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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A scattering of orders


Authors: Uri Abraham, Robert Bonnet, James Cummings, Mirna Džamonja and Katherine Thompson
Journal: Trans. Amer. Math. Soc. 364 (2012), 6259-6278
MSC (2010): Primary 06A07; Secondary 06A05, 06A06
Published electronically: July 2, 2012
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Abstract: 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 $ \mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $ \mathcal B$.

More generally, we say that a partial ordering is $ \kappa $-scattered if it does not contain a copy of any $ \kappa $-dense linear ordering. We prove analogues of Hausdorff's result for $ \kappa $-scattered linear orderings, and for $ \kappa $-scattered partial orderings satisfying the finite antichain condition.

We also study the $ \mathbb{Q}_\kappa $-scattered partial orderings, where $ \mathbb{Q}_\kappa $ is the saturated linear ordering of cardinality $ \kappa $, and a partial ordering is $ \mathbb{Q}_\kappa $-scattered when it embeds no copy of $ \mathbb{Q}_\kappa $. We classify the $ \mathbb{Q}_\kappa $-scattered partial orderings with the finite antichain condition relative to the $ \mathbb{Q}_\kappa $-scattered linear orderings. We show that in general the property of being a $ \mathbb{Q}_\kappa $-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.


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Additional Information

Uri Abraham
Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva, 84105 Israel
Email: abraham@math.bgu.ac.il

Robert Bonnet
Affiliation: Laboratoire de mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac CEDEX, France
Email: bonnet@in2p3.fr

James Cummings
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvannia 15213
Email: jcumming@andrew.cmu.edu

Mirna Džamonja
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
Email: m.dzamonja@uea.ac.uk

Katherine Thompson
Affiliation: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8 - 10/104, A-1040 Wien, Austria
Email: aleph{\textunderscore}nought@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05466-3
PII: S 0002-9947(2012)05466-3
Keywords: Scattered posets, scattered chains, classification, well-quasi-orderings, better-quasi-orderings, finite antichain condition
Received by editor(s): June 6, 2010
Received by editor(s) in revised form: September 13, 2010
Published electronically: July 2, 2012
Additional Notes: The second author was supported by Exchange Grant 2856 from the European Science Foundation Research Networking Programme “New Frontiers of Infinity”, and by the Ben-Gurion University Center for Advanced Studies in Mathematics.
The third author was partially supported by NSF Grant DMS-0654046.
The fourth author was supported by EPSRC through the grant EP/G068720.
The fifth atuhor was supported by Lise-Meitner Project number M1076-N13 from the FWF (Austrian Science Fund).
Dedicated: This paper is dedicated to the memory of our friend and colleague Jim Baumgartner
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.