concept lattice

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

$$X^{\prime }:=\{m\in M\mid (x,m)\in I\text{ for all }x\in G\}.$$

In other words, $X^{\prime }$ is the set of all attributes that are common to all objects in $X$. Similarly, for any set $Y\subseteq M$ of attributes, set

$$Y^{\prime }:=\{g\in G\mid (g,y)\in I\text{ for all }y\in M\}.$$

In other words, $Y^{\prime }$ is the set of all objects having all the attributes in $M$. We call a pair $(X,Y)\subseteq G\times M$ a concept of the context $(G,M,I)$ provided that

$$X^{\prime }=Y\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{Y}^{\prime }=X.$$

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 $\le$ on $𝔹(G,M,I)$ by $(X_1,Y_1)\le (X_2,Y_2)$ iff $X_1\subseteq X_2$. Then $\le$ makes $𝔹(G,M,I)$ a lattice, and in fact a complete lattice. $𝔹(G,M,I)$ together with $\le$ 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