We adopt the notation of Sikorski [3] with the following additions. A Boolean algebra 51 is trivial iff it has only one element. 31 is minjective iff 31 is an m-algebra and whenever we are given m algebras 33 and 6 with m-homomorphisms ƒ, g of 33 into 21 and 93 into £ respectively, and with g one-one, there is an m-homomorphism k of 6 into 21 such that ƒ = k o g (o denotes composition of functions). Obviously every trivial Boolean algebra is m-injective for any m. Halmos [l] raised the question

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... concerning what cr-injective Boolean algebras look like, and Linton [2] derived interesting consequences from the assumption that nontrivial <r-injectives exist. The theorem of the title follows easily from the following two lemmas, the first of which is well known, while the second has some independent interest. LEMMA 1. If 21 satisfies the m-chain condition, {A t }ter is a set of elements of 21, and UterAt exists, then there is a subset S of T with S^m such that U s esA s exists and equals (JterAt. PROOF. Let 93 be a maximal set of pairwise disjoint elements of 2t such that for every ££33 there is a tÇzT such that BC.A t (such a 93 exists by Zorn's lemma). With every ££33 one can associate an element /(£) of T such that BC.A t (B)-By the m-chain condition, 58 ^m, and hence also {t(B)} B eSd is m-indexed. Now U.Be«£ exists and equals UterAt. For, if this is not true then, by virtue of the fact that BQUterAt for each ££33, it follows that there is a CVA such that £P\C=A for all ££33, while CCMterAt. Then CC\A t^A for a certain /o£7\ and $8^J{Cr\At 0 } is a set properly including 33 with all the properties of 33. This contradiction shows that UB<S«£ exists and equals U* fc ?^4*. Obviously, then, ÖBe%A t (B) also exists and equals UterAt, as desired. LEMMA 2. For every m there is a complete Boolean algebra 21 such that every nontrivial a-homomorphic image of 21 has cardinality at least m. PROOF. Let 33 be a free Boolean algebra on m generators, and let 21 be a completion of 33. By [3, pp. 72, 156], 2Ï satisfies the cr-chain condition. Let J be a proper cr-ideal of 21. By Lemma 1, ƒ is principal; 526