Partition algebras for almost-disjoint families

Abstract

A set a ⊆ ω a \subseteq \omega is a partitioner of a maximal almost-disjoint faculty F F of subsets of ω \omega if every element of F F is almost contained in or almost-disjoint from a a . The partition algebra of F F is the quotient of the Boolean algebra of partitioners modulo the ideal generated by F F and the finite sets. We show that every countable algebra is a partition algebra, and that CH implies every algebra of cardinality ≤ 2 ℵ 0 \leq {2^{{\aleph _0}}} is a partition algebra. We also obtain consistency and independence results about the representability of Boolean algebras as partition algebras.