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arXiv · 1902.00864

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

Winfried Bruns·Pedro A. García-Sánchez·Luca Moci
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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.

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Winfried Bruns, Pedro A. García-Sánchez, Luca Moci. 2019-02-03. The monoid of monotone functions on a poset and quasi-arithmetic multiplicities for uniform matroids. https://arxiv.org/abs/1902.00864

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