A polyhedral fan is a finite set of polyhedral cones in 
such that every nonempty face of a member belongs to the
set and the intersection
of any two members is a face of each. Cones are allowed
to contain lines. The union of
the cones is the support of the fan; the fan is complete when its support is all
of 
(Fukuda et al. 2007).
The four closed quadrants in the plane, their four bounding rays, and the origin form a complete polyhedral fan. A single closed quadrant together with its two bounding rays and the origin forms a polyhedral fan that is not complete. Gröbner fans apply this structure to changes of initial ideal.
