Title: | On direct sums of $\text{B}^{(1)}$-groups -- II (English) |

Author: | De Vivo, Clorinda |

Author: | Metelli, Claudia |

Language: | English |

Journal: | Commentationes Mathematicae Universitatis Carolinae |

ISSN: | 0010-2628 (print) |

ISSN: | 1213-7243 (online) |

Volume: | 47 |

Issue: | 2 |

Year: | 2006 |

Pages: | 189-202 |

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Category: | math |

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Summary: | $\operatorname{B}^{(1)}$-groups are a class of torsionfree Abelian groups of finite rank, part of the main class of Butler groups. In the paper C. Metelli, {\it On direct sums of $\operatorname{B}^{(1)}$-groups\/}, Comment. Math. Univ. Carolinae {\bf 34} (1993), 587--591, the problem of direct sums of $\operatorname{B}^{(1)}$-groups was discussed, and a necessary and sufficient condition was given for the direct sum of two $\operatorname{B}^{(1)}$-groups to be a $\operatorname{B}^{(1)}$-group. While sufficiency holds, necessity was wrongly claimed; we solve here the problem, and in the process study a curious hierarchy among indecomposable direct summands of $\operatorname{B}^{(1)}$-groups. (English) |

Keyword: | Butler group |

Keyword: | $\operatorname{B}^{(1)}$-group |

Keyword: | tent |

Keyword: | direct decomposition |

Keyword: | finite algorithm |

MSC: | 20K15 |

idZBL: | Zbl 1106.20041 |

idMR: | MR2241526 |

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Date available: | 2009-05-05T16:56:42Z |

Last updated: | 2012-04-30 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/119586 |

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Related article: | http://dml.cz/handle/10338.dmlcz/118616 |

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Reference: | [FII] Fuchs L.: Infinite Abelian Groups. Vol.II.Academic Press London - New York (1973). Zbl 0257.20035, MR 0349869 |

Reference: | [A] Arnold D.M.: Abelian groups and representations of finite partially ordered sets.CMS books in Mathematics Springer New York (2000). Zbl 0959.16011, MR 1764257 |

Reference: | [AV] Arnold D.M., Vinsonhaler C.: Finite rank Butler groups: a survey of recent results.Abelian groups (Curaçao, 1991) Dekker New York (1993), 17-41. Zbl 0804.20043, MR 1217256 |

Reference: | [M] Metelli C.: On direct sums of ${an{B^{(1)}}}$-groups.Comment. Math. Univ. Carolinae 34 (1993), 587-591. MR 1243091 |

Reference: | [DVM4] De Vivo C., Metelli C.: Decomposing ${an{B^{(1)}}}$-groups: an algorithm..Comm. Algebra 30 (2002), 12 5621-5637. MR 1941915 |

Reference: | [DVM8] De Vivo C., Metelli C.: On degenerate ${an{B^{(2)}}}$-groups.Houston J. Math., to appear. MR 2247900 |

Reference: | [CDVM] Caruso I., De Vivo C., Metelli C.: Partition bases for ${an{B^{(1)}}}$-groups.to appear on Abelian Groups, Rings and Modules; Lecture Notes in Pure and Applied Mathematics, Taylor and Francis, 2005. |

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