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

Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef

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.

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

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.
Paper Structure (10 sections, 14 theorems, 24 equations)

This paper contains 10 sections, 14 theorems, 24 equations.

Table of Contents

  1. Definitions and Motivations
  2. Generalizing the Bassian Property
  3. Finitely Bassian groups
  4. Semi Bassian groups
  5. Fully generalized Bassian groups
  6. Absolutely generalized Bassian groups
  7. Generalizing the co-Bassian Property
  8. Finitely and semi co-Bassian groups
  9. Fully generalized co-Bassian groups
  10. Absolutely generalized co-Bassian groups

Key Result

Lemma 2.1

K Suppose $G$ has finite (torsion-free) rank, $N$ is a subgroup of $G$, $\pi:G\to \overline G:=G/N$ is the usual epimorphism and $\phi: G\to \overline G$ is an injection. Then, $N\subseteq T$, so that $\overline T=T/N$ is the torsion subgroup of $\overline G$. In addition, $\pi$ induces an isomorphi

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Lemma 2.1
  • proof
  • ...and 22 more