Modelling spatial relations and operations with partially ordered sets

Some Results on the Modelling of Spatial Data

“…The result of Theorem 3, in the restricted formulation for simplicial complexes, where a universal partition always exists, was proved by Kainz, Egenhofer and Greasley [13]. An obvious consequence of Theorem 3 is that ∀s 1 …”

“…In a poset that represents such a class there is an element for each of the atomic units of the spatial subdivision of S. We show that when a class S includes a spatial subdivision, the normal completion of its poset representation is a correct representation of S . This was the case discussed by Kainz, Egenhofer and Greasley [13] since they modeled spatial regions by means of simplicial complexes, that include naturally a spatial subdivision.…”

“…In particular, the use of the lattice completion (or normal completion) of a poset has been shown by Kainz, Egenhofer and Greasley [13] to be a fundamental technique to build meaningful representation of spatial subdivisions. They proposed to represent by means of the elements introduced by the normal completion operator, the set-intersection between sets of the class.…”

“…The use of posets as a modelling structure for realities in spatial databases is largely widespread [2,13,8,15,19,11]. Also, a discrete basis for the sets of the class, which is analogous to the universal partition we introduce in Sect.…”

“…Formal methods based on the mathematical theory of partially ordered sets (i.e. posets) have been used for the description of spatial relations since many years [13,8,19].…”