Generalizations of the Bassian and co-Bassian Properties for Abelian Groups

TL;DR

This paper extends the concepts of Bassian and co-Bassian groups by introducing various generalized forms, analyzing their properties, and highlighting the distinctions between these extended properties in the context of abelian groups.

Contribution

It introduces new generalized notions of Bassian and co-Bassian groups and explores their properties, providing a broader framework for understanding these concepts in abelian groups.

Findings

Extensions of Bassian and co-Bassian properties are completely distinct.

Some generalized concepts coincide with classical properties in the reduced case.

New characterizations and properties of these generalized groups are established.

Abstract

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 finitely (co-)Bassian groups, semi (co-)Bassian groups, fully generalized (co-)Bassian groups, absolutely generalized (co-)Bassian groups and establishing their crucial properties and characterizations. In fact, some of the concepts give nothing new by coinciding in the reduced case with the well-known (co-)Bassian property. However, in some of the definitions, the situation is slightly more complicated and we obtain a few new and interesting things by showing that the extensions of the Bassian and co-Bassian properties are totally distinct each other.