Series: Penn State Logic Seminar Date: Tuesday, November 11, 2003 Time: 2:30 - 3:45 PM Place: 324 Sackett Building (note unusual location) Speaker: Katherine Thompson, Carnegie Mellon University, Mathematics Title: kappa-Scattered and kappa-Dense Orders Abstract: The results in this talk extend a paper of Abraham and Bonnet which generalized the famous Hausdorff characterization the class of scattered linear orders by giving a poset hierarchy that characterizes the class of FAC scattered posets. Building on the methods of Abraham and Bonnet we define a larger poset hierarchy than theirs, to include a broader class of "scattered" posets that we call kappa-scattered. These posets cannot embed the unique linear order such that for every two subsets of size < kappa, one being strictly less than the other, there is an element in between. We call this set Q(kappa) or a strongly kappa-dense set. Such a set does not exist for all kappa, but it can be shown to exist for kappa satisfying kappa^<kappa = kappa. We prove that our hierarchy includes all kappa-scattered FAC posets and is included in the class of all FAC posets that do not embed a weakly kappa-dense subset. For kappa = aleph_0 this gives the Abraham-Bonnet theorem.