  We address a number of problems on Boolean Algebras. For example, we
construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B
such that the depth of the free product of B_1,B_2 over B is strictly larger
than the depths of B_1 and of B_2 than kappa. We give a condition (for lambda,
mu and theta) which implies that for some BA A_theta there are B_1=B^1_{lambda,
mu, theta} and B_2B^2_{lambda, mu, theta} such that Depth (B_t) <= mu and Depth
(B_1 oplus_{A_theta} B_1) >= lambda. We then investigate for a fixed A, the
existence of such B_1,B_2 giving sufficient and necessary conditions, involving
consistency results. Further we prove that e.g. if B is a BA of cardinality
lambda, lambda >= mu and lambda, mu are strong limit singular of the same
cofinality, then B has a homomorphic image of cardinality mu (and with mu
ultrafilters). Next we show that for a BA B, if d(B)^kappa <|B| then ind (B)>
kappa or Depth (B) >= log (|B|). Finally we prove that if square_lambda holds
and lambda = lambda^{aleph_0} then for some BAs B_n, Depth (B_n) <= lambda but
for any uniform ultrafilter D on omega, prod_{n< omega} B_n/D has depth >=
lambda^+ .
