Structure of the largest idempotent-product free sequences in semigroups

Abstract

Let S be a finite semigroup, and let E(S) be the set of all idempotents of S. Gillam, Hall and Williams proved in 1972 that every S-valued sequence T of length at least |S|−|E(S)|+1 is not (strongly) idempotent-product free, in the sense that it contains a nonempty subsequence the product of whose terms, in the order induced from the sequence T, is an idempotent, which affirmed a question of Erdős. They also showed that the value |S|−|E(S)|+1 is best possible.Here, motivated by Gillam, Hall and Williams' work, we determine the structure of the idempotent-product free sequences of length |S∖E(S)| when the semigroup S (not necessarily finite) is such that |S∖E(S)| is finite, and we introduce a couple of structural constants for semigroups that reduce to the classical Davenport constant in the case of finite abelian groups.