Commutative monoid duality

TL;DR

This paper introduces new duality frameworks for interacting particle systems based on commutative monoids and semirings, unifying existing dualities and discovering new ones for larger state spaces.

Contribution

It develops a unified approach to dualities in particle systems using algebraic structures, extending known dualities and identifying new ones for multi-element state spaces.

Findings

Unified treatment of additive and cancellative dualities for two-element state spaces

Discovery of several new dualities for systems with three or more states

Framework based on commutative monoids and semirings

Abstract

We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities.