  The geometric and algebraic theory of monomial ideals and multigraded modules
is initiated over real-exponent polynomial rings and, more generally, monoid
algebras for real polyhedral cones. The main results include the generalization
of Nakayama's lemma; complete theories of minimal and dense primary, secondary,
and irreducible decomposition, including associated and attached faces; socles
and tops; minimality and density for downset hulls, upset covers, and fringe
presentations; Matlis duality; and geometric analysis of staircases. Modules
that are semialgebraic or piecewise-linear (PL) have the relevant property
preserved by functorial constructions as well as by minimal primary and
secondary decompositions. And when the modules in question are subquotients of
the group itself, such as monomial ideals and quotients modulo them, minimal
primary and secondary decompositions are canonical, as are irreducible
decompositions up to the new real-exponent notion of density.
