  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, 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 defined 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 re-interpret and
generalise several existing results relating to compact primitive Boolean
spaces, using the language of trim partitions and extended to Boolean spaces
which may be neither PI nor compact.
