# Series-parallel order

A class of partial order. A partially ordered set has a series-parallel order if it can be constructed recursively from the one-point ordered set using the operations $+$ and $*$ defined on orders as follows. Let $P_1,{<}$ and $P_2,{<}$ be disjoint ordered sets. Each of $P_1 + P_2$ and $P_1 * P_2$ has as underlying set the disjoint union $P_1 \sqcup P_2$ and an order $\prec$ where in each case $\prec$ restricted to $P_1$ and to $P_2$ gives the original order. In $P_1 + P_2$ all pairs $p_1,p_2$ with $P_1 \in P_1$ and $p_2 \in P_2$ are incomparable (parallel); in $P_1 * P_2$, then all such pairs have $p_1 \prec p_2$ (series). We note that $+$ is commutative but $*$ is not.
Each operation is an example of the ordered sum, the operation $+$ corresponding to the trivial order on the indices $\{1,2\}$ and $*$ corresponding to the order $1<2$.
Series-parallel graphs are characterised by being "N-free": having no subset $\{a,b,c,d\}$ with only the comparisons $a \prec b$, $a \prec d$, $c \prec d$.
Series-parallel graphs have dimension at most $2$.