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  "body": "\n## Definition\n\nA *formal context* is a triple $(G, M, I)$ of a set $G$ of objects, a set $M$ of attributes and a [[binary relation|05AVARPXA2]] $I \\subseteq G \\times M$.\nOne reads $g \\mathrel{I} m$ as \"object $g$ has attribute $m$\".\nFor $A \\subseteq G$ and $B \\subseteq M$, put\n$$A' = \\set{m \\in M : g \\mathrel{I} m \\text{ for all } g \\in A}, \\qquad B' = \\set{g \\in G : g \\mathrel{I} m \\text{ for all } m \\in B}.$$\nA *formal concept* of $(G, M, I)$ is a pair $(A, B)$ with $A \\subseteq G$, $B \\subseteq M$, $A' = B$ and $B' = A$.\nThe set $A$ is its extent and $B$ is its intent.\nFormal 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 $B_2 \\subseteq B_1$.\nThe ordered set of all formal concepts is the *concept lattice* $\\mathfrak{B}(G, M, I)$.\n*Formal concept analysis* is the study of formal contexts through their concept lattices.\n\n## Remarks\n\nThe two maps $A \\mapsto A'$ and $B \\mapsto B'$ between the [[power sets|00E0YZCERH]] of $G$ and $M$ form an antitone [[Galois connection|Q5JHEPRYQ6]].\nThe extents are the closed sets of the closure operator $A \\mapsto A''$.\nThe concept lattice is a [[complete lattice|D7A0ZGZT5S]]: the meet of concepts $(A_i, B_i)$ is $\\left(\\bigcap_i A_i, \\left(\\bigcup_i B_i\\right)''\\right)$.\nConversely, every complete lattice $L$ is isomorphic to the concept lattice of the context $(L, L, \\le)$.\nExample: for $G = \\set{1, 2}$, $M = \\set{a, b}$ and $I = \\set{(1, a), (2, a), (2, b)}$, the concepts are $(\\set{1, 2}, \\set{a})$ and $(\\set{2}, \\set{a, b})$; for instance $\\set{1, 2}' = \\set{a}$ and $\\set{a}' = \\set{1, 2}$.\nNon-example: in the same context, $(\\set{1}, \\set{a})$ is not a concept, because $\\set{a}' = \\set{1, 2} \\ne \\set{1}$.\n",
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