Homomorphic images of $\sigma$-complete Boolean algebras

Abstract:

It is a well-known theorem of R. S. Pierce that, for every infinite cardinal $\alpha ,{\alpha ^{{\aleph _0}}} = \alpha$ if and only if there is a complete Boolean algebra $B$ s.t. card $B = \alpha$ (see [3, Theorem 25.4]). Recently, Comfort and Hager proved [1] that, for every infinite $\sigma$-complete Boolean algebra $B,{(\operatorname {card} B)^{{\aleph _0}}} = \operatorname {card} B$. We extend this result to the class of homomorphic images of $\sigma$-complete algebras, following closely Comfort’s and Hager’s proof. As a corollary, an improvement of Shelah’s theorem on the cardinality of ultraproducts of finite sets [2] is derived (Theorem 2).$^{1}$