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An infinite Boolean algebra (BA) is said to be hopfian if every onto endomorphism is one-to-one. It is shown that there are no countably infinite hopfian BA's. An atomic hopfian BA of power continuum is constructed. We restrict our attention to infinite Boolean algebras W = (A, +, *,-, 0, 1), and use End W for the endomorphism semigroup of 9, At(W) for the set of atoms of 91, and Sg(T) for the subalgebra of W generated by T c A. P (X) denotes the power set of X and 6(X), the field of finitecofinite subsets of o. We use MA(K) to denote Martin's axiom: If ' is a partial ordering satisfying the countable antichain condition, and if IF is a collection of dense open subsets of 'P with 1j' 1 < K, then there is an C-generic filter on i'3. (See AK of Martin and Solovay [2].) Recall that MA(K) is a theorem in ZFC. THEOREM 1. Let 9 be a BA of power K < 2o with infinitely nany atoms and assume MA(K) holds. Then there is 93 C W such that e3 (Eo) and there is an isomorphism +: End 93 -* End W such that, for h E End 9, h is one-to-one (onto) iff h + is one-to-one (onto, respectively). The following lemma, which is (1) of Theorem 3.6 of McKenzie and Monk [3], is proved using MA(K). LEMMA 2. Under the hypothesis of Theorem 1, there is a D c At(W), with DI = Ho such that for every a E A, the set S(a) = {d E DId < a} is a finite or cofinite subset of D. PROOF OF THEOREM 1. Let e = Sg(D) and notice e _ C(w). Clearly, I = {a E AIS(a) is finite) is a maximal ideal of W. For a E I, there is a unique finite S(a) C D and unique t(a) E A such that a = E S(a) + t(a) and dt(a) = 0 for all d E D. Let h E End 93. For each a E I, define f(a) = 2 th(d)ld e S(a)} + t(a). Presented to the Society, August 17, 1977 under the title On 'strongly' rigid Boolean algebras and cofinalities of Boolean algebras; received by the editors October 24, 1978 and, in revised form, November 29, 1978. AMS (MOS) subject classifications (1970). Primary 06A40, 02J05.

Hopfian Boolean algebras of power less than or equal to continuum