\begin{definition}[Definition:Multiplication 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 $\alpha \cdot \beta$ be defined as:
:$\alpha \cdot \beta:= \map \ot {\struct {S_1, \preccurlyeq_1} \otimes^a \struct {S_2, \preccurlyeq_2} }$
where $\otimes^a$ denotes the antilexicographic product operator.
The operation $\cdot$ is known as '''order type multiplication''' or '''multiplication of order types'''.
The expression $\alpha \cdot \beta$ is known as the '''order product''' of $\alpha$ and $\beta$.
\end{definition}