Partitions of primitive Boolean spaces

Abstract

A Boolean ring and its Stone space (Boolean space) are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Hanf showed that a primitive PI Boolean algebra can be uniquely defined by a structure diagram. In a previous paper we defined trim $P$-partitions of a Stone space, where $P$ is a PO system (poset with a distinguished subset), and showed how they provide a physical representation within the Stone space of these structure diagrams. In this paper we study the class of trim partitions of a fixed primitive Boolean space, which may not be compact, and show how they can be structured as a quasi-ordered set via an appropriate refinement relation. This refinement relation corresponds to a surjective morphism of the associated PO systems, and we establish a quasi-order isomorphism between the class of well-behaved partitions of a primitive space and a class of extended PO systems. We also define rank partitions, which generalise the rank diagrams introduced by Myers, and the ideal completion of a trim $P$-partition, whose underlying PO system is the ideal completion of $P$, and show that rank partitions are just the ideal completions of trim partitions. In the process, we extend a number of existing results regarding primitive Boolean algebras or compact primitive Boolean spaces to locally compact Boolean spaces.