A theorem on \(P_{\kappa}(\lambda)\)
type: Item
instance of: scholarly article
title: A theorem on \(P_{\kappa}(\lambda)\)
zbMATH Open document ID: 0633.03048
author: Thomas J. Jech
published in: Journal of the Mathematical Society of Japan
publication date: 1986-01-01T00:00:00Z
review text: The set \(S(\kappa,\kappa^+)\) is \(\{x\in P_{\kappa}(\kappa^+)|| x| =| x\cap \kappa |^+\}\). It was investigated in \textit{S. Baldwin} [Proc. Am. Math. Soc. 92, 90-92 (1984; Zbl 0552.03035)]. Here it is shown that if f is an ordinal-valued function on \(\kappa\) with Galvin-Hajnal norm \(\| f\| =\beta\), where \(\beta <\lambda\), then the order-type OT(\(\beta\cap x)\leq f(OT(\kappa \cap x))\) for almost all \(x\in {\mathcal P}_{\kappa}(\lambda)\); as a corollary, if (i) \(S(\kappa,\kappa^+)\) is stationary then (ii) the function \(f(\xi)=\xi^+\) on \(\kappa\) has norm \(\kappa^+\). The paper ends with some remarks and open questions on the consistency strengths of (i) and (ii).
reviewed by: F.R.Drake
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 4029547
zbMATH Keywords: ordinal-valued function
zbMATH Keywords: order-type
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03830421
OpenAlex ID: W1980650262
DOI: 10.2969/JMSJ/03830421
description: scientific article; zbMATH DE number 4029547