On direct sums of $\text{B}^{(1)}$-groups – II

-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

B^{(1)}

-groups.

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De Vivo, Clorinda, and Metelli, Claudia. "On direct sums of $\text{B}^{(1)}$-groups – II." Commentationes Mathematicae Universitatis Carolinae 47.2 (2006): 189-202. <http://eudml.org/doc/249839>.

@article{DeVivo2006,
abstract = {$\operatorname\{B\}^\{(1)\}$-groups are a class of torsionfree Abelian groups of finite rank, part of the main class of Butler groups. In the paper C. Metelli, On direct sums of $\operatorname\{B\}^\{(1)\}$-groups, Comment. Math. Univ. Carolinae 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.},
author = {De Vivo, Clorinda, Metelli, Claudia},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Butler group; $\operatorname\{B\}^\{(1)\}$-group; tent; direct decomposition; finite algorithm; Butler group; B-group; tent; direct decomposition; finite algorithm},
language = {eng},
number = {2},
pages = {189-202},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On direct sums of $\text\{B\}^\{(1)\}$-groups – II},
url = {http://eudml.org/doc/249839},
volume = {47},
year = {2006},
}

TY - JOUR
AU - De Vivo, Clorinda
AU - Metelli, Claudia
TI - On direct sums of $\text{B}^{(1)}$-groups – II
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 2
SP - 189
EP - 202
AB - $\operatorname{B}^{(1)}$-groups are a class of torsionfree Abelian groups of finite rank, part of the main class of Butler groups. In the paper C. Metelli, On direct sums of $\operatorname{B}^{(1)}$-groups, Comment. Math. Univ. Carolinae 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.
LA - eng
KW - Butler group; $\operatorname{B}^{(1)}$-group; tent; direct decomposition; finite algorithm; Butler group; B-group; tent; direct decomposition; finite algorithm
UR - http://eudml.org/doc/249839
ER -

References

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Fuchs L., Infinite Abelian Groups. Vol.II, Academic Press London - New York (1973). (1973) Zbl0257.20035 MR0349869
Arnold D.M., Abelian groups and representations of finite partially ordered sets, CMS books in Mathematics Springer New York (2000). (2000) Zbl0959.16011 MR1764257
Arnold D.M., Vinsonhaler C., Finite rank Butler groups: a survey of recent results, Abelian groups (Curaçao, 1991) Dekker New York (1993), 17-41. (1993) Zbl0804.20043 MR1217256
Metelli C., On direct sums of $a n B^{(1)}$ -groups, Comment. Math. Univ. Carolinae 34 (1993), 587-591. (1993) MR1243091
De Vivo C., Metelli C., Decomposing $a n B^{(1)}$ -groups: an algorithm., Comm. Algebra 30 (2002), 12 5621-5637. (2002) MR1941915
De Vivo C., Metelli C., On degenerate $a n B^{(2)}$ -groups, Houston J. Math., to appear. MR2247900
Caruso I., De Vivo C., Metelli C., Partition bases for $a n B^{(1)}$ -groups, to appear on Abelian Groups, Rings and Modules; Lecture Notes in Pure and Applied Mathematics, Taylor and Francis, 2005.

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