Structure of long idempotent-sum-free sequences over finite cyclic semigroups

Abstract

Let [Formula: see text] be a finite cyclic semigroup written additively. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. A sequence [Formula: see text] over [Formula: see text] is called idempotent-sum-free provided that no idempotent of [Formula: see text] can be represented as a sum of one or more terms from [Formula: see text]. We prove that an idempotent-sum-free sequence over [Formula: see text] of length over approximately a half of the size of [Formula: see text] is well structured. This result generalizes the Savchev–Chen Structure Theorem for zero-sum-free sequences over finite cyclic groups.