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

TL;DR

This paper generalizes the concept of $r$-removed $P$-orderings from Dedekind domains to ultrametric spaces, showing that sets with maximal $r$-removed perimeter form a strong greedoid, simplifying existing proofs and extending previous results.

Contribution

It introduces a new framework for $r$-removed $P$-orderings in ultrametric spaces and proves they form a strong greedoid, generalizing prior work and providing simplified proofs.

Findings

Sets of maximal $r$-removed perimeter form a strong greedoid.

The greedy algorithm constructs these sets.

Generalizes previous results from Dedekind domains and $P$-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}.