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  "body": "\n## Definition\n\nLet $L$ be a [[lattice|WH4SFAACAS]] with a least element $0$, and let $a \\in L$.\nAn element $a^* \\in L$ is a *pseudocomplement* of $a$ if for every $x \\in L$,\n$$a \\wedge x = 0 \\iff x \\le a^*.$$\nThe lattice $L$ is *pseudocomplemented* if every element has a pseudocomplement.\n\n## Remarks\n\nEquivalently, $a^*$ is the greatest element of the set $\\set{x \\in L : a \\wedge x = 0}$, so it is unique when it exists.\nIn a pseudocomplemented lattice, $a \\le a^{**}$ and $a^{***} = a^*$ for all $a$.\n\nExample: the open sets of a topological space $X$, ordered by inclusion.\nFor an open set $U$, put $U^* = \\operatorname{int}(X \\setminus U)$.\nAn open set $V$ satisfies $U \\cap V = \\emptyset$ exactly when $V \\subseteq X \\setminus U$, that is, when $V \\subseteq \\operatorname{int}(X \\setminus U)$, since $V$ is open.\n\nNon-example: in the diamond lattice $M_3$ with atoms $p, q, r$, the element $p$ has no pseudocomplement.\nThe elements $x$ with $p \\wedge x = 0$ are $0$, $q$ and $r$, and this set has no greatest element, since $q \\vee r = 1$ and $p \\wedge 1 = p \\ne 0$.\n",
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    "name": "Pseudocomplement",
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  "msc": [
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  "name": "Pseudocomplement",
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