Series-parallel partial order

Remarks

Example: the poset with $a < b < d$, $a < c < d$ and $b$, $c$ incomparable is $\set{a} \oplus (\set{b} + \set{c}) \oplus \set{d}$, so it is series-parallel. Non-example: the poset $N$ on $\set{a, b, c, d}$ with $a < c$, $b < c$, $b < d$ as the only strict relations is not series-parallel. It is connected, so it is not a disjoint union of two nonempty posets. It is not an ordinal sum $A \oplus B$ of two nonempty posets either. Nothing lies below $a$, so $a$ would be in $A$, and every element of $B$ would be above $a$. The only element above $a$ is $c$, so $B = \set{c}$, but $d$ would then be below $c$, which fails. A finite poset is series-parallel if and only if it has no subposet isomorphic to $N$. Total orders and antichains are series-parallel.