On a conjecture about Parikh matrices

Based on Salomaa's characterization of M-equivalence, Atanasiu conjectured that a certain natural generalization of ME-equivalence solves the injectivity problem of Parikh matrices for the ternary alphabet. This paper refutes his conjecture but continues to study the interesting proposed Thue system. Characterization of certain irreducible elementary transformations under this system is obtained. Furthermore, these transformations are further scrutinized in terms of their replaceability by simpler ones.