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

Guoqing Wang; Guoqing Wang

doi:10.3934/math.2020193

AIMS Mathematics

2020, Volume 5, Issue 4: 2992-3001. doi: 10.3934/math.2020193

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Research article

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

Guoqing Wang ^,

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P. R. China

Received: 22 January 2020 Accepted: 13 March 2020 Published: 20 March 2020
MSC : 11B75, 11A05

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.
- Kruyswijk-Olson theorem,
- Davenport constant,
- zero-sum,
- Erdős-Burgess constant,
- cyclic semigroups,
- combinatorial number theory
Citation: Guoqing Wang. A generalization of Kruyswijk-Olson theorem on Davenport constant in commutative semigroups[J]. AIMS Mathematics, 2020, 5(4): 2992-3001. doi: 10.3934/math.2020193

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Abstract

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.

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