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analysis

Source: miklos schweitzer 1995 q8

October 5, 2021
partial orderanalysis

Problem Statement

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:Pn→Pf : P^n\to P can be written in the form g(x1,x2,...,xn,c1,...,cm)g( x_1 , x_2 , ..., x_n , c_1 , ..., c_m ), where ci∈Pc_i\in P and g is a monotonic, idempotent function. (g is idempotent iff g(x,x,...,x)=x ∀x∈Pg(x , x , ..., x) = x\,\forall x\in P)
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