Globalization of partial monoid actions via abstract rewriting systems

TL;DR

This paper investigates conditions under which partial monoid actions can be extended to global actions using rewriting systems, providing explicit criteria and applications to various semigroup actions.

Contribution

It establishes that local confluence ensures globalizability of partial monoid actions and offers explicit criteria for specific monoids, expanding understanding of partial action globalization.

Findings

Local confluence suffices for globalizability

Explicit criteria for monoid G^0 actions

Applications to semigroup and algebra actions

Abstract

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.