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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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.

References
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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).

  • Dedicated: This paper is dedicated to the memory of our friend and colleague Jim Baumgartner
  • © 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