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

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