Partition algebras for almost-disjoint families

Abstract:

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