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:


1. <http://www.math.harvard.edu/~lurie/155notes/lecture19.pdf>
2. <https://research-repository.st-andrews.ac.uk/bitstream/10023/7963/1/BCCpaperv9.pdf>
3. <http://arxiv.org/pdf/1107.5070v2.pdf>




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Crossposted on [MSE](https://math.stackexchange.com/questions/1639834/order-preserving-bijection-fa-to-a).



Order-Perserving Bijection $f:A\to A^*$?
Unless I have misunderstood, the answer seems to be no. 


Let $A=\omega+1$ with the usual order, or indeed, any well-quasi-ordering with an object having infinitely many predecessors. Such an object would get mapped to an element of $A^\*$, which we can think of as a finite term over $A$, and every such term has only finitely many subwords. So the map from $A$ to $A^\*$ cannot be both order-preserving and bijective. 


