Abstract
Galois connections (or residuated mappings) are of growing interest in various domains related with or relevant from Classification. Among their many uses, we select some topics related with modelization and aggregat ion of dissimilarities and conceptual classification. We partially revisit them in a common frame provided by a recent study about Galois connections between closure spaces.
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
BARBUT, M. and MONJARDET, B. (1970): Ordre et classification, Algebre et combinatoire. Hachette, Paris.
BARTHÉLEMY, J.P., LECLERC, B. and MONJARDET, B. (1986): On the use of ordered sets in problems of comparison and consensus of classifications. Journal of Classification, Vol. 3, 187–224.
BIRKHOFF, G. (1967): Lattice Theory (3rd ed.). Amer. Math. Soc, Providence, R.I.
BLYTH, T.S. and JANOWITZ, M.F. (1972): Residuation Theory. Pergamon Press, Oxford.
BOCK, H.H. and DIDAY, E. (2000): Analysis of symbolic data. Studies in Classification, Data analysis and Knowledge organization, Springer, Berlin.
BOORMAN, S.A. and OLIVIER, D.C. (1973): Metrics on spaces of finite trees. Journal of Mathematical Psychology, Vol. 10, 26–59.
BRITO, P. (1995): Symbolic objects: order structure and pyramidal clustering. Annals of Operations Research, Vol. 55, 277–297.
CASPARD, N. and MONJARDET, B. (2001): The lattice of closure systems, closure operators and implicational systems on a finite set; a survey. Discrete Applied Math., to appear.
DIDAY, E. (1988): The symbolic approach in clustering and related methods of data analysis: the basic choices, in Bock (Ed.), Classification and Related Methods od Data Analysis, North-Holland, Amsterdam, 673–683.
DOMENACH, F., and LECLERC, B. (2001): Biclosed binary relations and Galois connections. Order, Vol. 18, 89–104.
DUQUENNE, V. (1987): Contextual implications between attributes and some representation properties for finite lattices, in Ganter et al. (Eds.), Beitrage zur Begriffsan alyse, Wissenchaftverlag, Mannheim, 213–240.
GANTER, B., and WILLE, R. (1999): Formal concept analysis. Springer-Verlag, Berlin.
GUÉNOCHE, A. (1993): Hiérarchies conceptuelles de données binaires. Math. Inf. Sci. hum., Vol. 121, 23–34.
JANOWITZ, M.F. (1978): An order theoretic model for cluster analysis. SIAM J. Appl. Math., Vol. 37, 55–72.
JANOWITZ, M.F. and WILLE, R. (1995): On the classification of monotone and equivariant cluster methods, in Cox, Hansen, Julesz (Eds.), Partitioning data sets, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 19, Amer. Math. Soc., Providence, RI, 117–141.
LECLERC, B. (1991): Aggregation of fuzzy preferences: a theoretic Arrow-like approach. Fuzzy sets and systems, Vol. 43, 291–309.
LECLERC, B. (1994): The residuation model for the ordinal construction of dissimilarities and other valued objects, in Van Cutsem (Ed.), Classification and Dissimilarity Analysis, Springer-Verlag, New York, 149–172.
LECLERC, B., and MONJARDET, B. (1995): Latticial theory of consensus, in Social choice, Welfare and Ethics, in Barnett, Moulin, Salles, Schofield (Eds.), Cambridge University Press, 145–159.
MEPHU NGUIFO, E. (1993): Une nouvelle approche basee sur Ie treillis de Galois pour l’apprentissage de concepts. Math. lni. Sci. hum., Vol. 124, 19–63.
ÖRE, O. (1944): Galois connections. Trans. Amer. Math. Soc., Vol. 55, 494–513.
POLAILLON, G. (1998): Interpretation and reduction of Galois lattices of complex data, in Rizzi, Vichi, Bock (Eds.), Advances in Data Science and Classification, Studies in Classification, Data Analysis and Konwledge Organization, Springer-Verlag, Berlin, 433–440.
WILLE, R. (1982): Restructuring lattice theory: an approach based on hierarchies of concepts, in Rival (Ed.), Ordered Sets, D. Reidel, Dordrecht, 445–470.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Domenach, F., Leclerc, B. (2003). On the Roles of Galois Connections in Classification. In: Schwaiger, M., Opitz, O. (eds) Exploratory Data Analysis in Empirical Research. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55721-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-55721-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44183-0
Online ISBN: 978-3-642-55721-7
eBook Packages: Springer Book Archive