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Binary Trees and the n-Cutset Property

Published online by Cambridge University Press:  20 November 2018

Peter Arpin and
John Ginsburg
Peter Arpin
Affiliation:
University of Winnipeg Winnipeg, Manitoba R3B 2E9
John Ginsburg
Affiliation:
University of Winnipeg Winnipeg, Manitoba R3B 2E9
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Abstract

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A partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n . Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω . If P — Max P is countable then P contains a complete binary tree of height ω

Keywords

partially ordered setcutsetn-cutset propertymaximal elementbinary tree06A10

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 1 , 01 March 1991, pp. 23 - 30
Copyright
Copyright © Canadian Mathematical Society 1991

References

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