Obstacles to extending Mirsky's theorem
type: Item
instance of: scholarly article
title: Obstacles to extending Mirsky's theorem
zbMATH Open document ID: 1008.06002
DOI: 10.1023/A:1016541101728
author: James H. Schmerl
published in: Order
publication date: 2002-09-18T00:00:00Z
review text: A poset \((P,\leq)\) is called locally finite if for any \(a,b\in P\), \(a\leq b\), the interval \([a,b]= \{x\in P:a\leq x\leq b\}\) is finite. In this paper the following theorem is proved. Theorem: For each infinite cardinal \(\kappa\) there is a locally finite poset \(P\) such that (1) \(P\) has no infinite chain and (2) \(P\) cannot be covered by \(\kappa\) antichains. The local finiteness reminds the reviewer of his study of many years ago, that is, a very special case of a semilattice \(S\) satisfying the condition: for any \(a,b\in S\), \(a\leq b\), \([a,b]\) is a finite chain [J. Gakugei Tokushima Univ. Natur. Sci. Math. 7, 9-17 (1956; Zbl 0073.01803)]. However, it contains an infinite chain and there is no direct relation with this paper.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 1803032
zbMATH Keywords: antichain
zbMATH Keywords: locally finite poset
zbMATH Keywords: chain
reviewed by: Takayuki Tamura
MaRDI profile type: Publication
full work available at URL: a:1016541101728
OpenAlex ID: W2176276234
description: scientific article; zbMATH DE number 1803032