Abstract
Formal Concept Analysis is a mathematical theory of concept hierarchies which is based on Lattice Theory. It has been developed to support humans in their thought and knowledge. The aim of this paper is to show how successful the lattice-theoretic foundation can be in applying Formal Concept Analysis in a wide range. This is demonstrated in three sections dealing with representation, processing, and measurement of conceptual knowledge. Finally, further relationships between abstract Lattice Theory and Formal Concept Analysis are briefly discussed.
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
Barbut, M., Monjardet, B.: Ordre et Classification. Algébre et Combinatoire. Collection Hachette Université. Paris, Librairie Hachette (1970)
Becker, P., Hereth Correia, J.: The ToscanaJ Suite for implementing conceptual information systems. In: [GSW 2005], pp. 324–348 (2005)
Becker, T.: Formal Concept Analysis and Algebraic Geometry. Dissertation, TU Darmstadt. Shaker, Aachen (1999)
Becker, T.: Features of interaction between Formal Concept Analysis and Algebraic Geometry. In: [GSW 2005], pp. 49–80 (2005)
Becker, T.: Polynomial embeddings and representations. In: [KS 2007], pp. 281–303 (2007)
Birkhoff, G.: Lattice theory, 1st edn. American Mathematical Society, Providende (1940)
Burmeister, P.: Formal Concept Analysis with ConImp: Introduction to the basic features. TU Darmstadt (2003), http://www.mathematik.tu-darmstadt.de/~Burmeister
Chomsky, N., Halle, M.: The sound pattern of English. Harper & Row, New York (1968)
Davey, B.A., Priestley, H.A.: Introduction to lattices and order, 2nd edn. Cambridge University Press, Cambridge (2002)
Denecke, K., Erné, M., Wismath, S.L. (eds.): Galois connections and applications. Kluwer, Dordrecht (2004)
Eklund, P., Wille, R.: Semantology as basis for Conceptual Knowledge Processing. In: Kuznetsov, S.O., Schmidt, S. (eds.) ICFCA 2007. LNCS (LNAI), vol. 4390, pp. 18–38. Springer, Heidelberg (2007)
Ganter, B., Stumme, G., Wille, R. (eds.): Formal Concept Analysis: foundations and applications. State-of-the-Art Survey. LNCS (LNAI), vol. 3626. Springer, Heidelberg (2005)
Ganter, B., Wille, R.: Implikationen und Abhängigkeiten zwischen Merkmalen. In: Degens, P.O., Hermes, H.-J., Opitz, O. (eds.) Die Klassifikation und ihr Umfeld, pp. 171–185. Indeks, Frankfurt (1986)
B. Ganter, R. Wille: Conceptual scaling. In: F. Roberts (ed.): Applications of combinatorics and graph theory in the biological and social sciences. Springer, New York 1989, 139–167.
Ganter, B., Wille, R.: Formal Concept Analysis: mathematical foundations. Springer, Heidelberg (1999)
Wille, R., Gehring, P.: Semantology: Basic Methods for Knowledge Representations. In: Schärfe, H., Hitzler, P., Øhrstrøm, P. (eds.) ICCS 2006. LNCS (LNAI), vol. 4068, pp. 215–228. Springer, Heidelberg (2006)
Guigues, J.-L., Duquenne, V.: Familles minimales d’implications informatives resultant d’un tableau de données binaires. Math. Sci. Humaines 95, 5–18 (1986)
Großkopf, A.: Conceptual structures of finite abelian groups. Dissertation, TU Darmstadt. Shaker, Aachen (1999)
Hartung, G.: A topological representation of lattices. Algebra Universalis 29, 273–299 (1992)
von Hentig, H.: Magier oder Magister? Über die Einheit der Wissenschaft im Verständigungsprozess. Suhrkamp, Frankfurt (1974)
Kant, I.: Logic. Dover, Mineola (1988)
Kearnes, K.A., Vogt, F.: Bialgebraic contexts from dualities. Australian Math. Society (Series A) 60, 389–404 (1996)
Kollewe, W., Skorsky, M., Vogt, F., Wille, R.: TOSCANA - ein Werkzeug zur begrifflichen Analyse und Erkundung von Daten. In: [WZ 1994], pp. 267–288 (1994)
Krantz, D., Luce, D., Suppes, P., Tversky, A.: Foundation of measurement, vol. 1, vol. 2, vol. 3. Academic Press, London (1971) (1989) (1990)
Kuznetsov, S., St Schmidt, E.: ICFCA 2007. LNCS (LNAI), vol. 4390. Springer, Heidelberg (2007)
Luksch, P., Wille, R.: Substitution decomposition of concept lattices. In: Contributions to General Algebra, vol. 5, pp. 213–220. Hölder-Pichler-Tempsky, Wien (1987)
Luksch, P., Wille, R.: Formal concept analysis of paired comparisons. In: Bock, H.H. (ed.) Classification and related methods of data analysis, pp. 567–576. North-Holland, Amsterdam (1988)
Piaget, J.: Genetic Epistomology. Columbia University Press, New York (1970)
Scheich, P., Skorsky, M., Vogt, F., Wachter, C., Wille, R.: Conceptual data systems. In: Opitz, O., Lausen, B., Klar, R. (eds.) Information and classification. Concepts, methods and applications, pp. 72–84. Springer, Heidelberg (1993)
Schiffmann, H., Falkenberg, Ph.: The organization of stimuli and sensory neurons. Physiology and Behavior 3, 197–201 (1968)
Scott, D.: Measurement structures and linear inequalities. Journal of Mathematical Psychologie 1, 233–244 (1964)
