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References
A. V. Borovik, I. M. Gelfand and White, N. (2003). Coxeter matroids. Birkhauser.
Aichholzer, O. and Aurenhammer, F. (1996). Classifying hyperplanes in hyper-cubes. SIAM J. Discrete Math., 9:225–232.
Bandelt, H.-J. and Pesch, E. (1989). Dismantling absolute retracts of reflexive graphs. European J. Combinatorics, 10:211–220.
Bell, J. L. (1986). Anew approach to quantum logic. Brit. J. Phil. Sci., 37:83–99.
Biacino, L. and Gerla, G. (1991). Connection structures. Notre Dame J. Formal Logic, 37:431–439.
Birkhoff, G. (1948). Lattice Theory, revised edition. Am. Math. Soc. Publications.
Birkhoff, G. and von Neumann, J. (1936). The logic of quantum mechanics. Ann. Math., 37:823–843.
Björner, A., Vergnas, M. Las, Sturmfels, B., White, N., and Ziegler, G. (1993). Oriented Matroids. Cambridge University Press.
Bland, R. G. and Vergnas, M. Las (1978). Orientability of matroids. J. Combin. Theory Ser. B, 24(1):94–123.
Brandstädt, A., Le, V., and Spinrad, J. (1999). Graph Classes. SIAM Monographs in Discrete Mathematics and Applications.
Brightwell, G. (2000). Gibbs measures and dismantlable graphs. J. Comb. Theory, Series B, 78:141–166.
Brown, K. S. (1989). Buildings. Springer-Verlag.
Čech, E. (1966). Topological Spaces. John Wiley.
Clarke, B. (1981).Acalculus of individuals based on “connection”. Notre Dame J. Formal Logic, pages 204–218.
Clarke, B. (1985). Individuals and points. Notre Dame J. Formal Logic, 26: 61–75.
Coecke, B., Moore, D., and Wilce, A. (2000). Operational quantum logic:an overview. In Coecke, B., Moore, D., and Wilce, A., editors, Current Research in Operational Quantum Logic: Algebras, Categories, Languages. Kluwer.
Cohen, D. W. (1989). An Introduction to Hilbert Space and Quantum Logic. Springer-Verlag.
Cohn, A., Bennett, B., Gooday, J., and Gotts, N. (1997). RCC: a calculus for region based qualitative spatial reasoning. GeoInformatica, 1:275–316.
Coppel, W. (1998). Foundations of Convex Geometry. Cambridge University Press.
Duchet, P. (1988). Convex sets in graphs II, minimal path convexity. J. Combin. Theory Series B, 44:307–316.
Duchet, P. and Meyniel, H. (1983). Ensembles convexes dans les graphes I. European J. Combinatorics, pages 127–132.
Dvurečenskij, A. and Pulmannová, S. (2000). New Trends in Quantum Structures. Kluwer.
Engelking, R. (1989). General Topology. Heldermann, Berlin, revised edition edition.
Evako, A., Kopperman, R., and Mukhin, Y. V. (1996). Dimensional properties of graphs and digital spaces. J. Math. Imaging and Vision, 6:109–119.
Evako, A. V. (1994). Dimension on discrete spaces. Int. J. Theoretical Physics, 33:1553–1568.
Faure, C.-A. and Frölicher, A. (2000). Modern projective geometry. Kluwer.
Folkman, J. and Lawrence, J. (1978). Oriented matroids. J. Combin. Theory Ser. B, 25(2):199–236.
Foulis, D. J. (1999). A half century of quantum logic: what have we learned? In Aerts, D. and Pykacs, J., editors, Quantum Structures and the Nature of Reality, pages 1–36. Kluwer.
Foulis, D. J. and Randell, C. (1971). Lexicographic orthogonality. J. Combinatorial Theory, pages 157–162.
Georgatos, K. (2003). On indistinguishability and prototypes. Logic J. of the IGPL, 11:531–545.
Hell, P. and Nešetřil, J. (2004). Graphs and Homomorphisma. Oxford University Press.
Hurewicz, W. and Wallman, H. (1948). Dimension Theory. Princeton University Press.
Kalmbach, G. (1983). Orthomodular Lattices. Academic Press.
Khalimsky, E., Kopperman, R., and Meyer, P. (1990). Computer graphics and connected topologies on ordered sets of points. Topology Appl., 35:1–17.
Knuth, D. E. (1991). Axioms and Hulls. Lecture Notes in Computer Science 606. Springer-Verlag.
Lane, S. Mac and Birkhoff, G. (1967). Algebra. Macmillan, 2nd edition.
Lawvere, F. (1991). Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. (Springer) Lect. Notes in Math., 1488:279–281.
Markopoulou, F. and Smolin, L. (1997). Causal evolution of spin networks. Nucl. Phys.B508, page 409.
Martin, N. N. and Pollard, S. (1996). Closure Spaces and Logic. Kluwer.
Oxley, J. (1992a). Infinite matroids. In White, N., editor, Matroid Applications, pages 73–90. Cambridge University Press.
