Homomorphic images of 𝜎-complete Boolean algebras

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It is a well-known theorem of R. S. Pierce that, for every infinite cardinal α , α ℵ 0 = α \alpha ,{\alpha ^{{\aleph _0}}} = \alpha if and only if there is a complete Boolean algebra B B s.t. card B = α B = \alpha (see [3, Theorem 25.4]). Recently, Comfort and Hager proved [1] that, for every infinite σ \sigma -complete Boolean algebra B , ( card ⁡ B ) ℵ 0 = card ⁡ B B,{(\operatorname {card} B)^{{\aleph _0}}} = \operatorname {card} B . We extend this result to the class of homomorphic images of σ \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 ^{1}