On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences

TL;DR

This paper investigates the structure of sets of lengths in monoids of plus-minus weighted zero-sum sequences over abelian groups, revealing highly structured sets for finite groups and arbitrary sets for infinite groups.

Contribution

It characterizes the sets of lengths in these monoids, showing finite groups have structured sets while infinite groups can realize any finite subset of integers.

Findings

Finite groups have highly structured sets of lengths.

Infinite groups can realize any finite subset of integers as a set of lengths.

The structure of sets of lengths depends on the finiteness of the group.

Abstract

Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that $\varepsilon_1 g_1 + \ldots + \varepsilon_{\ell} g_{\ell}=0$. We study sets of lengths in the monoid $\mathcal B_{\pm} (G)$ of plus-minus weighted zero-sum sequences over $G$. If $G$ is finite, then sets of lengths are highly structured. If $G$ is infinite, then every finite, nonempty subset of $\mathbb N_{\ge 2}$ is the set of lengths of some sequence $S \in \mathcal B_{\pm} (G)$.