### ${\rm DEPTH}^+$ and ${\rm LENGTH}^+$ of Boolean Algebras

by Garti and Shelah. [GaSh:974]

Suppose kappa = textrm {cf}(kappa), lambda > textrm {cf}(lambda)=
kappa^+
and lambda = lambda^kappa . We prove that there exist a sequence
<{mathbf {B}}_i:i< kappa > of Boolean algebras and an
ultrafilter D on kappa so that lambda = prod limits_{i< kappa}
Depth^+({mathbf {B}}_i)/D< Depth^+(prod limits_{i<
kappa}{mathbf B}_i/D)= lambda^+ . An identical result holds
also for Length^+ .
The proof is carried in ZFC and it holds even above large cardinals.

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