In this scenario, the points are known as the germs while the sets are known as grains. It is not uncommon to consider the
union of all grains in a germ-grain model, a collection
which is sometimes referred to as the grain cover (Kuronen and Leskelä 2012).
The grain cover is sometimes referred to as the basis of the model in question.

In older literature, it is not uncommon to define a germ-grain model by assuming any of several other conditions, e.g., that the grains fail to be independent,
by allowing the subsets to be random
closed sets (not necessarily compact) in , and by considering as a basis all unions of the form

(1)

where
is a non-Poissonmarked
point process (MPP) on with mark space ; this is in contrast to, e.g., the Boolean-Poisson model which
has a basis of the form

(2)

where
is a Poisson point process in and where is the compact set in assigned to each (Hanisch 1981). Though dated, this notion of a germ-grain
model can be further elaborated upon using more rigorous mathematical formality by
explicitly writing the MPP as

Hanisch, K. H. "On Classes of Random Sets and Point Process Models." Serdica Bulgariacae Mathematicae Publicationes7,
160-166, 1981.Heinrich, L. "On Existence and Mixing Problems of
Germ-Grain Models." Statistics23, 271-286, 1992.Kuronen,
M. and Leskelä, L. "Hard-Core Thinnings of Germ-Grain Models with Power-Law
Grain Sizes." 5 Apr 2012. http://arxiv.org/abs/1204.1208.Meester,
R. and Roy, R. Continuum
Percolation. New York: Cambridge University Press, 2008.