Definition:Order Sum/Informal Interpretation

Definition

We can consider the order sum $\struct {S \preccurlyeq} := \struct {S_1 \preccurlyeq_1} \oplus \struct {S_2 \preccurlyeq_2}$ as:

First the set $S_1$, ordered by $\preccurlyeq_1$

Then the set $S_2$, ordered by $\preccurlyeq_2$

Note that:

$\forall x \in S_1, y \in S_2: \tuple {x, 0} \preccurlyeq \tuple {y, 1}$

That is, every element of $S_1$ is comparable with every element of $S_2$.