Simultaneous orderings in function fields

For every Dedekind domain R, Bhargava defined the factorials of a subset S of R by introducing the notion of p-ordering of S, for every maximal ideal p of R. We study the existence of simultaneous ordering in the case S=R=OK, where OK is the ring of integers of a function field K over a finite field Fq. We show, that when OK is the ring of integers of an imaginary quadratic extension K of Fq(T), K=Fq(T)/(Y2-D(T)), then there exists a simultaneous ordering if and only if degD⩽1.