Zero-sum subsets of decomposable sets in Abelian groups

IF 0.3 Q4 MATHEMATICS, APPLIED

Algebra & Discrete Mathematics Pub Date : 2019-03-08 DOI:10.12958/adm1494

T. Banakh, A. Ravsky

引用次数: 1

Abstract

A subset D of an abelian group is decomposable if ∅≠D⊂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⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|

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阿贝尔群中可分解集合的零和子集

阿贝尔群的子集D是可分解的，如果∅≠D⊂D+D。本文给出了一个开放问题的部分答案，该问题询问阿贝尔群的每个有限可分解子集D是否包含∑Z=0的非空子集Z⊂D。对于每个n∈n，我们在2n−1阶循环群中给出了基数|D|=n的可分解子集D，使得∑D=0，但对于任何适当的非空子集T⊂D，∑T≠0。另一方面，我们证明了基数|D|≤7的每个可分解子集D⊂R包含基数|Z|≤12|D|的非空子集T \8834D，∑Z=0。对于每个n∈n，我们给出了基数|D|=2n的子集D⊂Z，使得∑Z=0对于基数|Z|=n的某个子集Z \8834D，并且∑T≠0对于基数|T|

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