Definition:Order Sum

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

The order sum $\struct {S_1, \preccurlyeq_1} \oplus \struct {S_2, \preccurlyeq_2}$ of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ is the ordered set $\struct {T, \preccurlyeq}$ where:
 * $T := S_1 \sqcup S_2 = \paren {S_1 \times \set 0} \cup \paren {S_2 \times \set 1}$
 * $\preccurlyeq$ is defined as:
 * $\forall \tuple {a, b}, \tuple {c, d} \in T: \tuple {a, b} \preccurlyeq \tuple {c, d} \iff \begin {cases} b = 0 \text { and } d = 1 \\ b = d = 0 \text { and } a \preccurlyeq_1 c \\ b = d = 1 \text { and } a \preccurlyeq_2 c \end {cases}$

That is:
 * all the elements of $S_1$ precede all the elements of $S_2$

while
 * $S_1$ and $S_2$ individually keep their original orderings.

Also see

 * Definition:Disjoint Union: the construct $S_1 \sqcup S_2$