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Báo cáo toán học: "Unextendible Sequences in Finite Abelian Groups"
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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Unextendible Sequences in Finite Abelian Groups. | Unextendible Sequences in Finite Abelian Groups Jujuan Zhuang Department of Mathematics Dalian Maritime University Dalian P. R. China jjzhuang1979@yahoo.com.cn Submitted Oct 25 2007 Accepted Jun 21 2008 Published Jun 30 2008 Mathematics Subject Classifications 2000 11B75 11P21 11B50. Abstract Let G Cn1 . Cnr be a finite abelian group with r 1 or 1 nil. nr and let S a1 . at be a sequence of elements in G. We say S is an unextendible sequence if S is a zero-sum free sequence and for any element g 2 G the sequence Sg is not zero-sum free any longer. Let L G log2n1e . log2nre and d G Ur 1 ni 1 in this paper we prove among other results that the minimal length of an unextendible sequence in G is not bigger than L G and for any integer k where L G k d G there exists at least one unextendible sequence of length k. 1 Introduction Let G be an additively written finite abelian group G Cn1 . Cnr its direct decomposition into cyclic groups where r 1 or 1 n11. nr. Set ei 0 . 0 1 0 . 0 i th for all i 2 1 r then e1 . er is a basis of G. We set L G dlog2 ni 1 . dlog2 nr1 and r d G - 1 . i 1 Let F G denote the free abelian monoid over G with monoid operation written multiplicatively and given by concatenation i.e. F G consists of all multi-sets over G and an element S 2 F G which we refer to as a sequence is written in the form k S n g n gvs S i 1 g2G THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N24 1 with gi 2 G where vg S 2 No is the multiplicity of g in S and k is the length of S denoted by SI k. A sequence T is a subsequence of S if vg T vg S for every g 2 G denoted by T S and ST-1 denote the sequence obtained by deleting the terms of T from S. By ơ S we denote the sum of S that is ơ S pi 1 gi Pg2G vg S g 2 G. For every I 2 1 kg let p1 S gi1 gil 1 ii ii kg and P S Uk 1 Pi S . Let S be a sequence in G we call S a zero-sum sequence if ơ S 0 a zero-sum free sequence if for any subsequence W of S ơ W 0. In inverse zero-sum problems for example when we study the structure of the