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


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.