$B$-orderings for all ideals $B$ of Dedekind domains and generalized factorials

Abstract

This paper extends Bhargava's theory of $\mathfrak{p}$-orderings of subsets $S$ of a Dedekind ring $R$ valid for prime ideals $\mathfrak{p}$ in $R$. Bhargava's theory defines for integers $k\ge1$ invariants of $S$, the generalized factorials $[k]!_S$, which are ideals of $R$. This paper defines $\mathfrak{b}$-orderings of subsets $S$ of a Dedekind domain $D$ for all nontrivial proper ideals $\mathfrak{b}$ of $D$. It defines generalized integers $[k]_{S,T}$, as ideals of $D$, which depend on $S$ and on a subset $T$ of the proper ideals $\mathscr{I}_D$ of $D$. It defines generalized factorials $[k]!_{S,T}$ and generalized binomial coefficients, as ideals of $D$. The extension to all ideals applies to Bhargava's enhanced notions of $r$-removed $\mathfrak{p}$-orderings, and $\mathfrak{p}$-orderings of order $h$.