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Co-Hopfian and boundedly endo-rigid mixed abelian groups

Mohsen Asgharzadeh, Mohammad Golshani, Saharon Shelah

Abstract

For a given cardinal $λ$ and a torsion abelian group $K$ of cardinality less than $λ$, we present, under some mild conditions (for example $λ=λ^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of cardinality $λ$ with $Tor(G)=K$. Essentially, we give a complete characterization of such pairs $(K, λ)$. Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals $λ> 2^{\aleph_{0}}$ for which there is a co-Hopfian abelian group of size $λ$.

Co-Hopfian and boundedly endo-rigid mixed abelian groups

Abstract

For a given cardinal and a torsion abelian group of cardinality less than , we present, under some mild conditions (for example ), boundedly endo-rigid abelian group of cardinality with . Essentially, we give a complete characterization of such pairs . Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals for which there is a co-Hopfian abelian group of size .
Paper Structure (4 sections, 30 theorems, 155 equations)

This paper contains 4 sections, 30 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ZFC construction of boundedly rigid mixed groups
  4. Co-Hopfian and boundedly endo-rigid abelian groups

Key Result

Theorem 1.3

Given a cardinal $\lambda$ such that $\lambda=\lambda^{\aleph_0} > 2^{\aleph_0}$ and a torsion group $K$ of cardinality less than $\lambda$, there is a boundedly rigid abelian group $G$ of cardinality $\lambda$ with $\mathop{\mathrm{tor}}\nolimits(G)=K.$

Theorems & Definitions (85)

  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.8
  • Definition 2.12
  • Remark 2.16
  • Definition 2.17
  • ...and 75 more