concept lattice

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

X:={mM(x,m)I for all xG}.

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

Y:={gG(g,y)I for all yM}.

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 relationMathworldPlanetmath on 𝔹(G,M,I) by (X1,Y1)(X2,Y2) iff X1X2. Then makes 𝔹(G,M,I) a latticeMathworldPlanetmath, and in fact a complete latticeMathworldPlanetmath. 𝔹(G,M,I) together with is called the concept latice of the context (G,M,I).

Title concept lattice
Canonical name ConceptLattice
Date of creation 2013-03-22 19:22:34
Last modified on 2013-03-22 19:22:34
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 68Q55
Classification msc 68P99
Classification msc 08A70
Classification msc 06B23
Classification msc 03B70
Classification msc 06A15
Defines object
Defines attribute
Defines context
Defines concept
Defines extent
Defines intent