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Six kinds of both of primitivity and periodicity of words, introduced by Ito and Lischke [M. Ito and G. Lischke, 53 (2007) 91–106; Corrigendum in 53 (2007) 642–643], give rise to defining six kinds of roots of a nonempty word. For 1 ≤ ≤ 6, a -root word is a word which has exactly different roots, and a -cluster is a set of -root words where the roots of 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...
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