Coproduct of Ordered Sets
This definition (effectively pasting posets side by side) deserves a separate page, but I don't know a name; possibly "Disjoint Ordering", if no literature defining it can be located
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Theorem
Let $\mathbf{OrdSet}$ be the category of ordered sets.
Let $\left({P, \preceq_1}\right)$ and $\left({Q, \preceq_2}\right)$ be ordered sets.
Let $P \sqcup Q$ be the disjoint union of $P$ and $Q$.
Let $\preceq$ be the ordering on $P \sqcup Q$ defined by:
- $\left({x, i}\right) \preceq \left({y, j}\right)$ if and only if $i = j$ and $x \preceq_i y$
where $i = 1$ or $i = 2$ depending on whether $x,y \in P$ or $x,y \in Q$.
Then $\left({P \sqcup Q, \preceq}\right)$ is the coproduct of $P$ and $Q$ in $\mathbf{OrdSet}$.
Proof
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Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 3.2$