## 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 B of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in B. More generally, we say that a partial ordering is κ-scattered if it does not contain a copy of any κ-dense linear ordering. We prove analogues of Hausdorff's result for κ-scattered linear orderings, and for κ-scattered partial orderings satisfying the finite antichain condition. We also study the ℚκ-scattered partial orderings, where ℚκ is the saturated linear ordering of cardinality κ, and a partial ordering is ℚκ-scattered when it embeds no copy of ℚκ. We classify the ℚκ-scattered partial orderings with the finite antichain condition relative to the ℚκ-scattered linear orderings. We show that in general the property of being a ℚκ-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.

Original language | English |
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Pages (from-to) | 6259-6278 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 364 |

Issue number | 12 |

DOIs | |

State | Published - 27 Aug 2012 |

## Keywords

- Better-quasi-orderings
- Classification
- Finite antichain condition
- Scattered chains
- Scattered posets
- Well-quasi-orderings

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics