The Marczewski hull property and complete boolean algebras.

Abstract

Let \(\mathcal{A}\) be an algebra of subsets of an underlying set \(A\) (which is not the entire power set of \(A\) in general), and let \({\mathcal I} \subseteq {\mathcal A}\) be an ideal over \(A\). The pair \(({\mathcal A}, {\mathcal I})\) is said to have the {\em hull property} iff whenever \(X \subseteq A\), there is a \(Y \in {\mathcal A}\) such that \(X \subseteq Y\) and \(Y\) is ``least'' mod \({\mathcal I}\), i.e., if \(Z \in {\mathcal A}\) and \(X \subseteq Z\), then \(Y \setminus Z \in {\mathcal I}\). It has been observed that in many cases for which \(({\mathcal A}, {\mathcal I})\) satisfies the hull property, the quotient Boolean algebra \({\mathcal A}/{\mathcal I}\) is a complete Boolean algebra. That, and the superficial similarity between the definitions themselves, along with the similar proofs that have sometimes resulted when using these properties, leads to the natural question of how the two properties ``\(({\mathcal A}, {\mathcal I})\) satisfies the hull property'' and ``\({\mathcal A}/{\mathcal I}\) is a complete Boolean algebra'' are related to each other. Examples will be produced which show that neither of these two properties implies the other. In addition, we examine the question of what additional hypotheses would cause one of these properties to imply the other.