\begin{definition}[Definition:Addition of Order Types]
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
Let $\alpha := \map \ot {S_1, \preccurlyeq_1}$ and $\beta := \map \ot {S_2, \preccurlyeq_2}$ denote the order types of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ respectively.
Let $\alpha + \beta$ be defined as:
:$\alpha + \beta:= \map \ot {\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2} }$
where $\oplus$ denotes the order sum operator.
The operation $+$ is known as '''order type addition''' or '''addition of order types'''.
\end{definition}