Incomplete algebras

Abstract

The quotient of a Boolean algebra modulo an ideal may turn out to have a higher degree of completeness than one has a right to expect. Thus, for instance, the reduced Borel algebra and the reduced measure algebra of the unit interval are not only σ-algebras, which is all that the general theory can predict, but even complete. A few observations of this kind are likely to tip the balance of expectations too far over to the optimistic side. The purpose of this section is to provide a counterbalance in the form of some counterexamples. In other words, we shall obtain a few negative results: we shall see that certain quotient algebras are not complete.