We give sufficient and necessary conditions to be fulfilled by a filter and an ideal in order that the -quotient space of the -ideal product space preserves -properties () (“in the sense of the Łos theorem”). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.
We compare the forcing-related properties of a complete Boolean algebra with the properties of the convergences (the algebraic convergence) and on generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that is a topological convergence iff forcing by does not produce new reals and that is weakly topological if satisfies condition (implied by the -cc). On the other hand, if is a weakly topological convergence, then is a -cc algebra...
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