Two-cardinal versions of weak compactness: partitions of pairs
The authors study various partition properties on \(P_{\kappa}^{\lambda}\). They consider ideals associated with these partition properties, so for instance the ideal \(I_{\kappa,\lambda}\) (the noncofinal ideal), \(\mathrm{NSS}_{\kappa,\lambda}\) (the smallest seminormal ideal), \(\mathrm{NSh}_{\kappa,\lambda}\) (the \(\lambda\)-Shelah ideal). Their main result asserts that if \(\lambda^{<\lambda^{<\kappa}} = \lambda^{<\kappa}\), then \(p(\mathrm{NSh}_{\kappa,\lambda^{<\kappa}}))^+ \rightarrow (\mathrm{NSS}^+_{\kappa,\lambda})^2\), where \(p : P_{\kappa}(\lambda^{<\kappa}) \rightarrow P_{\kappa}(\lambda)\) is defined by \(p(x) = x \cap \lambda\). In their paper they review basic material and study elementary properties of the ideals \(\mathrm{NSJ}_{\kappa,\lambda}\), \(\mathrm{NJ}_{\kappa,\lambda}\), \(\mathrm{NSB}_{\kappa,\lambda}\) and \(\mathrm{NB}_{\kappa,\lambda}\). They investigate very strongly normal ideals and give an overview of the results under GCH.
scientific article; zbMATH DE number 5970593
scientific article