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

TL;DR

This paper characterizes the structure of long idempotent-sum free sequences over finite cyclic semigroups, extending classical results from cyclic groups to semigroups and revealing their well-organized nature.

Contribution

It generalizes the Savchev-Chen Structure Theorem from cyclic groups to finite cyclic semigroups for long idempotent-sum free sequences.

Findings

Long idempotent-sum free sequences are well-structured

The structure holds for sequences of length about half the size of the semigroup

Extends classical zero-sum theory to semigroups

Abstract

Let $\mathcal{S}$ be a finite cyclic semigroup written additively. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. A sequence $T$ over $\mathcal{S}$ is called {\sl idempotent-sum free} provided that no idempotent of $\mathcal{S}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum free sequence over $\mathcal{S}$ of length over approximately a half of the size of $\mathcal{S}$ is well-structured. This result generalizes the Savchev-Chen Structure Theorem for zero-sum free sequences over finite cyclic groups.