Hausdorff's theorem for posets that satisfy the finite antichain property

Uri Abraham, Robert Bonnet

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

Hausdorff characterized the class of scattered linear orderings as the least family of linear orderings that includes the ordinals and is closed under ordinal summations and inversions. We formulate and prove a corresponding characterization of the class of scattered partial orderings that satisfy the finite antichain condition (FAC). Consider the least class of partial orderings containing the class of well-founded orderings that satisfy the FAC and is closed under the following operations: (1) inversion, (2) lexicographic sum, and (3) augmentation (where 〈P, ≺(Combining low line)〉 augments 〈P, ≤〉 iff x ≺(Combining low line) y whenever x ≤ y). We show that this closure consists of all scattered posets satisfying the finite antichain condition. Our investigation also sheds some light on the natural (Hessenberg) sum of ordinals and the related product and exponentiation operations.

Original languageEnglish
Pages (from-to)51-69
Number of pages19
JournalFundamenta Mathematicae
Volume159
Issue number1
DOIs
StatePublished - 1 Jan 1999

Keywords

  • Hausdorff's theorem
  • Ordinals
  • Partial orderings
  • Scattered partial orderings

ASJC Scopus subject areas

  • Algebra and Number Theory

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