  A wealth of geometric and combinatorial properties of a given linear
endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope
$Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$.
This well-known line of study is particularly interesting in case $n\eqbd\rank
X \ll N$. We enhance this study to an algebraic level, and associate $X$ with
three algebraic structures, referred herein as {\it external, central, and
internal.} Each algebraic structure is given in terms of a pair of homogeneous
polynomial ideals in $n$ variables that are dual to each other: one encodes
properties of the arrangement ${\cal H}(X)$, while the other encodes by duality
properties of the zonotope $Z(X)$. The algebraic structures are defined purely
in terms of the combinatorial structure of $X$, but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to
either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it
requires no assumptions on the map $X$ (the only exception being that the
algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$
is unimodular), and provides new tools that can be used in enumerative
combinatorics, graph theory, representation theory, polytope geometry, and
approximation theory.
