Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ordering make $A^*$ well-quasi-ordered as well (reference to Corollary $1.7$ of [2]).

References:

https://research-repository.st-andrews.ac.uk/bitstream/10023/7963/1/BCCpaperv9.pdf

- http://arxiv.org/pdf/1107.5070v2.pdf

Crossposted on MSE.