decomposable sets
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2020 ◽  
Vol 485 (2) ◽  
pp. 123816 ◽  
Author(s):  
Valeriu Soltan
Keyword(s):  

2020 ◽  
Vol 30 (1) ◽  
pp. 15-25
Author(s):  
T. Banakh ◽  
◽  
A. Ravsky ◽  

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|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.


2017 ◽  
Vol 37 (3) ◽  
pp. 2837-2844 ◽  
Author(s):  
A. N. Iusem ◽  
M. I. Todorov
Keyword(s):  

2010 ◽  
Vol 372 (2) ◽  
pp. 525-537 ◽  
Author(s):  
M.A. Goberna ◽  
J.E. Martínez-Legaz ◽  
M.I. Todorov

Author(s):  
Andrzej Fryszkowski
Keyword(s):  

2000 ◽  
Vol 33 (3) ◽  
Author(s):  
Celina Rom
Keyword(s):  

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