  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.
