  Parikh matrices have been a powerful tool in arithmetizing words by numerical
quantities. However, the dependence on the ordering of the alphabet is
inherited by Parikh matrices. Strong M-equivalence is proposed as a canonical
alternative to M-equivalence to get rid of this undesirable property. Some
characterization of strong M-equivalence for a restricted class of words is
obtained. Finally, the existential counterpart of strong M-equivalence is
introduced as well.
