concept lattice

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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\textrm{ 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\textrm{ 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\quad\textrm{and}\quad 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 $\mathbb{B}(G,M,I)$ be the set of all concepts of $(G,M,I)$ . Define a binary relation $\leq$ on $\mathbb{B}(G,M,I)$ by $(X_{1},Y_{1})\leq(X_{2},Y_{2})$ iff $X_{1}\subseteq X_{2}$ . Then $\leq$ makes $\mathbb{B}(G,M,I)$ a lattice, and in fact a complete lattice. $\mathbb{B}(G,M,I)$ together with $\leq$ is called the concept latice of the context $(G,M,I)$ .

Defines:

object, attribute, context, concept, extent, intent

Type of Math Object:

Definition

Major Section:

Reference

Mathematics Subject Classification

68Q55 no label found68P99 no label found08A70 no label found06B23 no label found03B70 no label found06A15 no label found

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