  We extend the notion of standard pairs to the context of monomial ideals in
semigroup rings. Standard pairs can be used as a data structure to encode such
monomial ideals, providing an alternative to generating sets that is well
suited to computing intersections, decompositions, and multiplicities. We give
algorithms to compute standard pairs from generating sets and vice versa and
make all of our results effective. We assume that the underlying semigroup ring
is positively graded, but not necessarily normal. The lack of normality is at
the root of most challenges, subtleties, and innovations in this work.
