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From considering questions
about the existence of free α-complete Boolean algebras and free complete Boolean
algebras, one is led naturally to the following problem: Given a Boolean algebra B, is
it possible to embed B as a subalgebra in a complete Boolean algebra B∗ in such a
way that homomorphisms of B into complete Boolean algebras can be extended to
complete homomorphisms on B∗? In general, the answer is “no”; this paper
establishes that B can be so embedded if and only if every homomorphic
image of B is atomic. Severaf other equivalent conditions on B are also
developed.
To express these ideas more precisely, we say that the complete Boolean algebra
B∗ is a free complete extension of the Boolean B provided that there exist an
isomorphism i of B into B∗ such that
(i) if h is a homomorphism of B into a complete Boo# ean algebra C, then there
is a complete homomorphism h∗ of B∗ into C such that h∗∘ i = h;
(ii) B∗ has no regular complete proper subalgebra which contains i[B]—that is,
i[B] completely generates B∗. A Boolean algebra B is said to be superatomic if every
homomorphic image of B is atomic (or, equivalently, if every subalgebra of B is
atomic). Our principal result, then, is that a Boolean algebra B has a free complete
extension if and only if B is superatomic.
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