A scattering of orders

Uri Abraham, Robert Bonnet, James Cummings, Mirna Džamonja, Katherine Thompson
2012 Transactions of the American Mathematical Society  
A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class B of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in B. More generally, we say that a partial ordering is κ-scattered if it does not contain a copy of any κ-dense linear ordering. We prove analogues of Hausdorff's result for κ-scattered
more » ... for κ-scattered linear orderings, and for κ-scattered partial orderings satisfying the finite antichain condition. We also study the Q κ -scattered partial orderings, where Q κ is the saturated linear ordering of cardinality κ, and a partial ordering is Q κ -scattered when it embeds no copy of Q κ . We classify the Q κ -scattered partial orderings with the finite antichain condition relative to the Q κ -scattered linear orderings. We show that in general the property of being a Q κ -scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions.
doi:10.1090/s0002-9947-2012-05466-3 fatcat:nrovdabbobefzbfab4ou35ifi4