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

## TL;DR

This paper proves that every finite distributive lattice can be represented as the set of minimizers of an ${
m M}^{
atural}$-concave set function, extending the known relationship between lattices and submodular functions.

## Contribution

It establishes that any finite distributive lattice is isomorphic to the minimizer set of an ${
m M}^{
atural}$-concave function, broadening the understanding of discrete convex analysis.

## Key findings

- Any finite distributive lattice can be realized as a minimizer set of an ${
m M}^{
atural}$-concave function.
- The result extends the classical correspondence between distributive lattices and submodular functions.
- Provides a new connection between lattice theory and discrete convex analysis.

## Abstract

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.