poset sum of posets - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T12:33:09Z http://mathoverflow.net/feeds/question/69128 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69128/poset-sum-of-posets poset sum of posets Gérard Lang 2011-06-29T16:19:48Z 2011-06-29T18:40:59Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/69127/ordered-sum-of-posets" rel="nofollow">Ordered sum of posets</a> </p> </blockquote> <p>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</p>