  Since the introduction of the Parikh matrix mapping, its injectivity problem
is on top of the list of open problems in this topic. In 2010 Salomaa provided
a solution for the ternary alphabet in terms of a Thue system with an
additional feature called counter. This paper proposes the notion of a Parikh
rewriting system as a generalization and systematization of Salomaa's result.
It will be shown that every Parikh rewriting system induces a Thue system
without counters that serves as a feasible solution to the injectivity problem.
