An uncertainty principle for M\"obius inversion on posets

TL;DR

This paper establishes conditions under which M"obius inversion on locally finite posets implies that at least one of the involved functions has infinite support, generalizing Pollack's result with a poset-theoretic proof.

Contribution

It provides a new poset-theoretic proof and generalizes a known result about the support of functions linked by M"obius inversion.

Findings

Supports of functions linked by M"obius inversion are infinite under certain conditions

Generalizes Pollack's result to a broader class of posets

Includes various examples and non-examples

Abstract

We give conditions for a locally finite poset $P$ to have the property that for any functions $f:P\to {\bf C}$ and $g:P\to {\bf C}$ not identically zero and linked by the M\"obius inversion formula, the support of at least one of $f$ and $g$ is infinite. This generalises and gives an entirely poset-theoretic proof of a result of Pollack. Various examples and non-examples are discussed.