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  "body": "\n## Statement\n\nLet $k \\ge 1$ and order $\\N^k$ by the [[product order|K0AWJK6NZH]]: $(a_1, \\ldots, a_k) \\le (b_1, \\ldots, b_k)$ when $a_i \\le b_i$ for every $i$.\nLet $S$ be a subset of $\\N^k$.\nThen $S$ has only finitely many minimal elements, and every element of $S$ is greater than or equal to one of them.\n\n## Remarks\n\nEquivalently, $\\N^k$ with the product order is a [[well-quasi-ordering|NH25G6GC07]].\nThe proof is by induction on $k$.\nFrom any infinite sequence in $\\N^k$, extract a subsequence that is non-decreasing in the first coordinate.\nThen extract one that is also non-decreasing in the second coordinate, and so on.\nAfter $k$ steps the subsequence is non-decreasing in the product order, so no infinite antichain exists, and $\\N^k$ is well-founded.\n\nExample: for $S = \\set{(a, b) \\in \\N^2 : ab \\ge 6}$, the minimal elements are $(1, 6), (2, 3), (3, 2), (6, 1)$.\n\nThe main application is to [[monomial ideals|4G25S7PJ22]].\nIdentify the monomial $x_1^{a_1} \\cdots x_k^{a_k}$ with $(a_1, \\ldots, a_k)$; divisibility becomes the product order.\nSo every monomial ideal in $F[x_1, \\ldots, x_k]$ is generated by finitely many monomials.\n\n## Explanations\n\n### L1 human\n\nTake any collection of points with natural number coordinates in $k$ dimensions.\nSome of them are \"corner\" points, with no other point of the collection below and to the left in every coordinate.\nDickson's lemma says there are only finitely many corner points, and every point lies above one of them.\n\n### L2 human\n\nDickson's lemma is one of the first examples of a well-quasi-order beyond $\\N$ itself: a finite product of copies of $\\N$.\nIt is the finiteness step behind Hilbert's basis theorem in the proof with Gröbner bases, and it shows that the Buchberger algorithm terminates.\nHigman's lemma and Kruskal's tree theorem are its analogues for words and trees.\n",
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