ALGEBRAIC CHARACTERIZATIONS FOR REDUCTION SYSTEMS

Abstract

We give algebraic characterizations for a reduction system to be respectively Noetherian and confluent, and for a Noetherian reduction system to be confluent. The characterization of a confluent reduction system $(A,\to )$ is based on a relationship between the confluence of the system and the exactness of the colimit functor colim : $A{b}^{\mathcal{A}}\to Ab$ where $\mathcal{A}$ is the small category with objects the elements of $A$ and arrows $a\to b$ whenever $b$ is a consequence of $a$ in the system.