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Abstract

In §1 we define some properties of ideals by using games. These properties strengthen precipitousness. We call these stronger ideals. In §2 we show some limitations on the existence of such ideals over ${\cal P}_{\kappa}\lambda$. We also present a consistency result concerning the existence of such ideals over ${\cal P}_{\kappa}\lambda$. In §3 we show that such ideals satisfy stronger normality. We show a cardinal arithmetical consequence of the existence of strongly normal ideals. In § 4 we study some “large cardinal-like” consequences of stronger ideals.