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  "body": "\n## Definition\n\nLet $\\mathbf{2} = (\\set{0, 1}, \\wedge, \\vee, \\neg, 0, 1)$ be the [[two-element Boolean algebra|0WG43AC0EV]].\nA *term* is an expression built from variables with the binary symbols $\\wedge, \\vee$, the unary symbol $\\neg$ and the constants $0, 1$.\nAn *equation* is a pair of terms, written $s = t$.\nLet $A$ be a set with operations $\\wedge, \\vee, \\neg$ and constants $0, 1$.\nAn equation *holds* in $A$ if both sides take the same value for every assignment of elements of $A$ to the variables.\nIn the canonical definition, a *Boolean algebra* is such a set $A$ in which every equation that holds in $\\mathbf{2}$ also holds.\n\n## Remarks\n\nThis class coincides with the class given by the usual axioms of a [[Boolean algebra|06GKBTAA5E]].\nOne direction is clear, since the usual axioms hold in $\\mathbf{2}$; the other uses the fact that every equation true in $\\mathbf{2}$ follows from those axioms.\nIn the language of universal algebra, the Boolean algebras form the variety generated by $\\mathbf{2}$.\n\nExample: the power set of a set $X$, with intersection, union and complement.\nA subset $S$ of $X$ is the same as a function $X \\to \\set{0, 1}$, and the operations act point by point.\nSo an equation true in $\\mathbf{2}$ holds at each point of $X$, and hence in the power set.\n\nNon-example: the chain $0 \u003c m \u003c 1$ with $\\wedge = \\min$, $\\vee = \\max$, $\\neg 0 = 1$, $\\neg m = 0$ and $\\neg 1 = 0$.\nThe equation $x \\vee \\neg x = 1$ holds in $\\mathbf{2}$, but for $x = m$ it gives $m \\vee 0 = m \\ne 1$.\n",
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