Strong Greedoid Structure of $r$-Removed $P$-Orderings

Abstract

Inspired by the notion of $r$-removed $P$-orderings introduced in the setting of Dedekind domains by Bhargava, we generalize it to the framework of arbitrary ultrametric spaces. We show that sets of maximal "$r$-removed perimeter" can be constructed by a greedy algorithm and form a strong greedoid. This gives a simplified proof of several theorems previously obtained by Bhargava, as well as generalises some results of Grinberg and Petrov who considered the case $r=0$ corresponding, in turn, to simple $P$-orderings.