If $(A, \preceq)$ is well quasi order, is $(A^\omega, \ll)$ well quasi order as well?

Question

Suppose $(A, \preceq)$ is well quasi order. $\ll$ is subsequence ordering on $A^\omega$ i.e. $u \ll v$ if $v$ has a subsequence $w$ such that $|u| = |w|$ and $u_i \preceq w_i$.

Then is it true that $(A^\omega, \ll)$ is well quasi order?

Answer 1

Not necessarily; according to this review by Arnie Miller, a counterexample can be found in R. Rado, Partial well-orderings of sets of vectors, Mathematika $\mathbf{1}$ $(1954)$, $89$-$95$. I do not at the moment have access to this paper, but I did finally manage to find enough information to construct the example or a very similar one.

Let $A=\{\langle m,n\rangle\in\omega\times\omega:m<n\}$, and for $\langle k,\ell\rangle,\langle m,n\rangle\in A$ define $\langle k,\ell\rangle\preceq\langle m,n\rangle$ if and only if either $k=m$ and $\ell\le n$, or $\ell<m$. If $\big\langle\langle m_k,n_k\rangle:k\in\omega\big\rangle$ is a sequence in $A$, then either $n_0<m_k$ for some $k\in\omega$, in which case $\langle m_0,n_0\rangle\preceq\langle m_k,n_k\rangle$, or $m_k\le n_0$ for all $k\in\omega$. It follows that there is an $m\le n_0$ such that $M=\{k\in\omega:m_k=m\}$ is infinite. Choose $k\in M$ so that $n_k$ is minimal, and let $\ell\in M$ be greater than $k$; then $\langle m_k,n_k\rangle\preceq\langle m_\ell,n_\ell\rangle$. Thus, $\langle A,\preceq\rangle$ is a wqo.

Now for $n\in\omega$ let

$$\sigma_n=\big\langle\langle n,n+k\rangle:k\in\Bbb Z^+\big\rangle\in A^\omega\;.$$

Suppose that $m,n\in\omega$ and $m<n$; clearly $\sigma_n\not\ll\sigma_m$. Moreover, $\langle m,m+k\rangle\not\preceq\langle n,n+\ell\rangle$ whenever $k\ge n-m$, so $\sigma_m\not\ll\sigma_n$. Thus, $\{\sigma_n:n\in\omega\}$ is an infinite antichain in $\langle A^\omega,\ll\rangle$, which is therefore not a wqo.

In a comment you asked about the case in which $\langle A,\preceq\rangle$ is a linear order. A linear wqo must be a well order, and I have seen an unreferenced assertion that $\langle A^\omega,\ll\rangle$ is a wqo if $\langle A,\preceq\rangle$ is a well order. However, I have not seen a reference, let alone a proof, and do not immediately see how to prove the assertion.