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  "body": "\n## Definition\n\nLet $B$ be a [[Boolean algebra|06GKBTAA5E]] with meet $\\wedge$, greatest element $1$ and order $\\le$.\nAn *interior operator* on $B$ is a map $I \\colon B \\to B$ such that for all $x, y \\in B$:\n\n1. $I(1) = 1$;\n2. $I(x) \\le x$;\n3. $I(I(x)) = I(x)$;\n4. $I(x \\wedge y) = I(x) \\wedge I(y)$.\n\nAn *interior algebra* is a pair $(B, I)$ of a Boolean algebra $B$ and an interior operator $I$ on $B$.\n\n## Remarks\n\nExample: let $X$ be a [[topological space|0CNQXVBHCX]] and let $B$ be the power set of $X$.\nLet $I(A)$ be the [[interior|0C38S0KMJK]] of $A$.\nThe interior of $X$ is $X$, and the interior of $A$ lies in $A$.\nThe interior of an open set is the set itself, and $I(A \\cap A') = I(A) \\cap I(A')$.\nSo the four clauses hold.\nNon-example: on a Boolean algebra with $0 \\ne 1$, the map $I(x) = 0$ for all $x$ satisfies clauses 2, 3 and 4, but fails clause 1.\nClause 4 implies that $I$ is monotone: if $x \\le y$, then $I(x) = I(x \\wedge y) = I(x) \\wedge I(y) \\le I(y)$.\nThe map $C(x) = \\neg I(\\neg x)$ is the dual closure operator; a Boolean algebra with such a closure operator is a *closure algebra*.\nInterior algebras are the algebraic semantics of the [[modal logic|G8VGTW9P4F]] S4, with $I$ interpreting the necessity operator $\\square$.\n",
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