Possible Duplicate:
Ordered sum of posets
Let I,RI be a poset and for any i let Pi,Ri be a poset. Let P be the sum set of the Pi's and let R be the relation on P defined by qRr iff there is i such that q and r are members of Pi and qRi r, or q is member of Pj, r is member of Pk and jRIk. It is clear that P equipped with R is a poset. And, in the particular case that I,RI and Pi,Ri are totally ordered sets, so is also the case for P,R. Moreover, a theorem of Schoenfliess asserts that "Every ordered set is the union of scattered sets over a densely ordered indexing set." Question: Does there exist a corresponding decomposition theorem in the case of general posets ? Gérard Lang
