Any Finite Distributive Lattice is Isomorphic to the Minimizer Set of an ${\rm M}^{\natural}$-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 Birkhoff's representation theorem. ${\rm M}^{{\natural}$-concavity} is a key concept in discrete convex analysis. It is known for set functions that the class of ${\rm M}^{{\natural}$-concavity} is a proper subclass of submodularity. 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.