Depth of Boolean Algebras

Abstract

Suppose $D$ is an ultrafilter on $\kappa$ and $\lambda^\kappa = \lambda$. We prove that if ${\bf B}_i$ is a Boolean algebra for every $i < \kappa$ and $\lambda$ bounds the depth of every ${\bf B}_i$, then the depth of the ultraproduct of the ${\bf B}_i$'s mod $D$ is bounded by $\lambda^+$. We also show that for singular cardinals with small cofinality, there is no gap at all. This gives a partial answer to a previous problem raised by Monk.