On Monoids of plus-minus weighted Zero-Sum Sequences: The Isomorphism Problem and the Characterization Problem

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 first characterize (in terms of $G$) when the monoid ${\mathcal{B}}_{\pm }(G)$ of plus-minus weighted zero-sum sequences is Mori resp. Krull resp. finitely generated. After that we study the Isomorphism and the Characterization Problem for monoids of plus-minus weighted zero-sum sequences.