Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-19T01:41:27.794Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Number of Maximal Elements in a Partially Ordered Set

Published online by Cambridge University Press:  20 November 2018

John Ginsburg
John Ginsburg*
Affiliation:
The University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

Keywords

06A10Partially ordered setcutsetmaximal element

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 3 , 01 September 1987 , pp. 351 - 357
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Erdos, P. and Rado, R., A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), pp. 427–489.Google Scholar
2. Ginsburg, J. and Sands, B., A length-width inequality for partially ordered sets with two-element cutsets, to appear.Google Scholar
3. Nowakowski, R., Cutsets of Boolean lattices, Discrete Math. 63 2, 3 (1987), pp. 231–240.Google Scholar
4. Rival, I. and Zaguia, N., Antichain cutsets, Order Vol. 1, No. 2 (1985), pp. 235–247.Google Scholar
5. Sauer, N., to appear.Google Scholar
6. Sauer, N. and Woodrow, R., Finite cutsets and finite antichains, Order Vol. 1, No. 1 (1984), pp. 35—46.Google Scholar
7. Sauer, N. and El Zahar, M., The length, the width and the cutset number of finite ordered sets, Order Vol. 2 No. 3 (1985), pp. 243–248.Google Scholar