A scattering of orders

Abraham, Uri, Bonnet, Robert, Cummings, James, Džamonja, Mirna and Thompson, Katherine (2012) A scattering of orders. Transactions of the American Mathematical Society, 364. pp. 6259-6278. ISSN 0002-9947

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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.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Vishal Gautam
Date Deposited: 18 Mar 2011 10:19
Last Modified: 24 Jul 2019 13:00
URI: https://ueaeprints.uea.ac.uk/id/eprint/19975
DOI: 10.1090/S0002-9947-2012-05466-3

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