Abstract
Formal contexts are under certain conditions representable by oriented pseudoline arrangements. Oriented pseudoline arrangements respect the hierarchical order of concept lattices. The formal context is reconstructable out of an oriented pseudoline arrangement. The representation of oriented pseudoline arrangements are quite easy to understand. In general there is no sufficient condition to decide if a context is representable (pictorial) or not. There is an infinite class of minor-minimal non-pictorial contexts. This tells us that excluding a finite number of examples is never sufficient to guarantee that a formal context is pictorial. Oriented pseudoline arrangements are isomorphic to oriented matroids of rank 3. We can therefore use an algorithmic approach which is generating oriented matroids of rank 3 and decide wether a context is representable by an oriented pseudoline arrangement.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bokowski, J., Guedes De Oliveira, A., and Richter-Gebert, J. (1991), Algebraic varieties characterizing matroids and oriented matroids, Advances in Mathematics, 87, 160–185.
Bokowski, J., and Kollewe, W. (1992), On representing contexts in line arrangements, Order, (to appear).
Bokowski, J., and Sturmfels, B. (1989), Computational synthetic geometry, Lecture Notes in Mathematics, 1355.
Cordovil, R. (1983), Oriented matroids of rank three and arrangements of pseudolines, Annals of Discrete Math., 17, 219–223.
Feger, H. (1989), Testdaten als Merkmalsvektoren, Beitrag zur Festschrift für Karl Josef Klauer, ‘Wissenschaft und Verantwortung’, Hogrefe, Göttingen.
Folkman, J., and Lawrence, J. (1978), Oriented matroids, J. Combinatorial Theory, B 25, 199–236.
Gabriel, K. R. (1981), Biplot display of multivariate matrices of data and diagnosis, in: V. Barnett (ed.), Interpreting multivariate data, Wiley, Chichester, 147–173.
Ganter, B., and Wille, R. (1989), Conceptual scaling, in: F. Roberts (ed.), Applications of combinatorics and graph theory to the biological and social sciences, Springer Verlag, New York, 139–167
Ganter, B., and Wille, R. (1992), Formale Begriffsanalyse, Manuskript 75p.
Grünbaum, B. (1972), Arrangements and spreads, American Math. Soc, Regional Conf. Ser. 10, Rhode Island.
Kollewe, W. (1989), Evaluation of a survey with methods of formal concept analysis, in: O. Opitz (ed.), Conceptual and numerical analysis of data, Springer Verlag, Berlin-Heidelberg, 123–134.
Ringel, G. (1956), Teilungen der Ebene durch Geraden und topologische Geraden, Math. Zeitschrift, 64, 79–102.
Slater, P. (1977), The measurement of interpersonal space by grid technique, Vol. I and II, Wiley, New York, 1977.
Spangenberg, N. and Wolff, K.E. (1988), Conceptual grid evaluation, in: H.H. Bock (ed.), Classification and related methods of data analysis, North-Holland, Amsterdam, 577–580
Wille, R. (1984), Liniendiagramme hierarchischer Begriffssysteme, in: H. H. Bock (Hrsg.), Anwendungen der Klassifikation: Datenanalyse und numerische Klassifikation, Indeks-Verlag, Frankfurt, 32–51.
Wille, R. (1987), Bedeutungen von Begriffsverbänden, in: B. Ganter, R. Wille, E. Wolff (Hrsg.), Beiträge zur Begriffsanalyse, B.I.-Wissenschaftsverlag, Mannheim-Wien-Zürich, 161–211
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin · Heidelberg
About this paper
Cite this paper
Kollewe, W. (1993). Representation of Data by Pseudoline Arrangements. In: Opitz, O., Lausen, B., Klar, R. (eds) Information and Classification. Studies in Classification, Data Analysis and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-50974-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-50974-2_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56736-3
Online ISBN: 978-3-642-50974-2
eBook Packages: Springer Book Archive