Isomorphisms of Sums of Countable Boolean Algebras

Denote by nA the sum of n copies of a Boolean algebra A. We prove that, for any countable Boolean algebra A, nA at mA with m > n implies nA a (n + \)A. In [2], W. Hanf constructed a Boolean algebra H isomorphic to the direct product H X H X H but not to H X H. This result of Hanf was strengthened by J. Ketonen in [4]: there exists even a countable Boolean algebra with this property. In [1], P. R. Halmos examines various problems concerning isomorphism of products of Boolean algebras and asks the corresponding questions for sums in place of products. An analogy of the above result of W. Hanf is proved in [6]: there exists a Boolean algebra B isomorphic to B + B + B but not to B + B. Even, every Boolean algebra is a homomorphic image of a Boolean algebra with this property. All the algebras with this property, constructed in [6], are very large. It is natural to ask whether there exists a countable Boolean algebra B isomorphic to B + B + B but not to B + B. Let us notice that a weaker question about the existence of two nonisomorphic countable Boolean algebras A, B with A + A isomorphic to B + B, was answered affirmatively in [5]. Nevertheless, there exists no countable Boolean algebra B isomorphic to B + B + B but not to B + B. We prove here a stronger result, namely that, for countable Boolean algebras, nB a mB with m > n implies «5 a (n + l)B. This is stated in Theorem 3 of the present paper. Theorems 1 and 2 concern the Schroder-Bernstein property and the pseudoindecomposability of countable Boolean algebras. They form steps in the proof of Theorem 3, but they could be interesting in themselves. 1. For Boolean algebras A, B, let us write A < B iff B » A X C for some C. Definition. A Boolean algebra A is called weakly pseudoindecomposable with respect to a Boolean algebra A' if A » B X C implies either A' < B or A' < C. A is called weakly pseudoindecomposable if it is weakly pseudoindecomposable with respect to itself. 2. Theorem 1. A weakly pseudoindecomposable countable Boolean algebra A has the Schroder-Bernstein property : A < B, B < A implies A » B. Proof. We prove the dual form of the theorem. Received by the editors February 7, 1979 and, in revised form, November 20, 1979. 1980 Mathematics Subject Classification. Primary 06A40, 54B10. © 1980 American Mathematical Society 0002-9939/80/0000-0552/302.00 389 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use