[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84633-en":3,"doc-seo-84633-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84633,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","An overlap-free morphism is a k-power-free morphism for any integer k ≥ 3","Overlap-free morphisms and k-power-free morphisms are studied for any integer k ≥ 3. The text introduces foundational notions from combinatorics on words: alphabets, words and lengths, factors, prefixes and suffixes, mirror images, conjugacy, and overlaps. It further defines pure overlaps, k-powers and pure k-powers, links these properties via factor and conjugation arguments, and establishes supporting remarks. Key structural tools include a standard decomposition proposition for word equations, Lemma 1.8 on internal factors, and the Fine–Wilf theorem with Keränen’s corollary.","arXiv :2607 .0 1837v 1 [ cs .FL] 2 Jul 2026  \nAn overlap-free morphism is a k-power-free morphism  \nfor any integer k ≥ 3  \nFrancis Wlazinski  \nJuly 3, 2026  \nAbstract  \nIt’s all in the title.  \n1 Introduction and preliminaries  \nLet us recall some basic notions of Combinatorics of words we will use in this paper.  \n1.1 Words  \nIn the following, A and B are alphabets, that is, finite sets of symbols called letters. Since an alphabet with one element is of limited interest to us, we always assume that the cardinality of alphabets is at least two.  \nA word is an element in A∗ , the free monoid generated by A, whose identity element is the empty word, denoted ε, and whose composition law, usually unnoted, is simply the juxtaposition of symbols. We denote by A+ the set of non-empty words, that is, A+ = A∗ \\ {ε} . We also speak of product of words, just as we speak of juxtaposition of words. We will not always specify the alphabet used, as this will often be irrelevant.  \nGiven a non-empty word u = a 1 ... an , with ai ∈ A for every integer i from 1 to n, the length of u denoted by |u| is the integer n, that is, the number of letters of u. By convention, we have |ε| = 0 . The mirror image of u, denoted by u˜, is the word an . . . a2 a 1 .  \nA word u is a factor of a word v if there exist two (possibly empty) words p and s such that v = pus. We denote by Fcts(v) the set of all factors of v. If u ∈ Fcts(v), we also say that v contains the word u (as a factor) . If p = ε , u is a prefix of v. If s = ε , u is a suffix of v. If u  v , u is a proper factor of v. If u, p, and s are non-empty words, u is an internal factor of v.  \nTwo non-empty words u and v are conjugated if u = t 1t2 and v = t2t 1 for two (possibly empty) words t 1 and t2 . If t 1  ε and t2  ε, we say that v is a proper conjugated word of u.  \nLet w be a non-empty word and let i,j be two integers such that 0 ≤ i − 1 ≤ j ≤ |w| . We denote by w [i..j] the factor of w such that |w[i..j]| = j − i + 1 and w = pw [i..j]s for two words s and p satisfying |p| = i − 1. Note that, when j = i−1, we have w[i..j] = ε . When  \ni = j, we also denote by w [i] the factor w [i..i], which is the ith letter of w. In particular, w[1] and w [|w|] are respectively the first and the last letter of w.  \nAn overlap is a word of the form xuxux where x ∈ A and u ∈ A∗ . Note that an equivalent definition is obtained by taking x ∈ A+ . An overlap-free word is a word in which none of the factors are an overlap.  \nAn overlap is said to be pure if all its proper factors are overlap-free. Let us remark that, if avava is a pure overlap, then a ∈ A.  \nRepeating the reasoning that an overlap is either pure or contains an overlap, we obtain that a word that contains an overlap also contains a pure overlap (as a factor of the first) . Powers of a word are defined inductively by u0 = ε, and for every integer n ≥ 1, un = uun−1 . Given an integer k ≥ 2, since the case εk is of little interest, we call a k-power any word uk with u  ε . A 2-power (resp. a 3-power) is also called a square (resp. a cube) . Given an integer k ≥ 2, a word is k-power-free if it does not contain any k-power as factor. A primitive word is a word that is not a k-power of another word whatever the integer k ≥ 2. A (non-empty) k-power v k is called pure if any proper factor of v k is k-power-free. In particular, we say that v k is a pure k-power of a word w if v k ∈ Fcts(w) and v k is pure. As for pure overlap, repeating the fact that a non-pure k-power contains a k-power, which is itself pure or not, we obtain that any k-power contains a pure k-power. Moreover, if v kis a pure k-power then v is primitive but the converse does not hold.  \nRemark 1.1 Every conjugated word of a primitive word is primitive.  \nRemark 1.2 A word u is an overlap (resp. a pure-overlap, a k-power or a pure k-power) if and only if u˜ is an overlap (resp. a pure-overlap, a k-power or a pure k-power) .  \nRemark 1.3 For any non-empty word u and any integer k ≥ 3, a fac","cbCaimRvjklldE0n","https://ap.wps.com/l/cbCaimRvjklldE0n","pdf",598952,1,17,"English","en",105,"# Abstract\n# Introduction and preliminaries\n## Words and basic operations\n## Overlaps and pure overlaps\n## Powers and k-power-free notions\n## Word equations and auxiliary results","[{\"question\":\"What is an overlap-free word in this work?\",\"answer\":\"An overlap-free word is one that has no factors of the form xuxux, where x is a letter (equivalently in A+) and u is any word.\"},{\"question\":\"How are k-powers and pure k-powers defined?\",\"answer\":\"A k-power is a word u^k (with u non-empty). A pure k-power is a k-power whose every proper factor is k-power-free.\"},{\"question\":\"Which results are cited as key tools in the preliminaries?\",\"answer\":\"The text uses a proposition giving solutions to two word equations, Lemma 1.8 about internal factors of vv, and the Fine–Wilf theorem (with Keränen’s corollary) about powers sharing long common structure.\"}]",1784197341,43,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"an-overlap-free-morphism-is-a-k-power-free-morphism-for-any-integer-k-3","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/an-overlap-free-morphism-is-a-k-power-free-morphism-for-any-integer-k-3/84633/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is an overlap-free word in this work?","Question",{"text":75,"@type":76},"An overlap-free word is one that has no factors of the form xuxux, where x is a letter (equivalently in A+) and u is any word.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are k-powers and pure k-powers defined?",{"text":80,"@type":76},"A k-power is a word u^k (with u non-empty). A pure k-power is a k-power whose every proper factor is k-power-free.",{"name":82,"@type":73,"acceptedAnswer":83},"Which results are cited as key tools in the preliminaries?",{"text":84,"@type":76},"The text uses a proposition giving solutions to two word equations, Lemma 1.8 about internal factors of vv, and the Fine–Wilf theorem (with Keränen’s corollary) about powers sharing long common structure.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]