A note on the M\"obius uncertainty principle for posets

TL;DR

This paper explores two generalizations of Pollack's M"obius inversion uncertainty principle for locally finite posets, providing criteria for its validity and examining specific poset cases.

Contribution

It offers a simplified sufficient criterion and a necessary criterion for the uncertainty principle, disproves Goh's conjecture in general, and confirms it for lattices, introducing a new generalization.

Findings

Disproves Goh's conjecture for general posets

Confirms Goh's conjecture for lattice-structured posets

Shows the uncertainty principle holds for certain posets like finite subsets of natural numbers

Abstract

We consider two generalizations of Pollack's uncertainty principle for M\"obius inversion to locally finite posets. The first generalization was previously studied by Goh. Here, we provide a simplified sufficient criterion for the uncertainty principle to hold. We also provide a necessary criterion for the same which, in particular, disproves Goh's conjectural characterization of posets for which an uncertainty principle holds. Nevertheless, we prove that Goh's conjecture indeed holds when the poset forms a lattice. The second generalization is new and applies to posets with reduced incidence algebras of a certain form. Here, we make some preliminary observations, including the fact that the uncertainty principle holds for the poset of finite subsets of natural numbers and the poset of finite dimensional subspaces of $\mathbb{F}_q^\infty$. Our proofs in these settings are quite different from the proof for the poset of natural numbers under divisibility.