A characterization of non-linear maps satisfying orthogonality properties

Maps not necessarily linear but monotone (order preserving) between function spaces are analyzed. Characterizations of maps T from functions on X to those on Y with the property that the image Tf(y) depends on the value of f at one point are established. Then T has a functional representation, namely, Tf(y) is equal to a function composed with f(x). In particular, for X extremally disconnected, T satisfies the above property for non-negative functions on X if and only if is finitely disjointness preserving (), orthogonally additive (), and satisfies a continuity condition. In the absence of continuity conditions, the above order theoretic conditions are equivalent to a local condition, specifically, whenever on a neighborhood of x. Results for more general domains are provided as well as consequences for bijections.