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  "body": "\n## Definition\n\nLet $A$ be a set and let $\u003c$ be a [[strict partial order|SWB1ZXFVKY]] on $A$.\nTwo elements $a, b \\in A$ are *incomparable* when neither $a \u003c b$ nor $b \u003c a$.\nThe relation $\u003c$ is a *strict weak ordering* when incomparability is transitive.\nThat is, for all $a, b, c \\in A$: if $a, b$ are incomparable and $b, c$ are incomparable, then $a, c$ are incomparable.\n\n## Remarks\n\nLet $f \\colon A \\to \\R$ be any function and set $a \u003c b$ when $f(a) \u003c f(b)$.\nThis is a strict weak ordering: $a$ and $b$ are incomparable exactly when $f(a) = f(b)$, and equality of values is transitive.\nOn $A = \\set{a, b, c}$, the strict partial order whose only relation is $a \u003c b$ is not a strict weak ordering.\nHere $a$ and $c$ are incomparable and $c$ and $b$ are incomparable, but $a$ and $b$ are comparable.\nIn a strict weak ordering, incomparability is an [[equivalence relation|0SGTMBEWRF]], and $\u003c$ induces a strict total order on its classes.\nThe non-strict form is a total preorder: a [[preorder|0RXS8R6K37]] $\\lesssim$ in which $a \\lesssim b$ or $b \\lesssim a$ for all $a, b$.\nStrict weak orderings and total preorders on $A$ correspond one to one, by $a \\lesssim b$ if and only if not $b \u003c a$.\n\n## Explanations\n\n### L1 human\n\nA weak ordering is a ranking with ties.\nTied elements form groups, and the groups stand in a line.\n",
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