Coproduct of Ordered Sets

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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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