Modelling Spatial Relations and Operations with Partially Ordered Sets

' Formal methods for the description of spatial relations can be based on mathematical theories of order. queries about inclusion and coxtainment of spatiai areas. afteia u.iijni-or.tion to the basic concepts ol posets and lattices, their applications to modeilin! rputrut relations and operations for sparial regions in terms oicon. nover approach to characterize certain topological relations baseo on ulut-t-i". ola simplicial complex is a model for spatiar iegio-ns that combrnes ilh 6;i;;Ja uno order rerations and ailows spatiar queries to be answered in a uninea wiv.-' l. Introduction The modelling of,spatial relations in geographical information systems (GIS) is based on fundamental mathematical theor]es. over the past two decades it has been extensivelv invesrigated^l:*_ l: apply topol:gy {".-"a.rii"g rp"tial rerationships (corbett 1979, white 1990, r9g4,-p.a"k and Kuhn 19g6, Egenhofer and Franzosa 1991)' The findings of these investigations have significantly contributed to the state-of-the-art data models used in modern GIS, such as Arc/Info (Morehouse l9g5), TIGRIS (Herring 1987), System 9 (charrwoo d ei at.19g7) and smailworld GIS (Newell er a/. 1991), and the U.s. SDTS spatiar data exchange standard (Rossmeissl and Rugg l99r). At the same time, only littre attention has been p)ra i"-i-rr" application of complementary or alternative mathematical concepts to GIS and spatial analysis such as metric or order. Arthough white's (1 9g4) landmark paper on the'use of topology for cartography mentioned partially ordered sets. and a"spite potential for interesting spatial applications of partially ordered sets, this topic has gained only little attention from GIS researchers. Saalfeld (19g3 a, b, 19b5), Meixrer uni suutr"to (r9g5), Greasley (1988' 1990) and Kainz.(r9gg, r9g9, r990) investigated some aspects of partiar order relations in GIS. Nevertheless, to daie there exist n"o "o.pr"rr".rri* r""i"r" of partially ordered sets and assessment of their usefulness in GIS applications. This article attempts to fill this gap by providing an introduction to the matt "matlc, of partially ordered sets and demonstrating theii application to modelling *it"r relationships in The use of partially ordered sets is motivated here by an example to model spatial subdivisions of land. A spatial subdivision is commonly thought of u, u hierarchy in t current address: Department of Geoinformalic-s, International Institute for Aerospace Survey and Earth Sciencei (ITC) p.o. s"-i; Ni-2s00 AA Enschede, The Netherlands.