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On divisor-closed submonoids and minimal distances in finitely generated monoids. (English) Zbl 1451.13005

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 \(H\).

MSC:

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

Citations:

Zbl 1113.11002

Software:

Normaliz; Python
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Full Text: DOI arXiv

References:

[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 1456.20065
[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
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