A scattering of orders
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- by Uri Abraham, Robert Bonnet, James Cummings, Mirna Džamonja and Katherine Thompson PDF
- Trans. Amer. Math. Soc. 364 (2012), 6259-6278 Request permission
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
- MR Author ID: 289375
- ORCID: 0000-0002-7913-0427
- Email: jcumming@andrew.cmu.edu
- Mirna Džamonja
- Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom
- ORCID: setImmediate$0.3709267400444315$1
- 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_nought@yahoo.com
- 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). - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 6259-6278
- MSC (2010): Primary 06A07; Secondary 06A05, 06A06
- DOI: https://doi.org/10.1090/S0002-9947-2012-05466-3
- MathSciNet review: 2958935
Dedicated: This paper is dedicated to the memory of our friend and colleague Jim Baumgartner