Geometric data models form the backbone of virtually all spatial information systems, such as GIS, CAD, and CAM. Yet a lot of spatial information from textual sources, including historical documents or social media, is predominantly of qualitative, especially mereotopological, rather than geometric-quantitative nature. While mereotopological theories have been extensively studied in Logic, Computer Science, Cognitive Science, and Geographic Information Science, most are unidimensional mereotopologies in the sense that only entities of a single dimension are permitted to co-exist. Integrating mereotopological information with geometric data requires a multidimensional mereotopology, which permits entities of different dimensions to co-exist, similarly to how geometric and algebraic topological data models permit points, simple lines, polylines, cells, polygons, and polyhedra to co-exist. It further requires complex spatial objects to be represented as sets of atomic entities such that spatial relations between complex objects can be computed from the relations of the atomic entities in their decomposition.
This paper provides a comprehensive study of CODI, a first-order logic ontology of multidimensional mereotopology. An axiomatization of mereological closure operations of intersection, difference, and sums for CODI is proposed in which these operations apply to all pairs of spatial entities regardless of their dimension. It is proved that for atomic models – and thus all finite models – the extended theory is indeed able to decompose all spatial entities into a partition of atomic parts. A full representation of the models as sets of Boolean algebras verifies this. The closure operations are further shown to satisfy important mereological principles from unidimensional mereotopology and to preserve many of the mathematical properties of set intersection and set difference.
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