Erdős-Burgess constant of commutative semigroups

Abstract

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