Well quasi-orders generated by a word-shuffle rewriting

Given a set I of words, the set L"@?"""I^@e of all words obtained by the shuffle of (copies of) words of I is naturally provided with a partial order: for u,v in L"@?"""I^@e, u@?"I^*v if and only if v is the shuffle of u and another word of L"@?"""I^@e. In [F. D'Alessandro, S. Varricchio, Well quasi-orders, unavoidable sets and derivation systems, in: Word Avoidability Complexity and Morphisms (WACAM), RAIRO Theoretical Informatics and Applications 40 (3) (2006) 407-426 (special issue)], the authors have opened the problem of the characterization of the finite sets I such that @?"I^* is a well quasi-order on L"@?"""I^@e. In this paper we give an answer in the case when I consists of a single word w.

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