Formal concept analysis

Definition

A formal context is a triple $(G, M, I)$ of a set $G$ of objects, a set $M$ of attributes and a binary relation $I \subseteq G \times M$. One reads $g \mathrel{I} m$ as "object $g$ has attribute $m$". For $A \subseteq G$ and $B \subseteq M$, put $$A' = \set{m \in M : g \mathrel{I} m \text{ for all } g \in A}, \qquad B' = \set{g \in G : g \mathrel{I} m \text{ for all } m \in B}.$$ A formal concept of $(G, M, I)$ is a pair $(A, B)$ with $A \subseteq G$, $B \subseteq M$, $A' = B$ and $B' = A$. The set $A$ is its extent and $B$ is its intent. Formal concepts are ordered by $(A_1, B_1) \le (A_2, B_2)$ if and only if $A_1 \subseteq A_2$, which holds if and only if $B_2 \subseteq B_1$. The ordered set of all formal concepts is the concept lattice $\mathfrak{B}(G, M, I)$. Formal concept analysis is the study of formal contexts through their concept lattices.