Generalized Möbius inversion beyond posets for complex systems

Investigate whether decompositions obtained from generalizations of Möbius inversion to skeletal categories (including monoids and groupoids) and to bialgebras, beyond locally finite partial orders, faithfully and fruitfully capture higher-order structure in complex systems.

Background

The paper’s framework primarily treats mereologies as locally finite posets, enabling Möbius inversion to define microscopic interactions from macroscopic observables. The authors note that Möbius inversion admits broader generalizations to skeletal categories (such as monoids and groupoids) and to bialgebras.

They highlight the potential of exploring mereologies outside posets and ask whether such generalized decompositions can meaningfully describe higher-order interactions and emergent behaviors in complex systems.