Erd\H{o}s-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 Erd\H{o}s-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 $\ell$ contain a nonempty subsequence the sum of whose terms is an idempotent of $\mathcal{S}$. We make a study of ${\rm I}(\mathcal{S})$ when $\mathcal{S}$ is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions such that ${\rm I}(\mathcal{S})$ is finite, and in particular, we obtain sharp bounds of ${\rm I}(\mathcal{S})$ in case ${\rm I}(\mathcal{S})$ is finite, and determine the precise values of ${\rm I}(\mathcal{S})$ in some cases which unifies some well known results on the precise values of Davenport constant in the setting of commutative semigroups.