Changing the depth of an ordered set by decomposition

E. C. Milner, K. Prikry

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The depth of a partially ordered set (P, >) is the smallest ordinal γ such that (P, >) does not embed γ*. The width of (P, γ) is the smallest cardinal number μ such that there is no antichain of size μ+1 in P. We show that if γ > ω and γ is not an infinite successor cardinal, then any partially ordered set of depth γ can be decomposed into cf(γ) parts so that the depth of each part is strictly less than γ. If γ = ω or if γ is an infinite successor cardinal, then for any infinite cardinal ⋋ there is a linearly ordered set of depth γ such that for any ⋋-decomposition one of the parts has the same depth γ. These results are used to solve an analogous problem about width. It is well known that, for any cardinal ⋋, there is a partial order of width ω which cannot be split into ⋋ parts of finite width. We prove that, for any cardinal ⋋ and any infinite cardinal ν there is a partial order of width ν+ which cannot be split into ⋋ parts of smaller width.

Original languageEnglish (US)
Pages (from-to)773-785
Number of pages13
JournalTransactions of the American Mathematical Society
Volume290
Issue number2
DOIs
StatePublished - Aug 1985

Keywords

  • Category
  • Depth
  • Extendable ordinal
  • Graph
  • Partial order
  • Partition relation
  • Width

Fingerprint Dive into the research topics of 'Changing the depth of an ordered set by decomposition'. Together they form a unique fingerprint.

Cite this