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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\le$ be a [[preorder|0VABYWJWH9]] on $X$.\nThe preorder is a *well-quasi-ordering* when for every infinite [[sequence|0G9EKM0E15]] $x_1, x_2, x_3, \\ldots$ of elements of $X$ there are indices $i \u003c j$ with $x_i \\le x_j$.\n\n## Remarks\n\nExample: every finite set with any preorder is well-quasi-ordered.\nCheck: an infinite sequence in a finite set has $x_i = x_j$ for some $i \u003c j$, and then $x_i \\le x_j$ by reflexivity.\n\nNon-example: the positive integers ordered by divisibility are well-founded but not well-quasi-ordered.\nThe sequence of primes $2, 3, 5, 7, \\ldots$ has no $i \u003c j$ with $p_i \\mid p_j$.\n\nEquivalent form: every infinite sequence has an infinite subsequence $x_{i_1} \\le x_{i_2} \\le \\cdots$ with $i_1 \u003c i_2 \u003c \\cdots$.\nAnother equivalent form: the strict part of $\\le$ is well-founded and every [[antichain|03HF498P1J]] is finite.\nA sequence with no $i \u003c j$ such that $x_i \\le x_j$ is called bad, so a well-quasi-ordering is a preorder with no bad sequences.\nThe abbreviation wqo is common.\n",
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