  We unify the study of quotients of matroids, polymatroids, valuated matroids
and strong maps of submodular functions in the framework of Murota's discrete
convex analysis. As a main result, we compile a list of ten equivalent
characterizations of quotients for M-convex sets, generalizing existing
formulations for (poly)matroids and submodular functions. We also initiate the
study of quotients of M-convex functions, constructing a hierarchy of four
separate characterizations. Our investigations yield new insights into the
fundamental operation of induction, as well as the structure of linking sets
and linking functions, which are generalizations of linking systems and
bimatroids.
