# 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.

## Metaproperties

### Dependence only on reduction graph

This property of a rewriting system depends only on the reduction graph associated with the rewriting system, viz it can be reduces to testing the reduction graph for a directed graph property

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.

### Free product-closedness

This property of a rewriting system is free product-closed. In other words, if we have two rewriting systems satisfying the property, the natural free product of these rewriting systems also satisfies the property

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.