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  "body": "\n## Definition\n\nLet $I$ be a set and let $(P_i, \\le_i)$ be a [[partially ordered set|0QV2JS9RJ3]] for each $i \\in I$.\nThe *product order* on the [[Cartesian product|09CEDJYBKP]] $\\prod_{i \\in I} P_i$ is the relation $\\le$ given by\n$$x \\le y \\iff x_i \\le_i y_i \\text{ for all } i \\in I.$$\n\n## Remarks\n\nIt is a partial order: reflexivity, antisymmetry and transitivity hold in each coordinate, hence for tuples.\nFor two factors, $(a, b) \\le (a', b')$ means $a \\le a'$ and $b \\le b'$.\nExample: in $\\N \\times \\N$, $(1, 2) \\le (3, 2)$, since $1 \\le 3$ and $2 \\le 2$.\nNon-example: $(1, 2)$ and $(2, 1)$ are incomparable, since the first coordinate rises and the second falls.\nFor finitely many factors, the product order is contained in the lexicographic order on the same set, which is total when all factors are total.\nThe power set of a set $I$, ordered by inclusion, is isomorphic to the product order on $\\set{0, 1}^I$.\n",
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