Co-Hopfian and boundedly endo-rigid mixed abelian groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917610912743424 |
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| author | Asgharzadeh, Mohsen Golshani, Mohammad Shelah, Saharon |
| author_facet | Asgharzadeh, Mohsen Golshani, Mohammad Shelah, Saharon |
| contents | 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 $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_17210 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Co-Hopfian and boundedly endo-rigid mixed abelian groups Asgharzadeh, Mohsen Golshani, Mohammad Shelah, Saharon Logic Group Theory Rings and Algebras 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 $λ$. |
| title | Co-Hopfian and boundedly endo-rigid mixed abelian groups |
| topic | Logic Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2210.17210 |