# 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 $\mathbb{B}(G,M,I)$ be the set of all concepts of $(G,M,I)$. Define a binary relation^{} $\le $ on $\mathbb{B}(G,M,I)$ by $({X}_{1},{Y}_{1})\le ({X}_{2},{Y}_{2})$ iff ${X}_{1}\subseteq {X}_{2}$. Then $\le $ makes $\mathbb{B}(G,M,I)$ a lattice^{}, and in fact a complete lattice^{}. $\mathbb{B}(G,M,I)$ 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 |