# nLab locally confluent

Let $(X,\rightarrow)$ be an abstract rewriting system ie. a set equipped with a binary relation $\rightarrow$. Write $\rightarrow^{*}$ for the reflexive-transitive closure of $\rightarrow$. We say that $\rightarrow$ is locally confluent iff for every $a,b,c \in X$ such that $a \rightarrow b$ and $a \rightarrow c$, there exists $d \in X$ such that $b \rightarrow^{*} d$ and $c \rightarrow^{*} d$.

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