Díaz-Ramírez, J. D.; García-García, J. I.; Sánchez-R.-Navarro, A.; Vigneron-Tenorio, A. A geometrical characterization of proportionally modular affine semigroups. (English) Zbl 1521.20129 Result. Math. 75, No. 3, Paper No. 99, 22 p. (2020). Summary: A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality \(f_1 x_1 + \cdots + f_n x_n \bmod b \le g_1 x_1 + \cdots + g_n x_n\), where \(g_1, \dots ,g_n, f_1,\ldots ,f_n \in \mathbb{Z}\), and \(b \in \mathbb{N}\). In this work, a geometrical characterization of these semigroups is given. On the basis of this geometrical approach, some algorithms are provided to check if a semigroup \(S\) in \(\mathbb{N}^n\), with \(\mathbb{N}^n{\setminus} S\) a finite set, is a proportionally modular affine semigroup. Cited in 2 Documents MSC: 20M14 Commutative semigroups 20-04 Software, source code, etc. for problems pertaining to group theory 20-08 Computational methods for problems pertaining to group theory Keywords:affine semigroup; modular Diophantine inequalities; numerical semigroup; proportionally modular numerical semigroup Software:NumericMonoid; PropModSemig.m; Mathematica; nsgtree; GitHub PDFBibTeX XMLCite \textit{J. D. Díaz-Ramírez} et al., Result. Math. 75, No. 3, Paper No. 99, 22 p. (2020; Zbl 1521.20129) Full Text: DOI arXiv References: [1] Bruns, W.; Gubeladze, J., Polytopes: Rings, and K-theory (2009), Dordrecht: Springer, Dordrecht · Zbl 1168.13001 [2] De Loera, J.A., Malkin, P.N., Parrilo, P.A.: Computation with polynomial equations and inequalities arising in combinatorial optimization. In: Mixed Integer Nonlinear Programming, IMA Volumes in Mathematics and its Applications, vol. 154, pp. 447-481. Springer, New York (2012) · Zbl 1242.90191 [3] Díaz-Ramírez, J.D., García-García, J.I., Sánchez-R.-Navarro, A., Vigneron-Tenorio, A.: PropModSemig.m. http://fqm366.uca.es/propmodsemig-tar (2019) [4] Failla, G.; Peterson, C.; Utano, R., Algorithms and basic asymptotics for generalized numerical semigroups in \(\mathbb{N}^p\), Semigroup Forum, 92, 2, 460-473 (2016) · Zbl 1384.20047 · doi:10.1007/s00233-015-9690-8 [5] Fromentin, J., Hivert, F.: Computing the number of numerical monoid of a given genus. https://github.com/hivert/NumericMonoid (2013-2018) [6] Fromentin, J.; Hivert, F., Exploring the tree of numerical semigroups, Math. Comput., 85, 301, 2553-2568 (2016) · Zbl 1344.20075 · doi:10.1090/mcom/3075 [7] García-García, JI; Marín-Aragón, D.; Vigneron-Tenorio, A., An extension of Wilf’s conjecture to affine semigroups, Semigroup Forum, 96, 2, 396-408 (2018) · Zbl 1456.20065 · doi:10.1007/s00233-017-9906-1 [8] García-García, JI; Moreno-Frías, MA; Vigneron-Tenorio, A., Proportionally modular affine semigroups, J. Algebra Appl., 17, 1, 1850017 (2018) · Zbl 1439.13064 · doi:10.1142/S0219498818500172 [9] Rosales, JC; García-Sánchez, PA, Numerical Semigroups. Developments in Mathematics (2009), New York: Springer, New York [10] Rosales, JC; García-Sánchez, PA; García-García, JI; Urbano-Blanco, JM, Proportionally modular Diophantine inequalities, J. Number Theory, 103, 2, 281-294 (2003) · Zbl 1039.20036 · doi:10.1016/j.jnt.2003.06.002 [11] UCA Supercomputer Service: http://supercomputacion.uca.es/ [12] Wolfram Research, Inc.: Mathematica, Version 11.2, Champaign, IL (2017) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.