Exact sequences of commutative monoids and semimodules

Basic homological lemmas well known for modules over rings and, more generally, in the context of abelian categories, have been extended to many other concrete and abstract-categorical contexts by various authors. We propose a new such extension, specifically for commutative monoids and semimodules; these two contexts are equivalent since the forgetful functors from varieties of semimodules to the variety of commutative monoids preserve all limits and colimits.