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The monoid of monotone functions on a poset and quasi-arithmetic multiplicities for uniform matroids. (English) Zbl 07286488
Summary: 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 Gröbner 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.
06A06 Partial orders, general
06A07 Combinatorics of partially ordered sets
05B35 Combinatorial aspects of matroids and geometric lattices
Full Text: DOI
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