In a recent paper, I proposed to study complex systems through a mereological lens by applying the Möbius inversion theorem. I also covered this in a recent blogpost. 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