Abstract
The aim of this paper is to analyse the closure system of extents of direct products of convex-ordinal scales. By definition the extents of a convex-ordinal scale are exactly the convex subsets of its underlying ordered set. The first characterization describes the extent system of a direct product of convex-ordinal scales as the smallest closure system that contains the convex subsets of the direct product of the underlying ordered sets and is invariant under dualization of arbitrary factors. Another description is inspired by the fact that the extents of direct products of convex-ordinal scales constitute a convex geometry. A result on extreme points of direct products of convex-ordinal scales leads to a geometric description of their extent spaces. The steps that are necessary to generate closures in extent spaces can be described by so-called pseudo-extents together with their closures. It turns out that the pseudo-extents of direct products of convex-ordinal scales are exactly the minimal sets that are not closed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birkhoff, G. (1967)Lattice Theory (3rd ed.) Amer. Math. Soc., Providence, R.I.
Birkhoff, G. and Bennett, M. K. (1985) The convexity lattice of a poset,Order 2, 223–242.
Duquenne, V. and Guiges, J. L. (1986) Families minimales d'implications informatives résultant d'un tableau de données binaires,Math. Sci. Hum. 95, 5–18.
Edelman, P. H. (1980) Meet-distributive lattices and the anti-exchange closure,Algebra Universalis 10, 239–244.
Edelman, P. H. and Jamison, R. E. (1985) The theory of convex geometries,Geom. Dedicata 19, 247–270.
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, New York, pp. 139–167.
Ganter, B. and Wille, R. Formale Begriffsanalyse (B.I.-Wissenschaftsverlag, to appear).
Jamison-Waldner, R. E. (1980) A convexity characterization of ordered sets, Proc. of the 10th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Congressus Numerantium 29, 529–540.
Jamison, R. E. and Strahringer, S. Representing graphs in disjunctive products (in preparation).
Korte, B., Lovasz, L. and Schrader, R. (1991)Greedoids, Springer, Berlin-Heidelberg.
Strahringer, S. Dimensionality of ordinal structures (to appear inDiscrete Math.).
Strahringer, S. and Wille, R. (1991) Convexity in ordinal data, in: H. H. Bock and P. Ihm (eds.),Classification, Data Analysis, and Knowledge Organization, Springer, Berlin-Heidelberg, 113–120.
Strahringer, S. and Wille, R. (1992) Towards a structure theory for ordinal data, in: M. Schader, (ed.),Analyzing and Modeling Data and Knowledge, Springer, Heidelberg, 129–139.
Strahringer, S. and Wille, R. (1993) Conceptual clustering via convex-ordinal structures, in: Opitz, Lausen, Klar (eds.),Information and Classification. Concepts, Methods and Applications, Springer, Berlin-Heidelberg, 85–93.
Wille, R. (1982) Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (ed.),Ordered Sets, Reidel, Dordrecht-Boston, 445–470.
Wille, R. (1985) Tensorial decomposition of concept lattices,Order 2, 81–95.
Author information
Authors and Affiliations
Additional information
Communicated by I. Rival
Rights and permissions
About this article
Cite this article
Strahringer, S. Direct products of convex-ordinal scales. Order 11, 361–383 (1994). https://doi.org/10.1007/BF01108768
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01108768