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While the problem of identifying M-unambiguous words for a binary alphabet is solved using a palindromicly amicable relation, it is open for larger alphabets. We propose substitution rules that establish M-equivalence and solve the problem of M-unambiguity for a ternary alphabet. Our rules build on the principles of the palindromicly amicable relation and enable tracking of the differences of length-3 ordered subsequences. We characterize the set of M-unambiguous words and obtain a regular expression for the set. In addition, we examine the weak M-relation, a variant of M-equivalence, and present a solution for the generalization of weak M-relation regarding its degree.<\/jats:p>","DOI":"10.1142\/s0129054125410047","type":"journal-article","created":{"date-parts":[[2025,3,16]],"date-time":"2025-03-16T22:27:01Z","timestamp":1742164021000},"page":"73-92","source":"Crossref","is-referenced-by-count":0,"title":["Characterizations of\n                    <i>M<\/i>\n                    -Equivalence and Weak\n                    <i>M<\/i>\n                    -Relation"],"prefix":"10.1142","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-5890-4916","authenticated-orcid":false,"given":"Joonghyuk","family":"Hahn","sequence":"first","affiliation":[{"name":"Department of Computer Science, Yonsei University, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-8299-1405","authenticated-orcid":false,"given":"Hyunjoon","family":"Cheon","sequence":"additional","affiliation":[{"name":"College of Engineering, University of Seoul, Republic of Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7211-6657","authenticated-orcid":false,"given":"Yo-Sub","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Yonsei University, Republic of Korea"}]}],"member":"219","published-online":{"date-parts":[[2025,3,13]]},"reference":[{"key":"S0129054125410047BIB001","doi-asserted-by":"publisher","DOI":"10.1142\/S0129054107004735"},{"key":"S0129054125410047BIB002","first-page":"1","volume-title":"Discrete Mathematics and Computer Science","author":"Atanasiu A.","year":"2014"},{"key":"S0129054125410047BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2007.10.022"},{"key":"S0129054125410047BIB004","doi-asserted-by":"publisher","DOI":"10.3233\/FUN-2002-49401"},{"key":"S0129054125410047BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/s00236-018-0327-8"},{"key":"S0129054125410047BIB006","doi-asserted-by":"publisher","DOI":"10.1142\/S0129054109006498"},{"issue":"1","key":"S0129054125410047BIB007","first-page":"265","volume":"73","author":"\u015eerb\u0103nu\u0163\u0103 V. 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