On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$
Let $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$,...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-11-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/10.3934/math.2021066/fulltext.html |
| _version_ | 1818694746548207616 |
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| author | Guixin Deng |
| author_facet | Guixin Deng |
| author_sort | Guixin Deng |
| collection | DOAJ |
| description | Let $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$, denoted by $\textsf{D}(X)$, is defined to be the supremum of lengths of all minimal zero-sum sequences over $X$. In this paper, we study the minimal zero-sum sequences over $X=\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]\subset\mathbb{Z}^2$. We completely determine the structure of minimal zero-sum sequences of maximal length over $X$ and obtain that $\textsf{D}(X)=2(n+m).$ |
| first_indexed | 2024-12-17T13:34:29Z |
| format | Article |
| id | doaj.art-2afa0fca90be4f86a2ad84d93ad24793 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-17T13:34:29Z |
| publishDate | 2021-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-2afa0fca90be4f86a2ad84d93ad247932022-12-21T21:46:28ZengAIMS PressAIMS Mathematics2473-69882021-11-01621101110910.3934/math.2021066On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$Guixin Deng0Shuxin WangLet $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$, denoted by $\textsf{D}(X)$, is defined to be the supremum of lengths of all minimal zero-sum sequences over $X$. In this paper, we study the minimal zero-sum sequences over $X=\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]\subset\mathbb{Z}^2$. We completely determine the structure of minimal zero-sum sequences of maximal length over $X$ and obtain that $\textsf{D}(X)=2(n+m).$https://www.aimspress.com/article/10.3934/math.2021066/fulltext.htmlzero-summinimal zero-sumdavenport constant |
| spellingShingle | Guixin Deng On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$ AIMS Mathematics zero-sum minimal zero-sum davenport constant |
| title | On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$ |
| title_full | On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$ |
| title_fullStr | On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$ |
| title_full_unstemmed | On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$ |
| title_short | On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]$ |
| title_sort | on the davenport constant of a two dimensional box left kern 0 15em left 1 1 right kern 0 15em right times left kern 0 15em left m n right kern 0 15em right |
| topic | zero-sum minimal zero-sum davenport constant |
| url | https://www.aimspress.com/article/10.3934/math.2021066/fulltext.html |
| work_keys_str_mv | AT guixindeng onthedavenportconstantofatwodimensionalboxleftkern015emleft11rightkern015emrighttimesleftkern015emleftmnrightkern015emright |