Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 2958935
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. U. Abraham and R. Bonnet, Hausdorff's theorem for posets that satisfy the finite antichain property, Fundamenta Mathematicae 159 (1999), 51-69. MR 1669718 (2000a:06003)
  • 2. J. Baumgartner, L. Harrington, and E. Kleinberg, Adding a closed unbounded set, Journal of Symbolic Logic 41 (1976), 481-482. MR 0434818 (55:7782)
  • 3. R. Bonnet and M. Pouzet, Linear extensions of ordered sets. Ordered sets (Banff, Alberta, 1981), 125-170, NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, 83. Reidel, Dordrecht, 1982. MR 661293 (83h:06004)
  • 4. M. Džamonja and K. Thompson, A poset hierarchy, Central European Journal of Mathematics 4 (2006), 225-241. MR 2221106 (2007j:03059)
  • 5. R. Fraïssé, Theory of relations, Studies in Logic and the Foundations of Mathematics, 145. North-Holland, Amsterdam, 2000. MR 1808172 (2002d:03084)
  • 6. F. Hausdorff, Grundzüge einer Theorie der geordneten Mengen, Mathematische Annalen 65 (1908), 435-505. MR 1511478
  • 7. R. Laver, On Fraïssé's order type conjecture. Annals of Mathematics 93 (1971), 89-111. MR 0279005 (43:4731)
  • 8. W. Sierpiński, Sur une propriété des ensembles ordonnés, Fundamenta Mathematicae 36 (1949), 56-67. MR 0031528 (11:165b)
  • 9. W. Sierpiński, Sur un problème de la théorie des relations, Annali della Scuola Normale Superiore di Pisa (Classe di Scienze) 2 (1933), 285-287. MR 1556708
  • 10. E. Szpilrajn, Sur l'extension de l'ordre partiel, Fundamenta Mathematicae 16 (1930), 386-389.
  • 11. S. Todorčević, Very strongly rigid Boolean algebras, Publication de l'Institut Mathématique (Beograd) 27(41) 1980, 267-277. MR 621959 (82k:06017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 06A07, 06A05, 06A06

Retrieve articles in all journals with MSC (2010): 06A07, 06A05, 06A06

Additional Information

Uri Abraham
Affiliation: Department of Mathematics, Ben-Gurion University, Beer-Sheva, 84105 Israel

Robert Bonnet
Affiliation: Laboratoire de mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac CEDEX, France

James Cummings
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvannia 15213

Mirna Džamonja
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, United Kingdom

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

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.

American Mathematical Society