It is well known that for a given set $S$ with well-founded order relation $R$, the lexicographic order that extends $R$ on tuples of $S$ is also well-founded.
Also, the multiset order on the multiset extension of $S$ is also well-founded.
Are there any other extensions to sets besides the lexicographic extension and the multiset extension that have a natural ordering that preserve well-foundedness?
I seem to remember reading that lexicographic and multiset extensions are the only extensions that preserve well-foundedness, but I cannot find the reference.
I am only asking about well-foundedness, i.e., not assuming the orderings are necessarily total.
There are certainly other natural constructions than lexicographic orders that preserver well orders.
For example, we can compare $(a,b)$ with $(c,d)$ first by comparing $\max(a,b)$ with $\max(c,d)$ and fall back to the lexicographic ordering iff the maxima are the same. This has the nice property that if $S$ has order type $\omega$, then so does $S\times S$.