classes of generalized co-Hopfian and relatively co-Hopfian groups, respectively, we consider two expanded versions of the generalized co-Bassian groups and of the…
Two Generalizations of Hopfian Abelian Groups
Andrey R. Chekhlov, Peter V. Danchev, Brendan Goldsmith +1
This paper targets to generalize the notion of Hopfian groups in the commutative case by defining the so-called {\bf relatively Hopfian groups} and {\bf weakly Hopfian groups}, and…
Generalizations of the Bassian and co-Bassian Properties for Abelian Groups
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
Trying to finalize in some way the present subject, this paper targets to generalize substantially the notions of Bassian and co-Bassian groups by introducing the so-called finitel…
Semi-Generalized co-Bassian Groups
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
As a common non-trivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassi…
Semi-Generalized Bassian Groups
Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef
As a common non-trivial generalization of the concept of a proper generalized Bassian group, we introduce the notion of a semi-generalized Bassian group and initiate its comprehens…
Generalized Bassian and other Mixed Abelian Groups with Bounded p-Torsion
Peter V. Danchev, Patrick W. Keef
It is known that a mixed abelian group G with torsion T is Bassian if, and only if, it has finite torsion-free rank and has finite p-torsion (i.e., each Tp is finite). It is also k…
Abelian p-groups with minimal characteristic inertia
Peter V. Danchev, Patrick W. Keef
For Abelian p-groups, Goldsmith, Salce, et al., introduced the notion of minimal full inertia. In parallel to this, we define the concept of minimal characteristic inertia and expl…