### ${\rm DEPTH}^+$ 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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