SCHWEITZER 1995 Problem 8

Let P be a finite, partially ordered set with one largest element, which is the only upper bound of the set of minimal elements. Prove that any monotonic function $f : P^n\to P$ can be written in the form $g( x_1 , x_2 , ..., x_n , c_1 , ..., c_m )$ , where $c_i\in P$ and g is a monotonic, idempotent function. (g is idempotent iff $g(x , x , ..., x) = x\,\forall x\in P$ )