Vol. 15, No. 4, 1965

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Free complete extensions of Boolean algebras

George Wesley Day

Vol. 15 (1965), No. 4, 1145–1151

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 hi = 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.

Mathematical Subject Classification
Primary: 06.60
Received: 2 June 1964
Published: 1 December 1965
George Wesley Day