# 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 $(G,M,I)$.

 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