Definition:Simple Order Product

Definition
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

The simple (order) product $\struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq_s}$ where:
 * $T := S_1 \times S_2$, that is, the Cartesian product of $S_1$ and $S_2$
 * $\preccurlyeq_s$ is defined as:
 * $\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq_s \tuple {c, d} \iff a \preccurlyeq_1 c \text { and } b \preccurlyeq_2 d$