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  "body": "\n## Definition\n\nLet $m, n \\ge 1$ be integers.\nA *Boolean matrix* of size $m \\times n$ is an $m \\times n$ [[matrix|0VD9ZSWWJG]] $A = (a_{ij})$ with every entry $a_{ij} \\in \\set{0, 1}$.\nLet $A$ be a Boolean matrix of size $m \\times n$ and $B$ one of size $n \\times p$.\nTheir *Boolean product* $A \\odot B$ is the $m \\times p$ Boolean matrix with entries\n$$(A \\odot B)_{ik} = \\max_{1 \\le j \\le n} \\min(a_{ij}, b_{jk}),$$\nthat is, $(A \\odot B)_{ik} = 1$ exactly when there is some $j$ with $a_{ij} = b_{jk} = 1$.\n\n## Remarks\n\nEquivalently, the entries are computed in the Boolean semiring $\\set{0, 1}$ with $1 + 1 = 1$.\nFor finite sets $X = \\set{x_1, \\ldots, x_m}$ and $Y = \\set{y_1, \\ldots, y_n}$, a [[relation|05AVARPXA2]] $R \\subseteq X \\times Y$ has the Boolean matrix with $a_{ij} = 1$ exactly when $(x_i, y_j) \\in R$.\nIf $S \\subseteq Y \\times Z$ has matrix $B$, then the [[composite relation|0PYSR20HGF]] \"first $R$, then $S$\" has matrix $A \\odot B$.\nExample: for $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$, $A \\odot A = A$, since the entry $(1,2)$ is $\\max(\\min(1,1), \\min(1,1)) = 1$ and the entry $(2,1)$ is $\\max(\\min(0,1), \\min(1,0)) = 0$; the ordinary product would give the entry $2$ at $(1,2)$.\nNon-example: $\\begin{pmatrix} 1 \u0026 2 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not a Boolean matrix, because the entry $2$ is not in $\\set{0, 1}$.\n",
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