ON P-ORDERINGS, RINGS OF INTEGER-VALUEDPOLYNOMIALS, AND ULTRAMETRIC ANALYSIS

摘要

In this article, we introduce two new variations of a game called "P-ordering" [4],which we term an r-removed P-ordering and a P-ordering of order h, respectively.Given any subset S of a Dedekind domain D and a prime ideal P of D, the gameinvolves the construction of a special sequence in S having certain surprising com-binatorial properties. In particular, these two notions of P-ordering result in athree-parameter generalization of the usual factorial function n!, which we denoten!_(s,h)~{r}hereS, h,andrdenote the three parameters, whereSis any subset ofD,r is a nonnegative integer, and h E R U The usual factorial function is obtainedby setting S = D = Z, r = 0, and h .