Concept Lattices Constrained by Equivalence Relations

Formal concept analysis is a method of exploratory data analysis that aims at the extraction of natural clusters from object- attribute data tables. The clusters, called formal concepts, are naturally interpreted as human-perceived concepts in a traditional sense and can be partially ordered by a subconcept-superconcept hierarchy. The hierar- chical structure of formal concepts (so-called concept lattice) represents a structured information obtained automatically from the input data ta- ble. This paper presents a preliminary study in which we deal with the problem of how further information additionally supplied with the ba- sic object-attribute data table can be utilized. The additional informa- tion we consider has the form of a binary relation on the set of ob- jects. Primarily, we focus on equivalence relations. Equivalences can be used modeling similarity, indistinguishability, etc.—a kind of informa- tion quite often supplied/available with a collection of objects. We aim at emphasizing two aspects. First, the additional information can provide a criterion for the relevance/importance of formal concepts. Only con- cepts which are in an appropriate sense compatible with the additional information are considered important. Second, selecting only important concepts means a reduction of the overall concept lattice which helps to make the resulting set of formal concepts more readable.