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

Let $G$ be an additive abelian group. A sequence $S=g_1\cdot \dots \cdot g_{\mathrm{\ell }}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are ${\epsilon }_1,\dots ,{\epsilon }_{\mathrm{\ell }}\in \{-1,1\}$ such that ${\epsilon }_1{g}_1+\dots +{\epsilon }_{\mathrm{\ell }}{g}_{\mathrm{\ell }}=0$. 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 $S\in {\mathcal{B}}_{\pm }(G)$.