  The present paper is motivated by the need to generalize the construction of
the Scarf complex in order to give combinatorial resolutions of a much broader
class of modules than just the monomial ideals. For any subset $A\subseteq
\mathbb{R}^n$, let $\mathfrak{N}(A)$ denote the collection of all subsets
$B\subseteq A$ such that there is no $a\in A$ that is strictly less than the
supremum of $B$ in all coordinates. We show that if $A\subseteq \mathbb{Z}^n$
is generic (in a sense appropriate for this context), then $\mathfrak{N}(A)$ is
a locally finite simplicial complex. Moreover, if $A$ is generic, then the
barycentric subdivision of $\mathfrak{N}(A)$ is equivalent to a triangulation
of a PL hypersurface in $\mathbb{R}^n$. This gives us natural generalizations
of the notions of ``staircase surface'' and ``Buchberger graph,'' described by
Miller and Sturmfels, to arbitrary dimension. (This seems to be a new result,
even in the well-studied case that $A$ is a finite subset of $\mathbb{N}^n$.)
We give examples that show that when $A$ is infinite, $\mathfrak{N}(A)$ may
have complicated topology, but if there are at most finitely many elements of
$A$ below any given $b\in \mathbb{R}^n$, then $\mathfrak{N}(A)$ is locally
contractible. $\mathfrak{N}(A)$ can therefore be used to construct locally
finite free resolutions of sub-$k[\mathbb{N}^n]$-modules of the group algebra
$k[\mathbb{R}^n]$ ($k$ is a field). We prove various additional facts about the
structure of $\mathfrak{N}(A)$
