  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.
