P-orderings and polynomial functions on arbitrary subsets of Dedekind rings

We introduce the notion of a P-ordering of an arbitrary subset X of a Dedekind ring R, and use it to investigate the functions from X to R which can be represented by polynomials. In the case when R is a finite principal ideal ring, our results include canonical representations for those polynomials which vanish on X, a canonical representation for each polynomial function from X to R, necessary and sufficient congruence conditions for a function from X to R to be a polynomial function, and a formula for the number of such functions. When R is a Dedekind domain, we use the concept of P-ordering to give necessary and sufficient conditions for the existence of a regular basis for the ring Int(X, R), and moreover, we give an explicit construction of such a basis whenever it exists. Finally, we deduce many of the classical theorems of Kempner, Carlitz, Pólya, and others on polynomial mappings as special cases of our results.