Strong greedoid structure of $r$-removed $P$-orderings

Abstract

Inspired by the notion of \emph{$r$-removed $P$-orderings} introduced in the setting of Dedekind domains by Bhargava \cite{Bha09-1} we study its generalization in the framework of arbitrary (generalised) 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 in \cite{Bha09-1} and also generalises the results of \cite{GP21} which considered the case $r=0$ corresponding, in turn, to simple $P$-orderings of \cite{Bha97}.