A weighted M\"obius function

TL;DR

This paper introduces a weighted variant of the M"obius function on finite abelian ll-groups and proves an analog of Hall's theorem regarding its vanishing properties.

Contribution

It defines a new weighted Mb4obius function for finite abelian ll-groups and establishes an analog of Hall's theorem for this variant.

Findings

Defined a weighted Mb4obius function for ll-groups

Proved an analog of Hall's theorem on the vanishing of this function

Extended classical Mb4obius function properties to a new setting

Abstract

For a fixed odd prime $\ell$, we define a variant of the classical M\"{o}bius function on the poset of isomorphism classes of finite abelian $\ell$-groups, then we prove an analog of Hall's theorem on the vanishing of the M\"{o}bius function.