Six kinds of both of primitivity and periodicity of words, introduced by Ito and Lischke [M. Ito and G. Lischke, Math. Log. Quart. 53 (2007) 91-106; Corrigendum in Math. Log. Quart. 53 (2007) 642-643], give rise to defining six kinds of roots of a nonempty word. For 1 ≤ k ≤ 6, a k-root word is a word which has exactly k different roots, and a k-cluster is a set of k-root words u where the roots of u fulfil a given prefix relationship. We show that out of the 89 different clusters that can be considered at all, in fact only 30 exist, and we give their quasi-lexicographically smallest elements. Also we give a sufficient condition for words to belong to the only existing 6-cluster. These words are also called Lohmann words. Further we show that, with the exception of a single cluster, each of the existing clusters contains either only periodic words, or only primitive words.
Mots-clés : periodicity of words, primitivity of words, roots of words, classification of words
@article{ITA_2014__48_3_267_0, author = {Lischke, Gerhard}, title = {Root clustering of words}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {267--280}, publisher = {EDP-Sciences}, volume = {48}, number = {3}, year = {2014}, doi = {10.1051/ita/2014009}, mrnumber = {3302488}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ita/2014009/} }
TY - JOUR AU - Lischke, Gerhard TI - Root clustering of words JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2014 SP - 267 EP - 280 VL - 48 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ita/2014009/ DO - 10.1051/ita/2014009 LA - en ID - ITA_2014__48_3_267_0 ER -
%0 Journal Article %A Lischke, Gerhard %T Root clustering of words %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2014 %P 267-280 %V 48 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ita/2014009/ %R 10.1051/ita/2014009 %G en %F ITA_2014__48_3_267_0
Lischke, Gerhard. Root clustering of words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014) no. 3, pp. 267-280. doi : 10.1051/ita/2014009. http://archive.numdam.org/articles/10.1051/ita/2014009/
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