Separability of M-Equivalent Words by Morphisms

Two words are M-equivalent iff they are indistinguishable by Parikh matrices. Even for the ternary alphabet, an incontestable characterization of the M-equivalence relation is long overdue, ever since the introduction of Parikh matrices by Mateescu et al. in 2001. Recent works by Atanasiu attempted to distinguish M-equivalent words by the Parikh matrices of their images under some morphism. This paper addresses various aspects of this approach. In particular, it is shown that no morphism is capable of completely separating M-equivalent words over a given alphabet. However, if the class of words is restricted in length, then such morphism exists, whose codomain is connected to the notion of t-spectrum.