Cardinality and cofinality of homomorphs of products of Boolean algebras (Q798347)

The paper concerns itself with the problem when \(| A|^{\omega}=| A|\) for an infinite Boolean algebra, and various related questions. \(| A|^{\omega}=| A|\) holds if A is \(\sigma\) -complete, and more generally, if A is a homomorphic image of a \(\sigma\) -complete algebra. The authors investigate the following situation: Let B be a homomorphic image of a product of \(\kappa\) algebras \(A_ i\) such that \(| A_ i| <| B|\); is \(| B| =| B|^{\omega}?\) The question leads to large cardinals. It is shown that the answer is affirmative if \(\kappa\) is less than the least measurable cardinal, and a counterexample exists assuming (roughly) a strongly compact cardinal.