  This work addresses models (e.g. potential models of directed orbital
systems- the manganates) in which an effective reduction dimensionality occurs
as a result of a new symmetry which is intermediate between that of global and
local gauge symmetry. This path towards dimensional reduction is examined in
simple O(n) spin models and lattice gauge theories. A high temperature
expansion is employed to map special anisotropic high dimensional models into
lower dimensional variants. We show that it is possible to have an effective
reduction in the dimension without the need of compactifying some dimensions.
These models are frustrated and display a symmetry intermediate between local
and global gauge symmetries. Some solutions are presented. Our dimensional
reductions are a generlization of the trivial dimensional reduction that occur
in pure two dimensional gauge theories. It will be further seen that the
absence of a ``phase interference'' effect plays an important role in high
dimensional problems. By identifying another (``permutational'') symmetry
present in the large n limit, we will further show how to generally map global
high dimensional spin systems onto a one dimensional chain and discuss
implications.
