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  "body": "\n## Definition\n\nLet $A$ be a set with a binary operation $+$ and a unary operation $n$.\nThe structure $(A, +, n)$ satisfies *Huntington's axioms* if for all $x, y, z \\in A$:\n\n1. $x + y = y + x$.\n2. $(x + y) + z = x + (y + z)$.\n3. $n(n(x) + y) + n(n(x) + n(y)) = x$.\n\nThe third axiom is called *Huntington's equation*.\n\n## Remarks\n\nA structure satisfies Huntington's axioms if and only if it comes from a [[Boolean algebra|0XADJKM699]] with $x + y = x \\vee y$ and $n(x) = \\neg x$.\nThe other operations are recovered as $x \\wedge y = n(n(x) + n(y))$, $1 = x + n(x)$ and $0 = n(1)$, where the last two do not depend on $x$.\n\nExample: $\\set{0, 1}$ with $x + y = x \\vee y$ and $n(x) = \\neg x$.\nBy De Morgan's laws, the left side of axiom 3 is $(x \\wedge \\neg y) \\vee (x \\wedge y) = x \\wedge (\\neg y \\vee y) = x$.\n\nNon-example: $\\set{0, 1}$ with $x + y = x \\vee y$ and $n(x) = x$.\nAxioms 1 and 2 hold, but axiom 3 becomes $(x \\vee y) \\vee (x \\vee y) = x$, which fails for $x = 0$, $y = 1$.\n",
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