Quotients of M-convex sets and M-convex functions

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 anal-ysis. 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 func-tions, which are generalizations of linking systems and bimatroids.