Best $g$ Approximation Poset maps

Question

Let $P, Q$ be posets, $g: P\rightarrow Q$ be a monotone map. Let $x\in Q$. By a $g$-approximation of $x$ we mean an element $y\in P$ with $x\leq g(y)$. A best $g$-approximation of $x$ is a $g$-approximation $y$ of $x$ so that for any $z\in P$ with $x\leq g(z)$, $y\leq z$.

Find an example of posets $P$, $Q$ and a surjective monotone map $g:P\rightarrow Q$ such that some $x\in Q$ does not have a best $g$-approximation.

Answer 1

Let $P = (\mathbb{R}, \le)$ and $Q = (\mathbb{R}_{\ge 0}, \le)$. Define $g : P \to Q$ by $g(x) = \text{e}^x$. Then $0 \in Q$ has no best approximation by $g$.