zbMATH — the first resource for mathematics

On divisor-closed submonoids and minimal distances in finitely generated monoids. (English) Zbl 07040044
For a finitely generated commutative cancellative monoid \(H\), the authors study the set \(\Delta^*(H)\) of minimal distances occurring in the theory of non-unique factorizations, developed in the book by A. Geroldinger and F. Halter-Koch [Non-unique factorizations. Algebraic, combinatorial and analytic theory. Boca Raton, FL: Chapman & Hall/CRC (2006; Zbl 1113.11002)]. They show that the divisor-closed submonoids \(A\) of \(H\) form a finite lattice (Theorem 4), determine the sets of generators for them, and show how to compute \(\Delta^*(A)\). In the case when \(H\) is an affine semigroup a geometric approach is used to describe its divisor-closed submonoids (Theorem 15). This is used to present an algorithm to compute \(\Delta^*(H)\) for every finitely generated \(H\).
13A05 Divisibility and factorizations in commutative rings
20M13 Arithmetic theory of semigroups
11R27 Units and factorization
20M32 Algebraic monoids
68W30 Symbolic computation and algebraic computation
20M14 Commutative semigroups
52B11 \(n\)-dimensional polytopes
Normaliz; Python
Full Text: DOI
[1] (Anderson, D. D., Factorization in Integral Domain. Factorization in Integral Domain, Lecture Notes in Pure and Appl. Math., vol. 189, (1997), Marcel Dekker: Marcel Dekker New York)
[2] Anderson, David F.; Chapman, Scott T.; Kaplan, Nathan; Torkornoo, Desmond, An algorithm to compute ω-primality in a numerical monoid, Semigroup Forum, 82, 1, 96-108, (2011) · Zbl 1218.20038
[3] Barron, Thomas; O’Neill, Christopher; Pelayo, Roberto, On dynamic algorithms for factorization invariants in numerical monoids, Math. Comput., 86, 307, 2429-2447, (2017) · Zbl 1385.20019
[4] Brøndsted, A., An Introduction to Convex Polytopes, (1983), Springer Science+Business Media: Springer Science+Business Media New York · Zbl 0509.52001
[5] Bruns, W.; Ichim, B.; Römer, T.; Söger, C., The normaliz project, available at
[6] Chang, S. T.; Chapman, S. T.; Smith, W. W., On minimum delta set values in block monoids over cyclic groups, Ramanujan J., 14, 155-171, (2007) · Zbl 1128.20047
[7] Chapman, S. T.; García-Garcí a, J. I.; García-Sánchez, P. A.; Rosales, J. C., Computing the elasticity of a Krull monoid, Linear Algebra Appl., 336, 191-200, (2001) · Zbl 0995.20040
[8] Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Ponomarenko, V.; Rosales, J. C., The catenary and tame degree in finitely generated commutative cancellative monoids, Manuscr. Math., 120, 3, 253-264, (2006) · Zbl 1117.20045
[9] Chapman, S. T.; Schmid, W. A.; Smith, W. W., Minimal distances in Krull monoids, Bull. Lond. Math. Soc., 40, 613-618, (2008) · Zbl 1198.20049
[10] Gao, W.; Geroldinger, A., Systems of Sets of Lengths, II, Abh. Math. Sem. Univ. Hamburg, 31-49, (2000) · Zbl 1036.11054
[11] García-García, J. I.; Moreno, M. A., On morphisms of commutative monoids, Semigroup Forum, 84, 333-341, (2012) · Zbl 1252.20059
[12] García-Garcí a, J. I.; Moreno, M. A.; Vigneron, A., Computation of delta sets of numerical monoids, Monatshefte Math., 178, 3, 457-472, (2015) · Zbl 1343.20061
[13] García-García, J. I.; Moreno, M. A.; Vigneron, A., Computation of the ω-primality and asymptotic ω-primality with applications to numerical semigroups, Isr. J. Math., 206, 1, 395-411, (2015) · Zbl 1337.20067
[14] García-García, J. I.; Vigneron-Tenorio, A., Computing families of Cohen-Macaulay and Gorenstein rings, Semigroup Forum, 88, 3, 610-620, (2014) · Zbl 1319.13016
[15] García-García, J. I.; Marín-Aragón, D.; Vigneron-Tenorio, A., A characterization of some families of Cohen-Macaulay, Gorenstein and/or Buchsbaum rings, Discrete Appl. Math, (2018), Available online 26 March 2018 · Zbl 06893042
[16] García-García, J. I.; Marín-Aragón, D., Integer Smith normal form and some applications written in Python, Available at
[17] Geroldinger, A., Systeme von Längenmengen, Abh. Math. Sem. Univ. Hamburg, 60, 115-130, (1990) · Zbl 0721.11042
[18] Geroldinger, A.; Halter-Koch, F., Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol. 278, (2006), Chapman & Hall/CRC · Zbl 1113.11002
[19] Geroldinger, A.; Hamidoune, Y. O., Zero-sumfree sequences in cyclic groups and some arithmetical applications, J. Théor. Nr. Bordx., 14, 221-239, (2002) · Zbl 1018.11011
[20] Geroldinger, A.; Zhong, Q., The set of minimal distances in Krull monoids, Acta Arith., 173, 97-120, (2016) · Zbl 1360.20053
[21] Grillet, P. A., Commutative Semigroups, (2001), Kluwer Academic Publishers · Zbl 1040.20048
[22] Kainrath, F.; Lettl, G., Geometric Notes on Monoids, Semigroup Forum, vol. 61, 298-302, (2000) · Zbl 0964.20037
[23] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, (1990), Springer · Zbl 0717.11045
[24] O’Neill, Christopher; Ponomarenko, Vadim; Tate, Reuben; Webb, Gautam, On the set of catenary degrees of finitely generated cancellative commutative monoids, Int. J. Algebra Comput., 26, 3, 565-576, (2016) · Zbl 1357.20027
[25] Rosales, J. C.; García-Sánchez, P. A., On Cohen-Macaulay and Gorenstein simplicial affine semigroups, Proc. Edinb. Math. Soc. (2), 41, 3, 517-537, (1998) · Zbl 0904.20048
[26] 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
[27] Rosales, J. C.; García-Sánchez, P. A.; García-Garcí a, J. I., Atomic Commutative Monoids and Their Elasticity, Semigroup Forum, vol. 68, 64-86, (2004) · Zbl 1128.20049
[28] Schrijver, A., Theory of Linear and Integer Programming, (1999), John Wiley & Sons
[29] Sturmfels, B., Groebner Bases and Convex Polytopes, (1995), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.