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This paper further studies the notions of [Formula: see text]-equivalence and strong [Formula: see text]-equivalence. Certain structural properties of [Formula: see text]-equivalent ternary words are presented and then employed to (partially) characterize pairs of ternary words that are ME-equivalent (i.e. obtainable from one another by certain elementary transformations). Finally, a sound rewriting system in determining strong [Formula: see text]-equivalence is obtained for the ternary alphabet. <\/jats:p>","DOI":"10.1142\/s0129054118500065","type":"journal-article","created":{"date-parts":[[2018,3,19]],"date-time":"2018-03-19T22:20:23Z","timestamp":1521498023000},"page":"123-137","source":"Crossref","is-referenced-by-count":9,"title":["On M-Equivalence and Strong M-Equivalence for Parikh Matrices"],"prefix":"10.1142","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0808-0028","authenticated-orcid":false,"given":"Ghajendran","family":"Poovanandran","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Malaysia"}]},{"given":"Wen Chean","family":"Teh","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Malaysia"}]}],"member":"219","published-online":{"date-parts":[[2018,3,19]]},"reference":[{"key":"S0129054118500065BIB001","doi-asserted-by":"publisher","DOI":"10.1142\/S0129054107004735"},{"key":"S0129054118500065BIB002","first-page":"1","author":"Atanasiu A.","year":"2014","journal-title":"Discrete Mathematics and Computer Science. 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