  Parikh matrices have been extensively investigated due to their usefulness in
studying subword occurrences in words. Due to the dependency of Parikh matrices
on the ordering of the alphabet, strong M-equivalence was proposed as an
order-independent alternative to M-equivalence in studying words possessing the
same Parikh matrix. This paper introduces and studies the notions of strong 2.t
and strong 3.t transformations in determining when two ternary words are
strongly M-equivalent. The irreducibility of strong 2.t transformations are
then scrutinized, exemplified by a structural characterization of irreducible
strong 2.2 transformations. The common limitation of these transformations in
characterizing strong M-equivalence is then addressed.
