On tightly κ-filtered Boolean algebras

In this article we study the notion of tight ?-filteredness of a Boolean algebra for infinite regular cardinals ?. Tight à₀ aleph_0 -filteredness is projectivity. We give characterizations of tightly ?-filtered Boolean algebras which generalize the internal characterizations of projectivity given by Haydon, Šcepin, and Koppelberg (see [15] or [17]). We show that for each ? there is an rc-filtered Boolean algebra which is not tightly ?-filtered. This generalizes a result of Šcepin (see [15]). We prove that no complete Boolean algebra of size larger than à₂ aleph_2 is tightly à₁ aleph_1 -filtered. We give a new example of a model of set theory where frak P(w) frak P(omega) is tightly s-filtered. We study the effect of the tight s-filteredness of frak P(w) frak P(omega) on the automorphism group of frak P(w)/fin frak P(omega)/fin .