Structure Results for Function Lattices

Abstract

For partially ordered sets X and F let Yx denote the set of all order-preserving maps of X to Y partially ordered by f ≦ g if and only if f(x)≦ g (x) for each x ∈ X [1; 4; 6]. If X is unordered then Yx is the usual direct product of partially ordered sets, while if both X and Y are finite unordered sets then Yx is the commonplace exponent of cardinal numbers. This generalized exponentiation has an important vindication especially for those partially ordered sets that are lattices.