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  "body": "\n## Definition\n\nLet $A$ be a set.\nA *strict total order* on $A$ is a [[strict partial order|SWB1ZXFVKY]] $\u003c$ on $A$ such that, for all $a, b \\in A$ with $a \\ne b$, $a \u003c b$ or $b \u003c a$.\n\n## Remarks\n\nThe relation $\u003c$ on $\\R$ is a strict total order: it is irreflexive, transitive, and for distinct reals $x$ and $y$, $x \u003c y$ or $y \u003c x$.\nThe relation $\\subsetneq$ on the [[power set|0BC7KX51Z8]] of $\\set{1, 2}$ is a strict partial order but not a strict total order.\nThe sets $\\set{1}$ and $\\set{2}$ are distinct, and neither is a proper subset of the other.\nThe condition gives trichotomy: for all $a, b \\in A$, exactly one of $a \u003c b$, $a = b$, $b \u003c a$ holds.\nThe strict total orders on $A$ correspond one to one with the [[total orders|0316AYH1SC]] on $A$, by $a \\le b$ if and only if $a \u003c b$ or $a = b$.\nOther names are strict linear order and strict simple order.\n\n## Explanations\n\n### L1 human\n\nA strict total order ranks all elements in one line with no ties.\nGiven two different elements, it tells which one comes first.\n",
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