Shortlex-reducing rewriting system

Definition
A rewriting system is said to be shortlex-reducing if the following is true: there is a total ordering on the elements of the underlying alphabet such that for every rewrite, the right side is less than the left side, in the shortlex ordering on the set of words with respect to that total ordering.

Stronger properties

 * Length-reducing rewriting system

Weaker properties

 * Length-nonincreasing rewriting system
 * Acyclic rewriting system

Metaproperties
The rewriting system corresponding naturally to a free product of shortlex-reducing rewriting systems, is also shortlex-reducing. We can impose any total ordering on the new alphabet that, restricted to the alphabet of each factor, gives the total ordering on that factor.