The floor quotient partial order

TL;DR

This paper investigates the structure and properties of the floor quotient partial order on positive integers, focusing on its internal organization and the behavior of its Möbius function.

Contribution

It provides a detailed analysis of the internal structure and Möbius function of the floor quotient partial order, a novel relation on positive integers.

Findings

Characterization of the partial order structure

Explicit formulas or properties of the Möbius function

Insights into the order's combinatorial properties

Abstract

A positive integer $d$ is a floor quotient of $n$ if there is a positive integer $k$ such that $d = \lfloor n/k \rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies the internal structure of this partial order and its M\"{o}bius function.