Some partition relations for ideals on \(P_{\kappa}\lambda\)
type: Item
instance of: scholarly article
title: Some partition relations for ideals on \(P_{\kappa}\lambda\)
zbMATH Open document ID: 0733.03039
DOI: 10.1007/BF01903842
author: C. A. Johnson
published in: Acta Mathematica Hungarica
publication date: 1990-01-01T00:00:00Z
review text: The author develops techniques for proving partition relations for ideals on \(P_{\kappa}\lambda\). The class of seminormal ideals on \(P_{\kappa}\lambda\) is introduced, and it is shown that ideal-theoretic partition relations for this class are related to the distributivity of the quotient algebra with respect to a given ideal. The author extends a proof of Solovay to show that if cf \(\lambda\) \(>\kappa\) and \(\kappa\) is \(\lambda\)-Shelah then \(\lambda^{<\kappa}=\lambda.\) In earlier papers the author introduced an ideal-theoretic analogue, WC, of weak compactness on ideals on uncountable cardinals. Here a similar concept (also called WC) for ideals on \(P_{\kappa}\lambda\) is introduced and shown to be related to the ideal induced by the \(\lambda\)- Shelah property and to mild \(\lambda\)-ineffability. Results of Carr, Kunen, and Baumgartner are extended to show that ineffable and almost ineffable subsets of \(P_{\kappa}\lambda\) may be characterized in terms of regressive partition relations.
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 4212033
zbMATH Keywords: partition relations
zbMATH Keywords: ideals
zbMATH Keywords: seminormal ideals
zbMATH Keywords: quotient algebra
zbMATH Keywords: weak compactness on ideals
zbMATH Keywords: ineffability
zbMATH Keywords: regressive partition relations
reviewed by: J. M. Plotkin
MaRDI profile type: Publication
cites work: Q3342554
cites work: The structure of ineffability properties of \(P_{\kappa}\lambda\)
cites work: Pxδ‐Generalizations of Weak Compactness
cites work: Flipping properties and supercompact cardinals
cites work: Some combinatorial problems concerning uncountable cardinals
cites work: Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers
cites work: Distributive ideals and partition relations
cites work: More on distributive ideals
cites work: A combinatorial characterization of normal M-ultrafilters
cites work: A combinatorial property of pκλ
cites work: Weakly compact cardinals: A combinatorial proof
cites work: Strong axioms of infinity and elementary embeddings
full work available at URL: https://doi.org/10.1007/bf01903842
OpenAlex ID: W2047967114
description: scientific article; zbMATH DE number 4212033