Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all regressive functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$, that is $f(\alpha) < \alpha$ for $0 < \alpha < \omega_1$. Define a partial order on $F$ by $f \leq g$ if $f(\alpha) \leq g(\alpha)$ for all $\alpha < \omega_1$. Let $K$ be the subset of $F$, of functions with a finite range.

Question: Is there an order-preserving homomorphism $h : F \to K$, i.e if $f \leq g$ then $h(f) \leq h(g)$, and with the additional property that $f \leq h(f)$ ?

Yes, I did mean the functions $f$ must be regressive (when I had imprecisely said decreasing, in the original statement). For that matter, I had dropped one requirement (don't know if it would be essential), but you may assume in addition that the functions $f\in F$ preserve the order, that is $f(\alpha)\leq f(\beta)$ if $\alpha \leq \beta$ (perhaps the latter would be correctly termed non-decreasing). S.P.