On tightly κ-filtered Boolean algebras

Abstract.

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