Inequalities for the greedy dimensions of ordered sets

Henry A. Kierstead, William T. Trotter

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

Every linear extension L: [x1<x2<...<xm] of an ordered set P on m points arises from the simple algorithm: For each i with 0≤i<m, choose xi+1 as a minimal element of P-{xj:j≤i}. A linear extension is said to be greedy, if we also require that xi+1 covers xi in P whenever possible. The greedy dimension of an ordered set is defined as the minimum number of greedy linear extensions of P whose intersection is P. In this paper, we develop several inequalities bounding the greedy dimension of P as a function of other parameters of P. We show that the greedy dimension of P does not exceed the width of P. If A is an antichain in P and |P-A|≥2, we show that the greedy dimension of P does not exceed |P-A|. As a consequence, the greedy dimension of P does not exceed |P|/2 when |P|≥4. If the width of P-A is n and n≥2, we show that the greedy dimension of P does not exceed n2+n. If A is the set of minimal elements of P, then this inequality can be strengthened to 2 n-1. If A is the set of maximal elements, then the inequality can be further strengthened to n+1. Examples are presented to show that each of these inequalities is best possible.

Original languageEnglish (US)
Pages (from-to)145-164
Number of pages20
JournalOrder
Volume2
Issue number2
DOIs
StatePublished - Jun 1 1985
Externally publishedYes

Keywords

  • AMS (MOS) subject classifications (1980): 06A05, 06A10
  • Ordered sets
  • greedy dimensions
  • linear extension

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Theory and Mathematics

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