A generalization of Kruyswijk-Olson theorem on Davenport constant in commutative semigroups

AIMS Mathematics

Guoqing Wang¹ 

[1] Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P. R. China;

关键词: kruyswijk-olson theorem;    davenport constant;    zero-sum;    erdős-burgess constant;    cyclic semigroups;    combinatorial number theory;   

DOI  :  10.3934/math.2020193

来源: DOAJ

【 摘 要 】

Let $\mathcal{S}$ be a finite commutative semigroup written additively. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The Erdős-Burgess constant of the semigroup $\mathcal{S}$ is defined as the smallest positive integer $\ell$ such that any $\mathcal{S}$-valued sequence $T$ of length $\ell$ must contain one or more terms with the sum being an idempotent of $\mathcal{S}$. If the semigroup $\mathcal{S}$ is a finite abelian group, the Erdős-Burgess constant reduces to the well-known Davenport constant in Combinatorial Number Theory. In this paper, we determine the value of the Erdős-Burgess constant for a direct sum of two finite cyclic semigroups in some cases, which generalizes the classical Kruyswijk-Olson Theorem on Davenport constant of finite abelian groups in the setting of commutative semigroups.

【 授权许可】

Unknown