Geometric Representation of Concept Lattices

A fundamental method of data analysis is to visualize structures of data by geometric configurations in the plane. Since concept lattices are basic structures for data contexts, it is desirable to study systematically the geometric representation for concept lattices. In this paper we want to understand the notion of geometric representation to be based on the containment of figures; more specifically, a geometric representation of a concept lattice shall be given by a collection of figures in the plane corresponding to the concepts in such a way that the subconcept-superconcept-relation corresponds to the containment relation between the figures. It is most common to represent objects by points in which case an object has a given attribute if and only if the figure corresponding to the concept generated by this attribute contains the point which represents the object. Although the line diagram has been proven to be a general and successful tool for representing concept lattices (cf. [Wi84]), there are situations where the geometric representation is also useful and even more appropriate.