Ultraproducts, \(p\)-limits and antichains on the Comfort group order
type: Item
instance of: scholarly article
title: Ultraproducts, \(p\)-limits and antichains on the Comfort group order
zbMATH Open document ID: 1071.54004
published in: Topology and its Applications
publication date: 2004-10-01T00:00:00Z
review text: The authors of the paper under review consider the following question: Does \({\mathbf M}{\mathbf A}\) imply that the Comfort group order is not downward directed? The answer ``yes'' is equivalent to the existence of a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not \(r\)-compact for any free ultrafilter \(r\) on \(\omega\). The authors construct \(p\)-compact groups as follows: Let \(p\), \(q\) be incomparable selective ultrafilters. Then there exists a \(p\)-compact group \(G\) and a \(q\)-compact group \(H\) for which \(G\times H\) is not countably compact. This implies the following important results: if there are \(2^c\) incomparable selective ultrafilters, then the Comfort group order has an antichain of size \(2^c\). \({\mathbf M}{\mathbf A}\) implies the Comfort group order has an antichain of size \(2^c\).
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54G20
zbMATH DE Number: 2104477
zbMATH Keywords: \(p\)-limit
zbMATH Keywords: \(p\)-compact
zbMATH Keywords: Comfort group order
zbMATH Keywords: Countably compact group
zbMATH Keywords: Convergence
zbMATH Keywords: Topological product
zbMATH Keywords: Selective ultrafilters
author: Artur Hideyuki Tomita
author: Stephen Watson
reviewed by: Kiyoshi Iseki
MaRDI profile type: Publication
full work available at URL: j.topol.2004.02.012
OpenAlex ID: W2068390573
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DOI: 10.1016/J.TOPOL.2004.02.012
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description: scientific article; zbMATH DE number 2104477