  \emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a
generalization of Quillen exact categories which provide a framework for
defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting.
This formalism is well-suited to the study of categories whose objects have
strong combinatorial flavor.
  In this paper, we show that the categories of modules over semirings and
hyperrings - algebraic structures which have gained prominence in tropical
geometry - carry proto-exact structures.
  In the first part, we prove that the category of modules over a semiring is
equipped with a proto-exact structure; modules over an idempotent semiring have
a strong connection to matroids. We also prove that the category of algebraic
lattices $\mathcal{L}$ has a proto-exact structure, and furthermore that the
subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the
category of finite $\mathbb{B}$-modules as proto-exact categories, where
$\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relations
between $\mathcal{L}$ and geometric lattices (simple matroids) from this
perspective.
  In the second part, we prove that the category of modules over a hyperring
has 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 combinatorial
interpretation of exact sequences.
