Erdős-Burgess constant of commutative semigroups

Abstract

Let $\mathcal{S}$ be a nonempty commutative semigroup written additively. An element e of $\mathcal{S}$ is said to be idempotent if e + e = e. The Erdős-Burgess constant $\mathrm{I}(\mathcal{S})$ of the semigroup $\mathcal{S}$ is defined as the smallest positive integer $\mathcal{\ell }$ such that any $\mathcal{S}$-valued sequence T of length $\mathcal{\ell }$ contains a nonempty subsequence the sum of whose terms is an idempotent of $\mathcal{S}.$ We make a study of $\mathrm{I}(\mathcal{S})$ when $\mathcal{S}$ is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that $\mathrm{I}(\mathcal{S})$ is finite, and we obtain sharp bounds of $\mathrm{I}(\mathcal{S})$ in case $\mathrm{I}(\mathcal{S})$ is finite, and determine the precise value of $\mathrm{I}(\mathcal{S})$ in some cases which unifies some well known results on the precise value of Davenport constant in the setting of commutative semigroups.

Keywords:

• Cyclic semigroups
• Davenport constant
• Erdős-Burgess constant
• zero-sum

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M14
• 11B75

Acknowledgements

The author is grateful to the anonymous reviewer for many helpful suggestions, which have led to substantial improvements in the presentation and arguments of the paper.

Additional information

Funding

This work is supported by NSFC (grant no. 11971347).