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  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge, 0, 1)$ be a [[bounded lattice|72V1Z82M5F]] with order $\\le$.\nA *co-Heyting algebra* structure on $L$ is a binary operation $\\setminus$ on $L$ such that for all $a, b, c \\in L$,\n$$a \\setminus b \\le c \\iff a \\le b \\vee c.$$\nSo $a \\setminus b$ is the least element $c$ with $a \\le b \\vee c$.\nA *co-Heyting algebra* is a bounded lattice with such an operation.\n\n## Remarks\n\nThe operation is unique when it exists, since it is a least element.\nA bounded lattice is a co-Heyting algebra if and only if its order dual is a [[Heyting algebra|NG943XNQWH]].\nThe element $\\neg a = 1 \\setminus a$ is called the supplement of $a$; it is the least $c$ with $a \\vee c = 1$.\n\nExample: let $X$ be a topological space and $L$ its family of closed sets, ordered by inclusion.\nFor closed $A, B$, put $A \\setminus B = \\overline{A - B}$, the closure of the set difference.\nIf $A \\subseteq B \\cup C$ with $C$ closed, then $A - B \\subseteq C$, so $\\overline{A - B} \\subseteq C$.\nIf $\\overline{A - B} \\subseteq C$, then $A \\subseteq B \\cup (A - B) \\subseteq B \\cup C$.\nSo the defining equivalence holds.\n\nNon-example: the diamond lattice $M_3$, with atoms $p, q, r$, is bounded but has no such operation.\nEvery co-Heyting algebra is distributive, and $M_3$ is not, since $p \\vee (q \\wedge r) = p$ while $(p \\vee q) \\wedge (p \\vee r) = 1$.\nDirectly: the elements $c$ with $1 \\le p \\vee c$ are $q$, $r$ and $1$, and they have no least one.\n\nIn Mathlib the notion is `CoheytingAlgebra`, with difference `\\` and supplement `￢`.\n",
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