Well Quasi Orders and the Shuffle Closure of Finite Sets

Given a set I of words, the set L$_{\rm \vdash}^{\epsilon}~_{\rm I}$ 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$_{\rm \vdash}^{\epsilon}~_{\rm I}$, u$\vdash^{\rm *}_{I}$v if and only if v is the shuffle of u and another word of L$_{\rm \vdash}^{\epsilon}~_{\rm I}$. In [3], the authors have stated the problem of the characterization of the finite sets I such that $\vdash_{I}^{\rm *}$ is a well quasi-order on L$_{\rm \vdash}^{\epsilon}~_{\rm I}$. In this paper we give the answer in the case when I consists of a single word w.