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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.
##### MSC:
 06A06 Partial orders, general 06A07 Combinatorics of partially ordered sets 05B35 Combinatorial aspects of matroids and geometric lattices
##### Software:
GAP; Normaliz; NormalizInterface; numericalsgps
Full Text:
##### References:
 [1] Boussicault, A.; Féray, V.; Lascoux, A.; Reiner, V., Linear extension sums as valuations on cones, J. Algebraic Comb., 35, 573-610 (2012) · Zbl 1242.05274 [2] Brändén, P.; Moci, L., The multivariate arithmetic Tutte polynomial, Trans. Am. Math. Soc., 366, 10, 5523-5540 (2014) · Zbl 1300.05133 [3] Bruns, W.; Herzog, J., Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics (1998), Cambridge University Press · Zbl 0909.13005 [4] Bruns, W.; Gubeladze, J., Polytopes, Rings, and K-Theory, Springer Monographs in Mathematics (2009), Springer: Springer Dordrecht · Zbl 1168.13001 [5] Bruns, W.; Ichim, B.; Römer, T.; Sieg, R.; Söger, C., Normaliz. Algorithms for rational cones and affine monoids, available at [6] D’Adderio, M.; Moci, L., Arithmetic matroids, the Tutte polynomial and toric arrangements, Adv. Math., 232, 335-367 (2013) · Zbl 1256.05039 [7] Dedekind, R., Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler, Festschr. Hoch. Braunschw. u. ges. Werke (II), 103-148 (1897) · JFM 28.0186.04 [8] Delgado, M.; García-Sánchez, P. A.; Morais, J., NumericalSgps, a package for numerical semigroups (2018), Version 1.1.8 (Refereed GAP package) · Zbl 1365.68487 [9] Delucchi, E.; Moci, L., Products of arithmetic matroids and quasipolynomial invariants of CW-complexes, J. Comb. Theory, Ser. A, 157, 28-40 (2018) · Zbl 1385.05029 [10] Dupont, C.; Fink, A.; Moci, L., Universal Tutte characters via combinatorial coalgebras, Algebraic Combin., 1, 5, 603-651 (2018) · Zbl 1433.16037 [11] Féray, V.; Reiner, V., P-partitions revisited, J. Commut. Algebra, 4, 101-152 (2012) · Zbl 1261.06007 [12] Fink, A.; Moci, L., Matroids over a ring, J. Eur. Math. Soc., 18, 4, 681-731 (2016) · Zbl 1335.05031 [13] Fink, A.; Moci, L., Polytopes and parameter spaces for matroids over valuation rings (with A. Fink), Adv. Math., 343, 449-494 (2019) [14] GAP - Groups, Algorithms, and Programming, Version 4.9.0 (2018) [15] Gutsche, S.; Horn, M.; Söger, C., NormalizInterface, GAP wrapper for Normaliz (2016), Version 0.9.8 (GAP package) [16] Herzog, J., Generators and relations of abelian semigroups and semigroup rings, Manuscr. Math., 3, 175-193 (1970) · Zbl 0211.33801 [17] Hibi, T., Distributive lattices, affine semigroup rings and algebras with straightening laws, (Commutative Algebra and Combinatorics. Commutative Algebra and Combinatorics, Kyoto, 1985. Commutative Algebra and Combinatorics. Commutative Algebra and Combinatorics, Kyoto, 1985, Adv. Stud. Pure Math., vol. 11 (1987), North-Holland: North-Holland Amsterdam), 93-109 [18] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. Math., 96, 318-337 (1972) · Zbl 0233.14010 [19] Moci, L., A Tutte polynomial for toric arrangements, Trans. Am. Math. Soc., 364, 2, 1067-1088 (2012) · Zbl 1235.52038 [20] OEIS, The On-Line Encyclopedia of Integer Sequences [21] Oxley, J. G., Matroid Theory (1992), Oxford University Press: Oxford University Press Oxford · Zbl 0784.05002 [22] Pagaria, R., Orientable arithmetic matroids, Discrete Math., 343, Article 111872 pp. (2020) · Zbl 1437.05041 [23] Pagaria, R.; Paolini, G., Representations of torsion-free arithmetic matroids [24] Rosales, J. C.; García-Sánchez, P. A., Finitely Generated Commutative Monoids (1999), Nova Science Publishers, Inc.: Nova Science Publishers, Inc. New York · Zbl 0966.20028 [25] Stanley, R. P., Ordered Structures and Partitions, Memoirs of the American Mathematical Society, vol. 119 (1972), American Mathematical Society: American Mathematical Society Providence, R.I. · Zbl 0246.05007 [26] Stanley, R. P.; Rota, G.-C., Enumerative Combinatorics (1997), Cambridge University Press: Cambridge University Press Cambridge [27] Stephen, T.; Yusun, T., Counting inequivalent monotone Boolean functions, Discrete Appl. Math., 167, 15-24 (2014) · Zbl 1311.06011
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