Erd\H{o}s-Ginzburg-Ziv theorem for finite commutative semigroups

Abstract

Let $\mathcal{S}$ be a finite commutative semigroup written additively, and let $\exp(\mathcal{S})$ be its exponent which is defined as the least common multiple of all periods of the elements in $\mathcal{S}$. For every sequence $T$ of elements in $\mathcal{S}$ (repetition allowed), let $\sigma(T) \in \mathcal{S}$ denote the sum of all terms of $T$. Define the Davenport constant $D(\mathcal{S})$ of $\mathcal{S}$ to be the least positive integer $d$ such that every sequence $T$ over $\mathcal{S}$ of length at least $d$ contains a proper subsequence $T'$ with $\sigma(T')=\sigma(T)$, and define the Erd\H{o}s-Ginzburg-Ziv Theorem constant $E(\mathcal{S})$ to be the least positive integer $\ell$ such that every sequence $T$ over $\mathcal{S}$ of length at least $\ell$ contains a subsequence $T'$ with $|T|-|T'|=\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S})$ and $\sigma(T')=\sigma(T)$. When $\mathcal{S}$ is a finite abelian group, it is well known that $\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil\exp(\mathcal{S})=|\mathcal{S}|$ and $E(\mathcal{S})=D(\mathcal{S})+|\mathcal{S}|-1$. In this paper we investigate whether $E(\mathcal{S})\leq D(\mathcal{S})+\left\lceil\frac{|\mathcal{S}|}{\exp(\mathcal{S})}\right\rceil \exp(\mathcal{S})-1$ holds true for all finite commutative semigroups $\mathcal{S}$. We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.