Every finite distributive lattice is isomorphic to the minimizer set of an [formula omitted]-concave set function

Abstract

M♮-concavity is a key concept in discrete convex analysis. For set functions, the class of M♮-concavity is a proper subclass of submodularity. It is a well-known fact that the set of minimizers of a submodular function forms a distributive lattice, where every finite distributive lattice is possible to appear. It is a natural question whether every finite distributive lattice appears as the minimizer set of an M♮-concave set function. This paper affirmatively answers the question.