  In the context of modeling biological systems, it is of interest to generate
ideals of points with a unique reduced Groebner basis, and the first main goal
of this paper is to identify classes of ideals in polynomial rings which share
this property. Moreover, we provide methodologies for constructing such ideals.
We then relax the condition of uniqueness. The second and most relevant topic
discussed here is to consider and identify pairs of ideals with the same number
of reduced Groebner bases, that is, with the same cardinality of their
associated Groebner fan.
