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 $S=g_1\cdot\ldots\cdot g_{\ell}$ 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 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.