Zero-sum subsets of decomposable sets in Abelian groups
A subset \(D\) of an abelian group is decomposable if \( \emptyset\ne D\subset D+D\). In the paper we give partial answers to an open problem asking whether every finite decomposable subset \(D\) of an abelian group contains a non-empty subset \(Z\subset D\) with \(\sum Z=0\). For every \(n\in\mathb...
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Lugansk National Taras Shevchenko University
2020
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oai:ojs.admjournal.luguniv.edu.ua:article-14942021-01-03T08:39:36Z Zero-sum subsets of decomposable sets in Abelian groups Banakh, T. Ravsky, A. decomposable set, abelian group, sum-set 05E15 A subset \(D\) of an abelian group is decomposable if \( \emptyset\ne D\subset D+D\). In the paper we give partial answers to an open problem asking whether every finite decomposable subset \(D\) of an abelian group contains a non-empty subset \(Z\subset D\) with \(\sum Z=0\). For every \(n\in\mathbb{N}\) we present a decomposable subset \(D\) of cardinality \(|D|=n\) in the cyclic group of order \(2^n-1\) such that \(\sum D=0\), but \(\sum T\ne 0\) for any proper non-empty subset \(T\subset D\). On the other hand, we prove that every decomposable subset \(D\subset\mathbb{R}\) of cardinality \(|D|\le 7\) contains a non-empty subset \(T\subset D\) of cardinality \(|Z|\le\frac12|D|\) with \(\sum Z=0\). For every \(n\in\mathbb{N}\) we present a subset \(D\subset\mathbb{Z}\) of cardinality \(|D|=2n\) such that \(\sum Z=0\) for some subset \(Z\subset D\) of cardinality \(|Z|=n\) and \(\sum T\ne 0\) for any non-empty subset \(T\subset D\) of cardinality \(|T|<n=\frac12|D|\). Also we prove that every finite decomposable subset \(D\) of an Abelian group contains two non-empty subsets \(A,B\) such that \(\sum A+\sum B=0\). Lugansk National Taras Shevchenko University 2020-12-30 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1494 10.12958/adm1494 Algebra and Discrete Mathematics; Vol 30, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1494/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1494/619 Copyright (c) 2020 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2021-01-03T08:39:36Z |
| collection |
OJS |
| language |
English |
| topic |
decomposable set abelian group sum-set 05E15 |
| spellingShingle |
decomposable set abelian group sum-set 05E15 Banakh, T. Ravsky, A. Zero-sum subsets of decomposable sets in Abelian groups |
| topic_facet |
decomposable set abelian group sum-set 05E15 |
| format |
Article |
| author |
Banakh, T. Ravsky, A. |
| author_facet |
Banakh, T. Ravsky, A. |
| author_sort |
Banakh, T. |
| title |
Zero-sum subsets of decomposable sets in Abelian groups |
| title_short |
Zero-sum subsets of decomposable sets in Abelian groups |
| title_full |
Zero-sum subsets of decomposable sets in Abelian groups |
| title_fullStr |
Zero-sum subsets of decomposable sets in Abelian groups |
| title_full_unstemmed |
Zero-sum subsets of decomposable sets in Abelian groups |
| title_sort |
zero-sum subsets of decomposable sets in abelian groups |
| description |
A subset \(D\) of an abelian group is decomposable if \( \emptyset\ne D\subset D+D\). In the paper we give partial answers to an open problem asking whether every finite decomposable subset \(D\) of an abelian group contains a non-empty subset \(Z\subset D\) with \(\sum Z=0\). For every \(n\in\mathbb{N}\) we present a decomposable subset \(D\) of cardinality \(|D|=n\) in the cyclic group of order \(2^n-1\) such that \(\sum D=0\), but \(\sum T\ne 0\) for any proper non-empty subset \(T\subset D\). On the other hand, we prove that every decomposable subset \(D\subset\mathbb{R}\) of cardinality \(|D|\le 7\) contains a non-empty subset \(T\subset D\) of cardinality \(|Z|\le\frac12|D|\) with \(\sum Z=0\). For every \(n\in\mathbb{N}\) we present a subset \(D\subset\mathbb{Z}\) of cardinality \(|D|=2n\) such that \(\sum Z=0\) for some subset \(Z\subset D\) of cardinality \(|Z|=n\) and \(\sum T\ne 0\) for any non-empty subset \(T\subset D\) of cardinality \(|T|<n=\frac12|D|\). Also we prove that every finite decomposable subset \(D\) of an Abelian group contains two non-empty subsets \(A,B\) such that \(\sum A+\sum B=0\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2020 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1494 |
| work_keys_str_mv |
AT banakht zerosumsubsetsofdecomposablesetsinabeliangroups AT ravskya zerosumsubsetsofdecomposablesetsinabeliangroups |
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2025-07-17T10:30:59Z |
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2025-07-17T10:30:59Z |
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1837890132262256640 |