\begin{definition}[Definition:Order Sum]
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
The '''order sum''' $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq}$ where:
:$T := S_1 \sqcup S_2 = \paren {S_1 \times \set 0} \cup \paren {S_2 \times \set 1}$
:$\preccurlyeq$ is defined as:
::$\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq \tuple {c, d} \iff \begin {cases} b = 0 \text { and } d = 1 \\ b = d = 0 \text { and } a \preccurlyeq_1 c \\ b = d = 1 \text { and } a \preccurlyeq_2 c \end {cases}$
That is:
:all the elements of $S_1$ precede all the elements of $S_2$
while
:$S_1$ and $S_2$ individually keep their original orderings.
\end{definition}