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

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.