  Over any partially ordered abelian group whose positive cone is closed in an
appropriate sense and has finitely many faces, modules that satisfy a weak
finiteness condition admit finite primary decompositions. This conclusion rests
on the introduction of basic notions in the relevant generality, such as
closedness of partially ordered abelian groups, faces and their coprimary
modules, and finiteness conditions as well local and global support functors
for modules over partially ordered groups.
