Let omega_1 be the first uncountable ordinal, 
same as the set of all countable ordinals. 
Let F be the set of all decreasing 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 <= g if 
f(alpha) <= 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 -> K, i.e if f <= g then h(f) <= h(g), 
and with the additional property that f <= h(f) ?