# The monoid of monotone functions on a poset and quasi-arithmetic multiplicities for uniform matroids

## TL;DR

This paper analyzes the algebraic structure of the monoid of monotone functions on posets, explores its properties, and applies findings to quasi-arithmetic multiplicities in uniform matroids, providing new algebraic insights.

## Contribution

It characterizes the monoid of monotone functions on posets, studies its algebraic properties, and applies results to matroid multiplicities, including conjectures for general matroids.

## Key findings

- The monoid ring is normal and Cohen-Macaulay.
- A presentation and prime elements of the monoid are provided.
- Conjectures on irreducibles in matroid multiplicities are proposed.

## Abstract

We describe the structure of the monoid of natural-valued monotone functions on an arbitrary poset. For this monoid we provide a presentation, a characterization of prime elements, and a description of its convex hull. We also study the associated monoid ring, proving that it is normal, and thus Cohen-Macaulay. We determine its Cohen-Macaulay type, characterize the Gorenstein property, and provide a Gr\"obner basis of the defining ideal. Then we apply these results to the monoid of quasi-arithmetic multiplicities on a uniform matroid. Finally we state some conjectures on the number of irreducibles for the monoid of multiplicities on an arbitrary matroid.