  We classify the countable linear orders $X$ for which there is an order $A$
with at least two points such that the lexicographic product $AX$ is isomorphic
to $X$. Given such an $X$, we determine every corresponding order $A$, and
identify when $X$ is isomorphic to its square. More generally, we characterize
the countable orders that embed at least two disjoint convex copies of
themselves.
