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Submodularity is an important concept in combinatorial optimization, and itis often regarded as a discrete analog of convexity. It is a fundamental factthat the set of minimizers of any submodular function forms a distributivelattice. Conversely, it is also known that any finite distributive lattice isisomorphic to the minimizer set of a submodular function, through thecelebrated Birkhoff's representation theorem. ${\rm M}^{\natural}$-concavity isa key concept in discrete convex analysis. It is known for set functions thatthe class of ${\rm M}^{\natural}$-concavity is a proper subclass ofsubmodularity. Thus, the minimizer set of an ${\rm M}^{\natural}$-concavefunction forms a distributive lattice. It is natural to ask if any finitedistributive lattice appears as the minimizer set of an ${\rmM}^{\natural}$-concave function. This paper affirmatively answers the question.

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