Definition:Order Sum
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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:
while
- $S_1$ and $S_2$ individually keep their original orderings.
Informal Interpretation
We can consider the order sum $\struct {S \preccurlyeq} := \struct {S_1 \preccurlyeq_1} \oplus \struct {S_2 \preccurlyeq_2}$ as:
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 $S_2$.
Also see
- Definition:Disjoint Union: the construct $S_1 \sqcup S_2$
- Results about order sums can be found here.
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations