Existence of preordered coproduct as category If we view a preordered set as a category, what are the conditions for a product or coproduct to exist within it? This is mostly a curiosa question that arose as I read and I cannot figure it out. Best regards

Let $P$ be preordered set and $(a_i)_{i\in I}$ be a familiy of elements in $P$. An element $s$ is a product of $(a_i)$, if and only if:

* $a_i \leq s$ for all $i\in I$
* for all $s'$ with ($a_i \leq s'$ for all $i\in I$) we have $s\leq s'$



(That's because: parallel arrows are unique and every diagramm automatically commutes)

You should recognize this as the property of $s$ being a supremum of $(a_i)$. Dually, a coproduct of $(a_i)$ is an infimum. I say "a" or "an" because uniquess is only given up to isomorphism ($a\cong b \Leftrightarrow a\leq b\text{ and } b\leq a$).