# Erd\H{o}s-Burgess constant of the direct product of cyclic semigroups.

Let $\mathcal{S}$ be a nonempty semigroup endowed with a binary associative operation $*$. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e*e=e$. Originated by one question of P. Erd\H{o}s to D.A. Burgess: {\sl `If $\mathcal{S}$ is a finite semigroup of order $n$, does any $\mathcal{S}$-valued sequence $T$ of length $n$ contain a nonempty subsequence the product of whose terms, in some order, is idempotent?'}, we make a study of the associated invariant, denoted ${\rm I}(\mathcal{S})$ and called Erd\H{o}s-Burgess constant which is the smallest positive integer $\ell$ such that any $\mathcal{S}$-valued sequence $T$ of length $\ell$ contain a nonempty subsequence the product of whose terms, in any order, is an idempotent, when $\mathcal{S}$ is a direct product of arbitrarily many of cyclic semigroups. We give the necessary and sufficient conditions of ${\rm I}(\mathcal{S})$ being finite, and in particular, we give sharp lower and upper bounds of ${\rm I}(\mathcal{S})$ in case ${\rm I}(\mathcal{S})$ is finite. Moreover, we determine the precise values of ${\rm I}(\mathcal{S})$ in some cases. Also, we characterize the structure of long sequences which contains no nonempty subsequence with that property in the case when $\mathcal{S}$ is a cyclic semigroup.

Publisher URL: http://arxiv.org/abs/1802.08791

DOI: arXiv:1802.08791v2

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