Confluent rewriting system

Definition
A rewriting system is said to be confluent if it satisfies the following condition: whenever $$u \longrightarrow v$$ and $$u \longrightarrow w$$ are multi-step reductions in the rewriting system, then there exists a word $$z$$ such that there exist multi-step reductions $$v \longrightarrow z$$ and $$w \longrightarrow z$$.

In other words, any two things from the same source finally get together again.

The term confluent rewriting system can also be used for a rewriting system for a group. Note that the free group rewriting system is confluent. A group that possesses a confluent rewriting system is termed a confluent group.

Stronger properties

 * Weaker than::Strongly confluent rewriting system
 * Weaker than::Complete rewriting system

Weaker properties

 * Stronger than::Locally confluent rewriting system

Metaproperties
Whether a rewriting system is confluent or not, can be reduced to checking a property of the associated reduction graph )assuming we remove the identity rewrite.

A free product of confluent rewriting systems is confluent. This is essentially because reductions in the various free factors do not interfere with one another, and hence commute.