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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 $(G,M,I)$.

## Mathematics Subject Classification

68Q55*no label found*68P99

*no label found*08A70

*no label found*06B23

*no label found*03B70

*no label found*06A15

*no label found*

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