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  "body": "\n## Definition\n\nA *Heyting algebra* is a structure $(H, \\wedge, \\vee, \\to, 0, 1)$ such that:\n\n1. $(H, \\wedge, \\vee, 0, 1)$ is a bounded lattice, with [[partial order|0QV2JS9RJ3]] $a \\le b$ if and only if $a \\wedge b = a$, least element $0$ and greatest element $1$.\n2. $\\to$ is a binary operation on $H$.\n3. For all $a, b, c \\in H$, $c \\le (a \\to b)$ if and only if $c \\wedge a \\le b$.\n\nThe *pseudocomplement* of $a$ is $\\neg a = a \\to 0$.\n\n## Remarks\n\nClause 3 says that $a \\to b$ is the largest element $c$ with $c \\wedge a \\le b$, so $\\to$ is determined by the lattice.\n\nExample: let $X$ be a [[topological space|0CQHFPGJ45]] and let $H$ be the set of its [[open sets|02BH3MCBQB]], with $\\wedge = \\cap$, $\\vee = \\cup$, $0 = \\emptyset$, $1 = X$, and $U \\to V$ the [[interior|0RZH6W98HK]] of $(X \\setminus U) \\cup V$.\nFor an open set $W$, the inclusion $W \\subseteq (X \\setminus U) \\cup V$ holds if and only if $W \\cap U \\subseteq V$.\nSince $W$ is open, the first inclusion holds if and only if $W$ lies in the interior; this is clause 3.\n\nNon-example: the five-element lattice $M_3$ has a least element, a greatest element and three pairwise incomparable elements between them.\nIt is a bounded lattice but carries no operation $\\to$ satisfying clause 3.\nIndeed, every Heyting algebra is a [[distributive lattice|0HM0HQCK6N]], and $M_3$ is not distributive.\n\nA [[Boolean algebra|0XADJKM699]] is a Heyting algebra with $a \\to b = \\neg a \\vee b$, and a Heyting algebra is Boolean if and only if $a \\vee \\neg a = 1$ for all $a$.\nEvery finite distributive lattice is a Heyting algebra.\n",
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