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A new proof of the Gitik-Shelah theorem. (English) Zbl 0738.03019
For ideals \(I\) on \(\kappa\), \(J\) on \(X\), there are two natural definitions of a product ideal on \(\kappa\times X\). The author introduces the product property for \(J\): \(I\times J\subseteq(J\times I)^ T\) for all \(I\), and proves it to hold for the ideals of meagre and measure zero sets. Moreover, if \(J\) satisfies the product property, then there are large \(J\)-almost disjoint families of functions. These results are applied to give a purely combinatorial proof of the Gitik-Shelah Theorem on representations of the Boolean algebra \(P(\kappa)/I\) as \(B_ \lambda/J\) with \(J\) one of the above ideals, \(B_ \lambda\) the \(\sigma\)-field generated by the basic open sets in the product space \(2^ \lambda\).

MSC:
03E35 Consistency and independence results
06E05 Structure theory of Boolean algebras
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[1] M. Gitik and S. Shelah,Forcings with ideals and simple forcing notions, Isr. J. Math.68 (1989), 129–160. · Zbl 0686.03027 · doi:10.1007/BF02772658
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