\begin{definition}[Definition:Ordered Sum]
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let:
:the order type of $\struct {S, \preceq_1}$ be $\theta_1$
:the order type of $\struct {T, \preceq_2}$ be $\theta_2$.
Let $S \cup T$ be the union of $S$ and $T$.
We define the ordering $\preceq$ on $S$ and $T$ as:
:$\forall s \in S, t \in T: a \preceq b \iff \begin{cases}
a \preceq_1 b & : a \in S \land b \in S \\
a \preceq_2 b & : \neg \paren {a \in S \land b \in S} \land \paren {a \in T \land b \in T} \\
& : a \in S, b \in T
\end{cases}$
That is:
:If $a$ and $b$ are both in $S$, they are ordered as they are in $S$.
:If $a$ and $b$ are not both in $S$, but they ''are'' both in $T$, they are ordered as they are in $T$.
:Otherwise, that is if $a$ and $b$ are in ''both'' sets, their ordering in $S$ takes priority over that in $T$.
The ordered set $\struct {S \cup T, \preceq}$ is called the '''ordered sum''' of $S$ and $T$, and is denoted $S + T$.
The order type of $S + T$ is denoted $\theta_1 + \theta_2$.
\end{definition}