Special rewriting system

Symbol-free definition
A rewriting system is said to be special if all the rewrites have the empty word on the right side.

Note that though the term is used in the general context for rewriting system of a monoid, it can also be viewed in the context of a rewriting system for a group. Since the group rewrites ($$xx^{-1} \to e$$) all have the empty word on the right side, the rewriting system for a group is special if and only if the rewriting system for the corresponding monoid is.

Weaker properties

 * Stronger than::monadic rewriting system
 * Stronger than::length-reducing rewriting system
 * Stronger than::linear-time terminating rewriting system
 * Stronger than::finitely terminating rewriting system

Metaproperties
The property of being a special rewriting system can be described in terms of a certain property (right side being trivial) being true for each rewrite.

A free product of special rewriting systems is a special rewriting system. This follows from the local testability and from other facts.