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An uncertainty principle for Möbius inversion on posets

Marcel K. Goh

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 Möbius 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.

An uncertainty principle for Möbius inversion on posets

Abstract

We give conditions for a locally finite poset to have the property that for any functions and not identically zero and linked by the Möbius inversion formula, the support of at least one of and is infinite. This generalises and gives an entirely poset-theoretic proof of a result of Pollack. Various examples and non-examples are discussed.
Paper Structure (4 sections, 5 theorems, 21 equations)

This paper contains 4 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. Examples
  4. Non-examples

Key Result

Theorem 2.1

Let $P$ be a locally finite join-semilattice with bottom element. Suppose that for every $y\in P$, there exist infinitely many $z\in P$ such that i) $z$ covers $y$; and ii) for all $x\le y$, $\mu_P(x,z) = \mu_P(x,y)\mu_P(y,z) = -\mu_P(x,y)$. Then for any $f:P\to \mathbb{C}$ that is not identically z

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 1 more