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  "body": "\n## Definition\n\nLet $A$ be a set.\nA *strict partial order* on $A$ is a [[binary relation|08EJC6Q25H]] $\u003c$ on $A$ with these two properties.\n\n1. Irreflexivity: $a \u003c a$ holds for no $a \\in A$.\n2. [[Transitivity|0CYRECG2YQ]]: for all $a, b, c \\in A$, if $a \u003c b$ and $b \u003c c$, then $a \u003c c$.\n\n## Remarks\n\nThe relation \"is a proper subset of\" on the [[power set|0BC7KX51Z8]] of a set is a strict partial order.\nIt is irreflexive since no set is a proper subset of itself, and it is transitive since $X \\subsetneq Y \\subsetneq Z$ gives $X \\subsetneq Z$.\nThe relation $\\ne$ on $\\R$ is irreflexive but not transitive: $1 \\ne 2$ and $2 \\ne 1$, but $1 \\ne 1$ fails.\nA strict partial order is asymmetric: $a \u003c b$ and $b \u003c a$ together would give $a \u003c a$.\nFrom a [[partial order|011C29Q1S1]] $\\le$ one gets a strict partial order by $a \u003c b$ if and only if $a \\le b$ and $a \\ne b$.\nConversely, from a strict partial order one gets a partial order by $a \\le b$ if and only if $a \u003c b$ or $a = b$.\nThese two constructions are inverse to each other.\n\n## Explanations\n\n### L1 human\n\nA strict partial order is the relation \"strictly smaller than\" in an order where some pairs may be incomparable.\nThe relation $\u003c$ on numbers is the main example.\n",
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