We demonstrate how primary decomposition of commutative monoid congruences
fails to capture the essence of primary decomposition in commutative rings by
exhibiting a more sensitive theory of mesoprimary decomposition of congruences,
complete with witnesses, associated prime objects, and an analogue of
irreducible decomposition called coprincipal decomposition. We lift the
combinatorial theory of mesoprimary decomposition to binomial ideals in monoid
algebras. The resulting binomial mesoprimary decomposition is a new type of
intersection decomposition for binomial ideals that enjoys computational
efficiency and independence from ground field hypotheses. Furthermore, binomial
primary decomposition is easily recovered from mesoprimary decomposition, as is
binomial irreducible decomposition -- which was previously not known to exist
-- from binomial coprincipal decomposition.