Formal Concept Analysis over Graphs and Hypergraphs

Formal Concept Analysis (FCA) provides an account of classification based on a binary relation between two sets. These two sets contain the objects and attributes (or properties) under consideration. In this paper I propose a generalization of formal concept analysis based on binary relations between hypergraphs, and more generally between pre-orders. A binary relation between any two sets already provides a bipartite graph, and this is a well-known perspective in FCA. However the use of graphs here is quite different as it corresponds to imposing extra structure on the sets of objects and of attributes. In the case of objects the resulting theory should provide a knowledge representation technique for structured collections of objects. The generalization is achieved by an application of work on mathematical morphology for hypergraphs. 1 General Introduction 1.1 Formal Concept Analysis We recall the basic notions of Formal Concept Analysis from [GW99] with some notation taken from [DP11]. Definition 1 A Formal Context K consists of two sets U, V and a binary relation R ⊆ U × V . The elements of U are the objects of K and the elements of V are the properties. The relation can be written in the infix fashion a R b or alternatively as (u, v) ∈ R. The converse of R is denoted R̆. Definition 2 Given a formal context (U,R, V ) then R : PU → PV is defined for any X ⊆ U by R(X) = {v ∈ V : ∀x ∈ X (x R v)}. This operator provides the set of properties which every one of the objects in X possesses. Using the converse of R, the operator R̆ : PV → PU can be described explicitly by R̆(Y ) = {u ∈ U : ∀y ∈ Y (u R y)}