The Encyclopedia of Mobius Inversions in the Sciences

In a recent paper (https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.7.023016), I proposed to study complex systems through a mereological lens by applying the Möbius inversion theorem. I also covered this in a recent blogpost (https://abeljansma.nl/2025/01/28/mereoPhysics.html). Here, I will collect some of the most important applications of Möbius transformations in the sciences. I will update this table as I find more applications. If you have suggestions, please send me an email, or leave a comment below!

  Field of Study
  Macro quantity
  Mereology
  Micro quantity




  Statistics
  Moments
  Powerset
  Central moments


   
  Moments
  Partitions
  Cumulants


   
  Free moments
  Non-crossing partitions
  Free cumulants


   
  Path signature moments
  Ordered partitions
  Path signature cumulants


   
  Causal effects
  Antichains
  Causal synergy/redundancy


  Information Theory
  Entropy
  Powerset
  Mutual information


   
  Entropy
  Singletons
  Total correlation


   
  Surprisal
  Powerset
  Pointwise mutual information


   
  Joint Surprisal
  Powerset
  Conditional interactions


   
  Mutual Information
  Antichains
  Synergy/redundancy atom


  Biology
  Pheno- & Genotype
  Powerset
  Epistasis


   
  Gene expression profile
  Powerset
  Genetic interactions


   
  Population statistics
  Powerset
  Synergistic treatment effect


  Physics
  Energy
  Powerset
  Ising interactions


   
  Correlation functions
  Partitions
  Ursell functions


   
  Quantum corr. functions
  Partitions
  Scattering amplitudes


  Chemistry
  Molecular property
  Subgraphs
  Fragment contributions


   
  Molecular property
  Reaction poset
  Cluster contributions


  Game Theory
  Coalition value
  Powerset
  Harsanyi dividends


   
  Shapley value
  Supersets
  Normalised coalition synergy


  Artificial Intelligence
  Generative model probabilities
  Powerset
  Feature interaction


   
  Predictive model predictions
  Powerset
  Feature contribution


   
  Dempster-Shafer Belief
  Lattices
  Evidence weight


   
  KL-divergence
  Powerset
  $\Delta_{p|q}$ measure