Oxley, J. (1992b). Matroid Theory. Oxford University Press.
Pagliani, P. (1998). Intrinsic co-Heyting boundaries and informatin incompleteness in rough set analysis. In Polkowski, L. and Skowron, A., editors, RSTC’98, volume 1424 of LNAI, pages 123–130.
Penrose, R. (1971). Angular momentum: an approach to combinatorial space-time. In Quantum Theory and Beyond. Cambridge University Press.
Pfaltz, J. L. (1996). Closure lattices. Discrete Mathematics, 154:217–236.
Poincaré, H. (1905). La Valeur de la Science. Flammarion, Paris.
Poston, T. (1971). Fuzzy Geometry. PhD thesis, University of Warwick.
Pratt, I. and Lemon, O. (1997). Ontologies for plane, polygonal mereotopology. Notre Dame J. Formal Logic, 38(2):225–245.
Prenowitz, W. and Jantosciak, J. (1979). Join Geometries. Springer-Verlag.
Pták, P. and Pulmannová, S. (1991). Orthomodular Structures as Quantum Logics. Kluwer.
Pultr, A. (1963). An analogon of the fixed-point theorem and its application for graphs. Comment.Math. Univ. Carol.
Quilliot, A. (1983). Homomorphismes, points fixes, rétractions et jeux de pour-suite dans les graphes. PhD thesis, Paris.
Randell, D., Cui, Z., and Cohn, A. (1992). A spatial logic based on regions and connection. In Proc. 2nd Int. Conf. on Knowledge Representation and Reasoning, pages 165–176.
Reyes, G. and Zolfaghari, H. (1996). Bi-Heyting algebras, toposes and modalities. J. Philos. Logic, 25.
Ronan, M. (1989). Lectures on Buildings. Academic Press.
Rosenfeld,A. (1986). “Continuous” functions on digital pictures. Pattern Recog. Letters, 4:177–184.
Roy, A. and Stell, J. (2002). A qualitative account of discrete space. In Proc. 2nd Int. Conf. on Geographic Information Science, volume 2478 of LNCS, pages 276–290. Springer.
Smolin, L. (2001). Three Roads to Quantum Gravity. Weidenfeld and Nicholson.
Smyth, M. B. (1995). Semi-metrics, closure spaces and digital topology. Theoret. Comput. Sci., 151:257–276.
Smyth, M. B. (1997). Topology and tolerance. Electron. Notes in Theoret. Comput. Sci.,6.
Smyth, M. B. (2000). Region-based discrete geometry. J. Universal Comput. Sci., 6:447–459.
Smyth, M. B. and Tsaur, R. (2001–2002). Hyperconvex semi-metric spaces. Topology Proceedings, 26:791–810.
Smyth, M. B. and Webster, J. (2002). Finite approximation of stably compact spaces. Applied General Topology, 3:1–28.
Sorkin, R. (2002). Causal sets: discrete gravity. In Gomberoff, A. and Marolf, D., editors, Valdivia Summer School, 2002 (to appear).
Sossinsky, A. (1986). Tolerance space theory and some applications. Acta Applicandae Math., 5:137–167.
Stell, J. (2000). Boolean connection algebras: a new approach to the Region-Connection Calculus. Artificial Intelligence, 122:111–136.
Stell, J. and Worboys, M. (1997). The algebraic structure of sets of regions. Lect. Notes in Comp. Sci., 1329:163–174.
Stolfi, J. (1991). Oriented projective geometry. Academic Press.
Sumner, R. (1974). Dacey graphs. J. Australian Math. Soc., 18:492–502.
Tsaur, R. and Smyth, M. (2001). “Continuous” multifunctions in discrete spaces, with applications to fixed point theory. In Bertrand, G., Imiya, A., and Klette, R., editors, Digital and Image Geometry, volume 2243 of LNCS, pages 75–88. Springer.
Tsaur, R. and Smyth, M. (2004). Convexity in Helly graphs. In MFCSIT 2004 (to appear).
van de Vel, M. (1993). Theory of Convex Structures. Elsevier, Amsterdam.
Vergnas, M. Las (1980). Convexity in oriented matroids. J. Combin. Theory Ser. B, 29(2):231–243.
Webster, J. (1997). Topology and measure theory in the digital setting: on the approximation of spaces by inverse sequences of graphs. PhD thesis, Imperial College.
Webster, R. (1995). Convexity. Oxford University Press.
Whitney, H. (1935). On the abstract properties of linear dependence. American Journal of Mathematics, 57:509–533.
Wilce, A. (2004). Topological test spaces. To appear in: Int. J. Theor. Physics.
Zeeman, E. C. (1962). The topology of the brain and visual perception. In Fort, M. K., editor, Topology of 3-manifolds. Prentice Hall, NJ.
Ziegler, G. M. (1995). Lectures on Polytopes. Springer.
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Smyth, M., Webster, J. (2007). Discrete Spatial Models. In: Aiello, M., Pratt-Hartmann, I., Van Benthem, J. (eds) Handbook of Spatial Logics. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5587-4_12
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