  A theory of modules over posets is developed to define computationally
feasible, topologically interpretable data structures, in terms of birth and
death of homology classes, for persistent homology with multiple real
parameters. To replace the noetherian hypothesis in the general setting of
modules over posets, a "finitely encoded" condition is defined combinatorially
and developed algebraically. It captures topological tameness of persistent
homology. Poset-modules satisfying it can be specified by "fringe
presentations" that reflect birth-and-death descriptions of persistence. A
syzygy theorem characterizes finitely encoded modules as admitting
appropriately finite presentations and resolutions.
  The geometric and algebraic theory focuses on modules over real polyhedral
groups (real vector spaces with polyhedral positive cones) and a parallel
theory over discrete polyhedral groups (abelian groups with finitely generated
positive cones). Existence of primary decomposition is proved over arbitrary
polyhedral partially ordered abelian groups, but the real and discrete cases
carry enough geometry and, crucially in the real case, topology to induce
complete theories of minimal primary and secondary decomposition, associated
and attached faces, minimal generators and cogenerators, socles and tops,
minimal upset covers and downset hulls, Matlis duality, and minimal fringe
presentation. Real semialgebraic properties of data are preserved by functorial
constructions.
  Tops and socles become functorial birth and death spaces for multiparameter
persistence modules. They yield functorial "QR codes" and "elder morphisms" for
modules over real and discrete polyhedral groups that generalize and categorify
the bar code and elder rule for persistent homology in one parameter. The
disparate ways that QR codes and elder morphisms model bar codes coalesce, in
one parameter, to functorial bar codes.
