Moderate families in Boolean algebras (Q1194247)

Let \(B\) be a Boolean algebra, \(F\subseteq B\). \(F\) is moderate iff for each \(b\in B\), the family \(\{f\in F: f\land b\neq 0\) and \(f-b\neq 0\}\) is finite. In the first two sections of this paper the author shows some facts on moderate families, which are used in the sequel, and introduces moderate products. In the next three sections the author investigates moderate Boolean algebras, i.e. Boolean algebras for which each ideal is generated by a moderate family. It is shown that countable Boolean algebras are moderate and that the class of moderate algebras is closed under subalgebras, homomorphic images and moderate products. In the next section it is shown that each subalgebra of a retractive moderate algebra is moderate too. The techniques of this section are used to construct a Boolean algebra which is not embeddable in an interval algebra and all of whose non-zero factors are of cardinality \(>2^ \omega\). In section 7 the author studies moderate algebras in which each ideal is generated by the disjoint union of two regular ideals. Such algebras are said to have the regular splitting property. In the last section the author investigates Boolean algebras whose Stone spaces are crumbly spaces. He characterizes these algebras as those moderate algebras all of whose quotients have the regular splitting property.