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# Article

 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 . Category: math . 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 . Date available: 2009-05-05T16:56:42Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119586 . Related article: http://dml.cz/handle/10338.dmlcz/118616 . 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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