Let I $I$ be a poset and for any i $i$ let Pi $P_i$ be a poset. Let P $P$ be the sum over I $I$ of the sets Pi, $P_i$, and let
<P$<_P$ be the relation defined on P $P$ by $q<Pr<_Pr$ iff q $q$ and r $r$ are members of the same Pi $P_i$ and $q<ir<_ir$ or q $q$ is member of Pj r $P_j$ $r$ is member of Pk $P_k$ and $j<ik. <_ik$. Maybe that formally, I should have taken Qi $Q_i$ as the cartesian product of the singleton of i $i$ with Pi, $P_i$, and Q $Q$ as the union of the Qi's, $Q_i$'s, and compared couples as (i,q) $(i,q)$ and (j,r). $(j,r)$. Anyway, it is clear that P, $P$, equipped with our relation is a poset, that is the poset sum over I $I$ of the posets Pi. $P_i$. Particularly, in the case that I $I$ and Pi $P_i$ all are totally ordered sets , P $P$ is a totally ordered set. And, in this particular case, a theorem of Schoenfliess Schoenflies asserts that "every (totally) ordered set is the union of scattered sets over a densely ordrerd ordered indexing setset";Qestion
Question: does there exist a similar decomposition theorem in the general case of posets ?