Globalization of partial monoid actions via abstract rewriting systems

We study the globalization problem for a strong partial action $\alpha$ of a monoid $M$ on a semigroup $X$ via the associated rewriting system $(X_M^{+},\to )$. We show that the local confluence of $(X_M^{+},\to )$ is sufficient for the globalizability of $\alpha$ but, unlike the group case, it is not necessary. Focusing on the monoid $M=G^0$, where $G$ is a group, we obtain an explicit criterion for the globalizability of $\alpha$ and a criterion for the local confluence of $(X_M^{+},\to )$. Several applications to strong partial actions of the monoid $M=\{0,1\}$ on semigroups and algebras, as well as to strong partial actions of an arbitrary monoid $M$ on left zero and null semigroups, are presented.