\begin{definition}[Definition:Antilexicographic Order]
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.
The '''antilexicographic order''' $\struct {S_1, \preccurlyeq_1} \otimes^a \struct {S_2, \preccurlyeq_2}$ on $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq_a}$ where:
:$T := S_1 \times S_2$, that is, the Cartesian product of $S_1$ and $S_2$
:$\preccurlyeq_a$ is the relation defined on $T$ as:
::$\tuple {x_1, x_2} \preccurlyeq_a \tuple {y_1, y_2} \iff \tuple {x_2 \prec_2 y_2} \lor \paren {x_2 = y_2 \land x_1 \preccurlyeq_1 y_1}$
\end{definition}