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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:
 3e+35 Consistency and independence results 600000 Structure theory of Boolean algebras
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##### References:
 [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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