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  "body": "\n## Definition\n\nLet $X$ be a [[Stone space|ZV38HA5PAQ]], that is, a compact Hausdorff totally disconnected space.\nLet $(\\mathbf{A}_x)_{x \\in X}$ be a family of [[algebras|QJGPT2W0FT]] of the same signature.\nA *Boolean product* of this family is a [[subdirect product|88WN3AD2VQ]] $\\mathbf{A}$ of the [[direct product|WJ73XYBNE4]] $\\prod_{x \\in X} \\mathbf{A}_x$ with these two properties:\n\n1. for all $a, b \\in A$, the set $\\set{x \\in X : a(x) = b(x)}$ is [[clopen|DQ4E55T3BW]] in $X$;\n2. for all $a, b \\in A$ and every clopen set $N \\subseteq X$, the function $c$ with $c(x) = a(x)$ for $x \\in N$ and $c(x) = b(x)$ for $x \\notin N$ lies in $A$.\n\n## Remarks\n\nClause 2 is called the patchwork property.\nAn algebra is said to be representable as a Boolean product when it is isomorphic to a Boolean product of some family.\n\nExample: let $\\mathbf{B}$ be a Boolean algebra with Stone space $X$, and let $\\mathbf{2}$ be the two-element Boolean algebra.\nThe continuous maps $X \\to \\mathbf{2}$, with $\\mathbf{2}$ discrete, form a Boolean product of copies of $\\mathbf{2}$ isomorphic to $\\mathbf{B}$.\nTwo continuous maps to a discrete space agree on a clopen set, and pasting two continuous maps along a clopen set gives a continuous map.\n\nNon-example: let $X = \\set{p, q}$ be discrete, and let $A = \\set{(0, 0), (1, 1)}$ inside $\\mathbf{2} \\times \\mathbf{2}$.\nThis is a subdirect product and satisfies clause 1, but pasting $(0, 0)$ on $\\set{p}$ with $(1, 1)$ on $\\set{q}$ gives $(0, 1) \\notin A$, so clause 2 fails.\n",
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