Vorträge im Schwerpunkt Reelle Geometrie und Algebra
Donnerstag, 16. Dezember 2010, um 17:00 Uhr
in F426
(Schwerpunktskolloquium, Kuhlmann)
Prof. Mirna Dzamonja (University
of East Anglia, Norwich, UK)
A scattering of orders
Abstract:
I will present results from a joint paper with U. Abraham, R. Bonnet,
J. Cummings and K. Thompson. We say that a linear ordering is scattered
if it does not contain a copy of the rationals. Hausdorff characterized
the class of scattered linear orderings as the least family of linear
orderings that includes the well-orderings, and is closed under
reversals and well-ordered lexicographic sums. More generally, 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 linear orderings, and for κ-scattered partial orderings
satisfying the finite antichain condition.
We also study the Qκ-scattered partial orderings, where Qκ
is the saturated linear ordering of cardinality κ, and a
partial ordering is Qκ-scattered when it embeds no copy of Qκ. We classify the Qκ-scattered partial orderings with the finite antichain condition relative to the Qκ-scattered linear orderings. We show that in general the property of being a Qκ-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.
zuletzt
geändert am 29. November 2010
