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  "body": "\n## Definition\n\nThe class of *series-parallel partial orders* is the smallest class $\\mathcal{S}$ of finite [[partially ordered sets|0QV2JS9RJ3]] such that:\n\n1. every one-element poset is in $\\mathcal{S}$;\n2. if $P$ and $Q$ are in $\\mathcal{S}$, then their [[ordinal sum|6AJ7QQCTQQ]] $P \\oplus Q$ is in $\\mathcal{S}$;\n3. if $P$ and $Q$ are in $\\mathcal{S}$, then their [[disjoint union|TZWMEM7GKA]] $P + Q$ is in $\\mathcal{S}$;\n4. every poset isomorphic to a member of $\\mathcal{S}$ is in $\\mathcal{S}$.\n\n## Remarks\n\nExample: the poset with $a \u003c b \u003c d$, $a \u003c c \u003c d$ and $b$, $c$ incomparable is $\\set{a} \\oplus (\\set{b} + \\set{c}) \\oplus \\set{d}$, so it is series-parallel.\nNon-example: the poset $N$ on $\\set{a, b, c, d}$ with $a \u003c c$, $b \u003c c$, $b \u003c d$ as the only strict relations is not series-parallel.\nIt is connected, so it is not a disjoint union of two nonempty posets.\nIt is not an ordinal sum $A \\oplus B$ of two nonempty posets either.\nNothing lies below $a$, so $a$ would be in $A$, and every element of $B$ would be above $a$.\nThe only element above $a$ is $c$, so $B = \\set{c}$, but $d$ would then be below $c$, which fails.\nA finite poset is series-parallel if and only if it has no subposet isomorphic to $N$.\nTotal orders and antichains are series-parallel.\n",
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