Stahl, J., Wille, R.: Preconcepts and set representations of concepts. In: Gaul, W., Schader, M. (eds.) Classification as a tool of research, North-Holland, Amsterdam (1986)
Stumme, G., Wille, R. (Hrsg.): Begriffliche Wissensverarbeitung: Methoden und Anwendungen. Springer, Heidelberg (2000)
Urquhart, A.: A topological representation theory for lattices. Algebra Universalis 8, 45–58 (1978)
Vogt, F.: Bialgebraic contexts. Dissertation, TU Darmstadt. Shaker, Aachen (1994)
Vogt, F.: Subgroup lattices of finite Abelian groups. In: Baker, K.A., Wille, R. (eds.) Lattice theory and its applications, pp. 241–259. Heldermann Verlag, Lemgo (1995)
Vogt, F., Wachter, C., Wille, R.: Data analysis based on a conceptional file. In: Bock, H.H., Ihm, P. (eds.) Classification, data analysis, and knowledge organisation, pp. 131–142. Springer, Heidelberg (1991)
Vogt, F., Wille, R.: Ideas of Algebraic Concept Analysis. In: Bock, H.-H., Lenski, W., Richter, M.M. (eds.) Information systems and data analysis, pp. 193–205. Springer, Heidelberg (1994)
Vogt, F., Wille, R.: TOSCANA – A graphical tool for analyzing and exploring data. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 226–233. Springer, Heidelberg (1995)
Weizsäcker, C.F.: Möglichkeit und Bewegung. Eine Notiz zur aristotelischen Physik. In: C. F. Weizsäcker: Die Einheit der Natur. 3. Aufl. Hanser, München, 428-440 (1972)
Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. Reidel, Dordrecht-Boston (1982)
Wille, R.: Liniendiagramme hierarchischer Begriffssysteme. In: Bock, H.H.(ed.) (Hrsg.): Anwendungen der Klassifikation: Datenanalyse und numerische Klassifikation. pp. 32–51 Indeks-Verlag, Frankfurt (1984); Übersetzung ins Englische: Line diagrams of hierachical concept systems. International Classification 11, 77–86 (1984)
Wille, R.: Lattices in data analysis: How to draw them with a computer. In: Rival, I. (ed.) Algorithms and order, pp. 33–58. Kluwer, Dordrecht (1989)
Wille, R.: Concept lattices and conceptual knowledge systems. Computers & Mathematics with Applications 23, 493–515 (1992)
Wille, R.: Dyadic mathematics - abstractions of logical thought. In: Denecke, K., Erné, M., Wismath, S.L. (eds.) Galois Connections and Applications, pp. 453–498. Kluwer, Dordrecht (2004)
Wille, R.: Formal Concept Analysis as mathematical theory of concepts and concept hierarchies. In: [GSW 2005], pp. 1–33 (2005)
Wille, R.: Methods of Conceptual Knowledge Processing. In: Missaoui, R., Schmidt, J. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3874, pp. 1–29. Springer, Heidelberg (2006)
Wille, R.: The basic theorem on labelled line diagrams of finite concept lattices. In: [KS 2007], pp. 303–312 (2007)
Wille, R.: Formal Concept Analysis of one-dimensional continuum structures. Algebra Universalis (to appear)
Wille, R., Wille, U.: On the controversy over Huntington’s equations. When are such equations meaningful? Mathematical Social Sciences 25, 173–180 (1993)
Wille, R., Wille, U.: Uniqueness of coordinatizations of ordinal structures. Contributions to General Algebra 9, 321–324 (1995)
Wille, R., Wille, U.: Coordinatization of ordinal structures. Order 13, 281–284 (1996)
Wille, R., Wille, U.: Restructuring general geometry: measurement and visualization of spatial structures. In: Contributions to General Algebra 14, pp. 189–203. Johannes Heyn Verlag, Klagenfurt (2003)
Wille, R., Zickwolff, M. (eds.): Begriffliche Wissensverarbeitung: Grundfragen und Aufgaben. B.I.-Wissenschaftsverlag, Mannheim (1994)
Wille, U.: Eine Axiomatisierung bilinearer Kontexte. Mitteilungen des Mathematischen Seminars Gießen 200, 71–112 (1991)
Wille, U.: Geometric representation of ordinal contexts. Dissertation, Univ. Gießen. Shaker Verlag, Aachen (1996)
Wille, U.: Representation of ordinal contexts by ordered n-quasigroups. European Journal of Combinatorics 17, 317–333
Wille, U.: The role of synthetic geometry in representational measurement theory. Journal of Mathematical Psychology 41, 71–78 (1997)
Wille, U.: Characterization of ordered bilinear contexts. Journal of Geometry 64, 167–207 (1999)
Wille, U.: Linear measurement models - axiomatizations and axiomatizability. Journal of Mathematical Psychology 44, 617–650 (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wille, R. (2008). Formal Concept Analysis as Applied Lattice Theory. In: Yahia, S.B., Nguifo, E.M., Belohlavek, R. (eds) Concept Lattices and Their Applications. CLA 2006. Lecture Notes in Computer Science(), vol 4923. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78921-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-78921-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78920-8
Online ISBN: 978-3-540-78921-5
eBook Packages: Computer ScienceComputer Science (R0)