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  "body": "\n## Definition\n\nLet $(A, \\le_A)$ and $(B, \\le_B)$ be [[partially ordered sets|011C29Q1S1]].\nThe lexicographic order on the [[Cartesian product|0C81YBVTWH]] $A \\times B$ is the relation $\\le$ with $(a, b) \\le (a', b')$ when\n$$a \u003c_A a' \\quad \\text{or} \\quad (a = a' \\text{ and } b \\le_B b').$$\n\n## Remarks\n\nIt is a partial order. When $\\le_A$ and $\\le_B$ are [[total orders|0316AYH1SC]], it is a total order.\nExample: in $\\mathbb{N} \\times \\mathbb{N}$, $(1, 5) \u003c (2, 0)$ and $(2, 0) \u003c (2, 3)$.\nA dictionary orders words in the same way, letter by letter.\n\n## Explanations\n\n### L1 human\n\nThe lexicographic order is the dictionary order.\n\"cat\" comes before \"cow\", because the first letters agree and \"a\" comes before \"o\".\n",
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