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  "body": "\n## Definition\n\nAn *MV-algebra* is a structure $(A, \\oplus, \\neg, 0)$, where $\\oplus$ is a binary operation, $\\neg$ is a unary operation and $0 \\in A$, such that for all $x, y \\in A$:\n\n1. $(A, \\oplus, 0)$ is a commutative [[monoid|0N6R902FG1]].\n2. $\\neg \\neg x = x$.\n3. $x \\oplus \\neg 0 = \\neg 0$.\n4. $\\neg (\\neg x \\oplus y) \\oplus y = \\neg (\\neg y \\oplus x) \\oplus x$.\n\n## Remarks\n\nExample: the unit interval $[0, 1]$ with $x \\oplus y = \\min(1, x + y)$, $\\neg x = 1 - x$ and $0$ is an MV-algebra.\nClauses 1 to 3 are direct, since $\\neg 0 = 1$.\nFor clause 4, if $y \\ge x$ the left side is $\\neg 1 \\oplus y = y$, and if $y \u003c x$ it is $(x - y) \\oplus y = x$; so both sides equal $\\max(x, y)$.\n\nNon-example: the integers $(\\Z, +, -, 0)$ satisfy clauses 1 and 2, but fail clause 3, since $1 + (-0) = 1 \\ne 0$.\n\nEvery [[Boolean algebra|0XADJKM699]] is an MV-algebra with $\\oplus = \\vee$ and $\\neg$ the complement.\nThe constant $\\neg 0$ is written $1$, and $x \\odot y = \\neg(\\neg x \\oplus \\neg y)$ is the dual operation.\n",
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