  The lattice cohomology of a plumbed 3--manifold $M$ associated with a
connected negative definite plumbing graph is an important tool in the study of
topological properties of $M$, and in the comparison of the topological
properties with analytic ones when $M$ is realized as complex analytic
singularity link. By definition, its computation is based on the
(Riemann--Roch) weights of the lattice points of $\Z^s$, where $s$ is the
number of vertices of the plumbing graph. The present article reduces the rank
of this lattice to the number of `bad' vertices of the graph. (Usually the
geometry/topology of $M$ is codified exactly by these `bad' vertices via
surgery or other constructions. Their number measures how far is the plumbing
graph from a rational one.) The effect of the reduction appears also at the
level of certain multivariable (topological Poincar\'e) series as well. Since
from these series one can also read the Seiberg--Witten invariants, the
reduction theorem provides new formulae for these invariants too. The reduction
also implies the vanishing $\bH^q=0$ of the lattice cohomology for $q\geq \nu$,
where $\nu$ is the number of `bad' vertices. (This bound is sharp.)
