concept lattice

Let $G$ and $M$ be sets whose elements we call objects and attributes respectively. Let $I\subseteq G<span\; class="naked\_sign">\times </span><span\; class="naked\_aural">ばつ</span>M$. 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

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

In other words, $Y^{\prime }$ is the set of all objects having all the attributes in $M$. We call a pair $(X,Y)\subseteq G<span\; class="naked\_sign">\times </span><span\; class="naked\_aural">ばつ</span>M$ 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 ^{Mathworld Planetmath} $\le$ 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 ^{Mathworld Planetmath} , and in fact a ^{Mathworld Planetmath} . $B(G,M,I)$ together with $\le$ is called the concept latice of the context $(G,M,I)$.