On collections of almost disjoint families. (English) Zbl 0675.03029

Balcar, Bohuslav; Simon, Petr

Commentat. Math. Univ. Carol. 29, No. 4, 631-646 (1988).

With suitable definition of \(b_{\kappa}(>\kappa)\) and of (\(\tau\),\(\mu\),\(\lambda)\)-nowhere distributivity (written (\(\tau\),\(\cdot,\lambda)\) when \(\mu\) is arbitrary), the authors prove: For every uncountable cardinal \(\kappa\), the quotient Boolean algebra \({\mathcal P}_{\kappa}(\kappa)={\mathcal P}(\kappa)/[\kappa]^{<\kappa}\) is nowhere distributive for (1) \((\omega,\cdot,b_{\kappa})\) when \(\kappa\) is regular (and then the forcing with \({\mathcal P}_{\kappa}(\kappa)\) collapses \(b_{\kappa}\) to \(\omega)\), (2) \((\omega,\cdot,\kappa^+)\) when \(\kappa\) is singular with uncountable cofinality, (3) \((\omega_ 1,\cdot,\kappa^{\omega})\) when \(\kappa\) is singular with countable cofinality. These results extend previous ones of the authors.

Reviewer: Mo Shaokui

MSC:

03E05	Other combinatorial set theory
03E45	Inner models, including constructibility, ordinal definability, and core models
06E05	Structure theory of Boolean algebras

Keywords:

almost disjoint family; distributivity of Boolean algebras; completion of Boolean algebra