  Zonotopal algebra interweaves algebraic, geometric and combinatorial
properties of a given linear map X. Of basic significance in this theory is the
fact that the algebraic structures are derived from the geometry (via a
non-linear procedure known as "the least map"), and that the statistics of the
algebraic structures (e.g., the Hilbert series of various polynomial ideals)
are combinatorial, i.e., computable using a simple discrete algorithm known as
"the valuation function". On the other hand, the theory is somewhat rigid since
it deals, for the given X, with exactly two pairs each of which is made of a
nested sequence of three ideals: an external ideal (the smallest), a central
ideal (the middle), and an internal ideal (the largest).
  In this paper we show that the fundamental principles of zonotopal algebra as
described in the previous paragraph extend far beyond the setup of external,
central and internal ideals by building a whole hierarchy of new
combinatorially defined zonotopal spaces.
