Some Results on the Modeling of Spatial Data
by Luca Forlizzi, Enrico Nardelli

Abstract:

Formal methods based on the mathematical theory of partially oredered sets 
(i.e., posets) have been used in the database field for the modeling of
spatial data since many years.

In particular, the use of the lattice completion (or normal completion) of 
a poset has been shown by Kainz, Egenhofer and Greasley [14] to be 
a fundamental technique to build meaningful representations of spatial
subdivisions.

In fact, they proved that the new elements introduced by the normal
completion process can (and have to) be interpreted as being the
intersection of poset elements. This is fundamental, from a mathematical
point of view, since it means that the lattice resulting from the normal
completion is the closure of the given poset with respect to the
intersection operation.

In this paper we precisely clarify the limitations for the use of lattices
as models for spatial subdivisions, by proving sufficient and necessary
conditions. Our result gives therefore a sound teoretical basis for the use
of lattices built on simplicial complexes as a data model for spatial
databases.