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  "body": "\n## Definition\n\nLet $S$ be a set.\nLet $F$ be a [[Boolean algebra|0XADJKM699]] and let $\\iota \\colon S \\to F$ be a map.\nThe pair $(F, \\iota)$ is a *free Boolean algebra* on $S$ if the following holds.\nFor every Boolean algebra $B$ and every map $g \\colon S \\to B$, there is exactly one homomorphism $\\bar g \\colon F \\to B$ with $\\bar g \\circ \\iota = g$.\n\n## Remarks\n\nIt exists for every $S$ and is unique up to a unique isomorphism compatible with $\\iota$.\nFor finite $S = \\set{1, \\ldots, n}$, one model is the algebra of all [[Boolean functions|0X05GS525E]] $\\set{0, 1}^n \\to \\set{0, 1}$, with $\\iota(i)$ the $i$-th coordinate function; it has $2^{2^n}$ elements and $2^n$ atoms.\nFor any $S$, one model is the algebra of clopen subsets of the product space $\\set{0, 1}^S$.\nThe free Boolean algebra on a countably infinite set is atomless.\n\nExample: for $S = \\set{x}$, the algebra $\\set{0, x, \\neg x, 1}$ with $x$ and $\\neg x$ incomparable is free.\nA map $g$ extends by $\\neg x \\mapsto \\neg g(x)$, $0 \\mapsto 0$, $1 \\mapsto 1$, and this extension is forced.\n\nNon-example: the power set of $\\set{1, 2}$ with $\\iota(x) = \\set{1}$ and $\\iota(y) = \\set{2}$ is generated by the image of $\\iota$ but is not free on $\\set{x, y}$.\nFor the map $g(x) = g(y) = 1$ into the two-element algebra, a homomorphism $\\bar g$ would send $\\set{1} \\cap \\set{2} = \\emptyset$ to $1 \\wedge 1 = 1$, but a homomorphism sends $\\emptyset$ to $0$.\n",
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