concept lattice (original) (raw)

Let G and M be sets whose elements we call objects and attributes respectively. Let I⊆G×M. We say that object g∈G has attribute m∈M iff (g,m)∈I. The triple (G,M,I) is called a context. For any set X⊆G of objects, define

In other words, X′ is the set of all attributes that are common to all objects in X. Similarly, for any set Y⊆M of attributes, set

In other words, Y′ is the set of all objects having all the attributes in M. We call a pair (X,Y)⊆G×M a concept of the context (G,M,I) provided that

If (X,Y) is a concept, then X is called the extent of the concept and Y the intent of the concept.

Given a context (G,M,I). Let 𝔹⁢(G,M,I) be the set of all concepts of (G,M,I). Define a binary relation ≤ on 𝔹⁢(G,M,I) by (X1,Y1)≤(X2,Y2) iff X1⊆X2. Then ≤ makes 𝔹⁢(G,M,I) a lattice, and in fact a complete lattice. 𝔹⁢(G,M,I) together with ≤ is called the concept latice of the context (G,M,I).