Pseudometrics and preorders on sets of integer sequences induced by arithmetic functions functions

TL;DR

This paper extends the concepts of pseudometrics and preorders from integers to finite sequences, especially consecutive integers, using arithmetic functions to induce these structures.

Contribution

It generalizes existing pseudometric and preorder concepts to sequences of integers and explores their relationships via arithmetic functions.

Findings

Developed approaches to extend pseudometrics to sequences in nd

Analyzed relationships between functions and preorders

Identified conditions for equivalence and distinction among preorders

Abstract

Starting from pseudometrics and preorders on sets of integers, we extend the focus to sets of finite sequences of integers, in particular sequences of consecutive integers. We outline existing concepts for deriving centred pseudometrics and preorders in a given pseudometric space and their application to $\mathbb{Z}$ and develop approaches to generalize the ideas to $\mathbb{Z}^m$. Sequences of consecutive integers represent a special case here and are examined in more detail. Another main topic is the use of arithmetic functions in this context. The types of pseudometrics and preorders examined in this paper can be induced by suitable arithmetic functions. We derive fundamental conclusions about relationships between functions and preorders, as well as about equivalent and potentially distinct types of preorders.