Abstract
An ordered setP is said to have 2-cutset property if, for every elementx ofP, there is a setS of elements ofP which are noncomparable tox, with |S|⩽2, such that every maximal chain inP meets {x}∪S. We consider the following question: Does there exist ordered sets with the 2-cutset property which have arbitrarily large dimension? We answer the question in the negative by establishing the following two results.Theorem: There are positive integersc andd such that every ordered setP with the 2-cutset property can be represented asP=X∪Y, whereX is an ordinal sum of intervals ofP having dimension ⩽d, andY is a subset ofP having width ⩽c. Corollary: There is a positive integern such that every ordered set with the 2-cutset property has dimension ⩽n.
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Communicated by I. Rival
The author gratefully acknowledges a grant from NSERC in support of this work.
This paper is dedicated to Gillian and Ezra.
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Ginsburg, J. On the dimension of ordered sets with the 2-cutset property. Order 10, 37–54 (1993). https://doi.org/10.1007/BF01108707
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DOI: https://doi.org/10.1007/BF01108707