  Stone space partitions $\{X_{p}\mid p\in P\}$ satisfying conditions like
$\bar{X_{p}}=\bigcup_{q\leqslant p}X_{q}$ for all $p\in P$, where $P$ is a
poset or PO system (poset with a distinguished subset), arise naturally in the
study both of primitive Boolean algebras and of $\omega$-categorical
structures. A key concept for studying such partitions is that of a $p$-trim
open set which meets precisely those $X_{q}$ for which $q\geqslant p$; for
Stone spaces, this is the topological equivalent of a pseudo-indecomposable
set. This paper develops the theory of infinite partitions of Stone spaces
indexed by a poset or PO system where the trim sets form a neighbourhood base
for the topology. We study the interplay between order properties of the
poset/PO system and topological properties of the partition, examine extensions
and completions of such partitions, and derive necessary and sufficient
conditions on the poset/PO system for the existence of the various types of
partition studied. We also identify circumstances in which a second countable
Stone space with a trim partition indexed by a given PO system is unique up to
homeomorphism, subject to choices on the isolated point structure and
boundedness of the partition elements. One corollary of our results is that
there is a partition $\{X_{r}\mid r\in[0,1]\}$ of the Cantor set such that
$\bar{X_{r}}=\bigcup_{s\leqslant r}X_{s}\text{ for all }r\in[0,1]$.
