Abstract
Recently pattern mining has investigated closure operators in families of subsets of an attribute set that are not lattices. In particular, various authors have investigated closure operators starting from a context, in the Formal Concept Analysis (FCA) sense, in which objects are described as usual according to their relation to attributes, and in which a closed element is a maximal element of the equivalence class of elements sharing the same support, i.e. occurring in the same objects. The purpose of this paper is twofold. First we thoroughly investigate this framework and relate it to FCA, defining in particular a structure called a pre-confluence, weaker than a lattice, in which we can define a closure operator with respect to a set of objects. Second, we show that the requirements allowing us to define abstract concept lattices also allow us to define corresponding abstract Galois pre-confluences.
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
Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer (1999)
Caspard, N., Monjardet, B.: The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey. Discrete Appl. Math. 127(2), 241–269 (2003)
Diday, E., Emilion, R.: Maximal and stochastic galois lattices. Discrete Appl. Math. 127(2), 271–284 (2003)
Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Efficient mining of association rules using closed itemset lattices. Information Systems 24(1), 25–46 (1999)
Arimura, H., Uno, T.: Polynomial-delay and polynomial-space algorithms for mining closed sequences, graphs, and pictures in accessible set systems. In: SDM, pp. 1087–1098. SIAM (2009)
Boley, M., Horváth, T., Poigné, A., Wrobel, S.: Listing closed sets of strongly accessible set systems with applications to data mining. Theor. Comput. Sci. 411(3), 691–700 (2010)
Ventos, V., Soldano, H.: Alpha Galois Lattices: An Overview. In: Ganter, B., Godin, R. (eds.) ICFCA 2005. LNCS (LNAI), vol. 3403, pp. 299–314. Springer, Heidelberg (2005)
Blyth, T.S.: Lattices and Ordered Algebraic Structures. Universitext. Springer (2005)
Ferré, S., Ridoux, O.: An introduction to logical information systems. Information Processing and Management 40(3), 383–419 (2004)
Ganter, B., Kuznetsov, S.O.: Pattern structures and their projections. In: Delugach, H.S., Stumme, G. (eds.) ICCS 2001. LNCS (LNAI), vol. 2120, pp. 129–142. Springer, Heidelberg (2001)
Pernelle, N., Rousset, M.C., Soldano, H., Ventos, V.: Zoom: a nested Galois lattices-based system for conceptual clustering. J. of Experimental and Theoretical Artificial Intelligence 2/3(14), 157–187 (2002)
Soldano, H., Ventos, V.: Abstract Concept Lattices. In: Valtchev, P., Jäschke, R. (eds.) ICFCA 2011. LNCS (LNAI), vol. 6628, pp. 235–250. Springer, Heidelberg (2011)
Pasquier, N., Taouil, R., Bastide, Y., Stumme, G., Lakhal, L.: Generating a condensed representation for association rules. Journal Intelligent Information Systems (JIIS) 24(1), 29–60 (2005)
Guigues, J., Duquenne, V.: Famille non redondante d’implications informatives résultant d’un tableau de données binaires. Mathématiques et Sciences Humaines 95, 5–18 (1986)
Negrevergne, B., Termier, A., Rousset, M.C., Méhaut, J.F.: Paraminer: a generic pattern mining algorithm for multi-core architectures. Data Mining and Knowledge Discovery, 1–41 (2013)
Kuznetsov, S.O., Samokhin, M.V.: Learning closed sets of labeled graphs for chemical applications. In: Kramer, S., Pfahringer, B. (eds.) ILP 2005. LNCS (LNAI), vol. 3625, pp. 190–208. Springer, Heidelberg (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Soldano, H. (2014). Closed Patterns and Abstraction Beyond Lattices. In: Glodeanu, C.V., Kaytoue, M., Sacarea, C. (eds) Formal Concept Analysis. ICFCA 2014. Lecture Notes in Computer Science(), vol 8478. Springer, Cham. https://doi.org/10.1007/978-3-319-07248-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-07248-7_15
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07247-0
Online ISBN: 978-3-319-07248-7
eBook Packages: Computer ScienceComputer Science (R0)