We say that a partially ordered set $(P,<)$ is *self-additive* if the two natural embeddings of $P$ in $P\oplus P$ (the linear sum of $P$ and itself) are elementary.

We have the following.

Suppose $(L,<)$ is a self-additive

linearorder. Then for any linear order $X$ and any nonempty $Y\subseteq X$, the natural embedding of $\sum_{x\in Y} L$ in $\sum_{x\in X} L$ is elementary.

My question is:

Does this hold if we replace $L$ with a

partialorder (but keep $X$ linear)?

This is Theorem 13.93 in Rosenstein's book on linear orderings. The key ingredient of the proof is the Corollary 13.39, which says that if $L$ is a linear order and $K\subseteq L$ is convex, then for any $U\subseteq L^n$ definable with parameters in $L\setminus K$, the set $U\cap K^n$ is definable in $(K,<)$ without parameters.

The proof of 13.39 is quite hard to understand without digging through the notation introduced before, but it seems to be related to binarity of linear orders, which is (I believe) not true for arbitrary posets. Thus, I suspect that the analogue of 13.39 is not true (well, maybe not the naive analogue). Still, the analogue of 13.93 seems plausible.