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  "body": "\n## Definition\n\nA *Stone algebra* is a [[bounded lattice|72V1Z82M5F]] $(L, \\vee, \\wedge, 0, 1)$ such that:\n\n1. $L$ is a [[distributive lattice|0HM0HQCK6N]].\n2. Every $a \\in L$ has a [[pseudocomplement|WMR73J0S26]] $a^*$.\n3. $a^* \\vee a^{**} = 1$ for every $a \\in L$.\n\n## Remarks\n\nExample: every bounded chain $C$ with $0 \\ne 1$ is a Stone algebra.\nIts pseudocomplement is $0^* = 1$ and $a^* = 0$ for $a \\ne 0$.\nIf $a = 0$ then $a^* \\vee a^{**} = 1 \\vee 0 = 1$, and if $a \\ne 0$ then $a^* \\vee a^{**} = 0 \\vee 1 = 1$.\n\nNon-example: the lattice of open sets of $\\R$, with $U^* = \\operatorname{int}(\\R \\setminus U)$, satisfies clauses 1 and 2 but not 3.\nFor $U = (0, \\infty)$, one has $U^* = (-\\infty, 0)$ and $U^{**} = (0, \\infty)$, so $U^* \\cup U^{**} = \\R \\setminus \\set{0} \\ne \\R$.\n\nEvery Boolean algebra is a Stone algebra, with $a^* = \\neg a$ and $a^{**} = a$.\n",
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