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.
Categories: math.CO math.AC math.NT