Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. (We call a function $f:P\to Q$ order preserving if $x\leq y$ in $P$ implies $f(x)\leq f(y)$ in $Q$.) 


There is a natural ordering relation on $\text{Hom}(P,Q)$ given by $f\leq g$ if and only if $f(p) \leq\_Q g(p)$ for all $p\in P$.


Let $D$ be directed and $P$ be a poset. Is $\text{Hom}(P,D)$ necessarily directed?



Is the homomorphism poset directed if the codomain is directed?
The answer is negative.


Let $P=\omega\_1$, and $Q$ be the disjoint union of


* $A=A\_0\mathbin{\dot\cup}A\_1$, where each $A\_i$ is a copy of $\omega\_1$,
* $B$, which consists of finite subsets of $A$ that intersect both $A\_0$ and $A\_1$, ordered by inclusion.


If $a\in A$ and $b\in B$, we put $a\le b$ iff there is $a'\in b$ such that $a\le a'$. It is easy to see that $Q$ is directed.


Let $f$ and $g$ be the isomorphisms of $P$ to $A\_0$ and $A\_1$, respectively, and assume for contradiction that $f,g\le h$ for some order-preserving $h\colon P\to Q$. For every $\alpha\in P$, we must have $h(\alpha)\in B$. However, $B$ contains no strictly increasing chain of length $\omega+1$, hence $h(\alpha)$ is eventually constant. This is impossible, as every element of $Q$ has only countably many predecessors, whereas $f[P],g[P]$ are uncountable.


