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\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are ageneralization of Quillen exact categories which provide a framework fordefining algebraic K-theory and Hall algebras in a \emph{non-additive} setting.This formalism is well-suited to the study of categories whose objects havestrong combinatorial flavor.  In this paper, we show that the categories of modules over semirings andhyperrings - algebraic structures which have gained prominence in tropicalgeometry - carry proto-exact structures.  In the first part, we prove that the category of modules over a semiring isequipped with a proto-exact structure; modules over an idempotent semiring havea strong connection to matroids. We also prove that the category of algebraiclattices $\mathcal{L}$ has a proto-exact structure, and furthermore that thesubcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to thecategory of finite $\mathbb{B}$-modules as proto-exact categories, where$\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relationsbetween $\mathcal{L}$ and geometric lattices (simple matroids) from thisperspective.  In the second part, we prove that the category of modules over a hyperringhas a proto-exact structure. In the case of finite modules over the\emph{Krasner hyperfield} $\mathbb{K}$, a well-known relation between finite$\mathbb{K}$-modules and finite incidence geometries yields a combinatorialinterpretation of exact sequences.

Proto-exact categories of modules over semirings and hyperrings