Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an ${ m M}^{atural}$-Concave Set Function

Submodularity is an important concept in combinatorial optimization, and it is often regarded as a discrete analog of convexity. It is a fundamental fact that the set of minimizers of any submodular function forms a distributive lattice. Conversely, it is also known that any finite distributive lattice is isomorphic to the minimizer set of a submodular function, through the celebrated Birkhoffu0027s representation theorem. ${rm M}^{natural}$-concavity is a key concept in discrete convex analysis. It is known for set functions that the class of ${rm M}^{natural}$-concave is a proper subclass of submodular. Thus, the minimizer set of an ${rm M}^{natural}$-concave function forms a distributive lattice. It is natural to ask if any finite distributive lattice appears as the minimizer set of an ${rm M}^{natural}$-concave function. This paper affirmatively answers the question.