  Recent results of Kahle and Miller give a method of constructing primary
decompositions of binomial ideals by first constructing "mesoprimary
decompositions" determined by their underlying monoid congruences. These
mesoprimary decompositions are highly combinatorial in nature, and are designed
to parallel standard primary decomposition over Noetherean rings. In this
paper, we generalize mesoprimary decomposition from binomial ideals to
"binomial submodules" of certain graded modules over the corresponding monoid
algebra, analogous to the way primary decomposition of ideals over a Noetherean
ring $R$ generalizes to $R$-modules. The result is a combinatorial method of
constructing primary decompositions that, when restricting to the special case
of binomial ideals, coincides with the method introduced by Kahle and Miller.
