Free complete extensions of Boolean algebras
George Day · Pacific Journal of Mathematics · 1965
From considering questions about the existence of free incomplete 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 JB* in such a way that homomorphisms of B into complete Boolean algebras can be extended to complete homomorphisms on B*Ί 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.Several 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 J3* such that (i) if h is a homomorphism of B into a complete Boolean algebra C, then there is a complete homomorphism h* of B* into C such that h*oi -h (ii) £>* 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.The problem of determining which Boolean algebras have free complete extensions arose from a conjecture made by L. Rieger in [7].This conjecture is, in effect, that no infinite free Boolean algebra has a free complete extension this has been verified by H. Gaifman in [3] and, independently and by different methods, by A. Hales,in [4].Rieger proved, on the other hand, that for any cardinal numbers a and β there exists a unique free α-complete Boolean algebra on β generators.This result can be expressed by the statement that the free Boolean algebra on β generators has a unique free α-extension (a free a-extension of a Boolean algebra is defined in the same manner as a free complete extension, with the concepts of completeness and complete homomorphism replaced by those of α-completeness and