×

A characterization of arithmetical invariants by the monoid of relations. II: The monotone catenary degree and applications to semigroup rings. (English) Zbl 1323.20057

It has been shown by S. T. Chapman, P. A. García-Sánchez, D. Llena, V. Ponomarenko and J. C. Rosales [in Manuscr. Math. 120, No. 3, 253-264 (2006; Zbl 1117.20045)] that algebraic properties of the monoid of relations of a finitely generated atomic monoid \(M\) can be used for the determination of several arithmetical invariants of \(M\), including catenary degree and elasticity. In part I of the paper [A. Philipp, Semigroup Forum 81, No. 3, 424-434 (2010; Zbl 1213.20059)] the author showed how this can be done also for monoids which are not finitely generated. In this paper the same method is used to study the monotone catenary degree of \(M\). This is applied to the determination of arithmetical invariants of semigroup rings and generalized power series.

MSC:

20M13 Arithmetic theory of semigroups
20M14 Commutative semigroups
13A05 Divisibility and factorizations in commutative rings
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Barucci, V.: Numerical semigroup algebras. In: Multiplicative Ideal Theory in Commutative Algebra, pp. 39-53. Springer, New York (2006) · Zbl 1115.13027
[2] Barucci, V., Dobbs, D.E., Fontana, M.: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Mem. Am. Math. Soc. 125(598), x+78 (1997) · Zbl 0868.13003
[3] Blanco, V., García-Sánchez, P.A., Geroldinger, A.: Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids. Illinois J. Math. 56 (2012) · Zbl 1279.20072
[4] Chapman, S.T., García-Sánchez, P.A., Llena, D.: The catenary and tame degree of numerical monoids. Forum Math. 21(1), 117-129 (2009) · Zbl 1177.20070 · doi:10.1515/FORUM.2009.006
[5] Chapman, S.T., García-Sánchez, P.A., Llena, D., Rosales, J.C.: Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations. Discrete Appl. Math. 154(14), 1947-1959 (2006) · Zbl 1106.20046 · doi:10.1016/j.dam.2006.03.013
[6] 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. Manuscripta Math. 120(3), 253-264 (2006) · Zbl 1117.20045 · doi:10.1007/s00229-006-0008-8
[7] Contejean, E., Devie, H.: An efficient incremental algorithm for solving systems of linear Diophantine equations. Inf. Comput. 113(1), 143-172 (1994) · Zbl 0809.11015 · doi:10.1006/inco.1994.1067
[8] Domenjoud, E.: Solving systems of linear Diophantine equations: an algebraic approach. In: Mathematical Foundations of Computer Science, 1991, (Kazimierz Dolny, 1991). Lecture Notes in Computer Science, vol. 520, pp. 141-150. Springer, Berlin (1991) · Zbl 0782.11008
[9] Foroutan, A.: Monotone chains of factorizations. In: Focus on Commutative Rings Research, pp. 107-130. Nova Sci. Publ., New York (2006) · Zbl 1165.13302
[10] Foroutan, A., Geroldinger, A.: Monotone chains of factorizations in C-monoids. In: Arithmetical Properties of Commutative Rings and Monoids. Lecture Notes in Pure Applied Mathematics, vol. 241, pp. 99-113. Chapman & Hall/CRC, Boca Raton (2005) · Zbl 1095.20040
[11] Foroutan, A., Hassler, W.: Chains of factorizations and factorizations with successive lengths. Commun. Algebra 34(3), 939-972 (2006) · Zbl 1087.11067 · doi:10.1080/00927870500441916
[12] Geroldinger, A., Grynkiewicz, D.J., Schaeffer, G.J., Schmid, W.A.: On the arithmetic of Krull monoids with infinite cyclic class group. J. Pure Appl. Algebra 214(12), 2219-2250 (2010) · Zbl 1208.13003 · doi:10.1016/j.jpaa.2010.02.024
[13] Geroldinger, A., Grynkiewicz, D.J., Schmid, W.: The catenary degree of Krull monoids I. J. Théor. Nombres Bordeaux 23(1), 137-169 (2011) · Zbl 1253.11101 · doi:10.5802/jtnb.754
[14] Geroldinger, A., Halter-Koch, F.: Nonunique factorizations. In: Pure and Applied Mathematics, vol. 278. Algebraic, Combinatorial and Analytic Theory. Chapman & Hall/CRC, Boca Raton (2006) · Zbl 1113.11002
[15] Geroldinger, A., Hassler, W.: Arithmetic of Mori domains and monoids. J. Algebra 319(8), 3419-3463 (2008) · Zbl 1195.13022 · doi:10.1016/j.jalgebra.2007.11.025
[16] Geroldinger, A., Kainrath, F.: On the arithmetic of tame monoids with applications to Krull monoids and Mori domains. J. Pure Appl. Algebra 214(12), 2199-2218 (2010) · Zbl 1207.20055 · doi:10.1016/j.jpaa.2010.02.023
[17] Geroldinger, A., Liebmann, A., Philipp, A.: Long zero-sum free sequences. Periodica Math. Hung. 64, 213-225 (2012) · Zbl 1263.11038 · doi:10.1007/s10998-012-3378-6
[18] Hassler, W.: Faktorisierung in eindimensionalen Integritätsbereichen. PhD thesis, Karl-Franzens Universität Graz (Advisor: F. Halter-Koch) (2000) · Zbl 0809.11015
[19] Kainrath, F.: Arithmetic of Mori domains and monoids: the global case (manuscript) · Zbl 1394.20035
[20] Malcolmson, P., Okoh, F.: Factorization in subalgebras of the polynomial algebra. Houston J. Math. 35(4), 991-1012 (2009) · Zbl 1183.13030
[21] Malcolmson, P., Okoh, F.: Power series extensions of half-factorial domains. J. Pure Appl. Algebra 213(4), 493-495 (2009) · Zbl 1155.13309 · doi:10.1016/j.jpaa.2008.07.014
[22] Omidali, M.: The catenary and tame degree of certain numerical semigroups. Forum Mathematicum 24, 627-640 (2012) · Zbl 1252.20057 · doi:10.1515/form.2011.078
[23] Ordaz, O., Philipp, A., Santos, I., Schmid, W.: On the Olson and the strong Davenport constants. Journal de Théorie des Nombres Bordeaux 23, 715-750 (2011) · Zbl 1252.11011 · doi:10.5802/jtnb.784
[24] Philipp, A.: A characterization of arithmetical invariants by the monoid of relations. Semigroup Forum 81, 424-434 (2010) · Zbl 1213.20059 · doi:10.1007/s00233-010-9218-1
[25] Philipp, A.: Nonunique factorizations—a semigroup-theoretic algorithmic approach with applications to non-principal orders in algebraic number fields. PhD thesis, Karl-Franzens Universität Graz (Advisors: F. Halter-Koch und A. Geroldinger) (2010) · Zbl 1155.13309
[26] A. Philipp. A precise result on the arithmetic of non-principal orders in algebraic number fields. J. Algebra Appl. 11 (2012) · Zbl 1303.11126
[27] Rosales, J.C., García-Sánchez, P.A.: Nonnegative elements of subgroups of \[{ Z}^nZn\]. Linear Algebra Appl. 270, 351-357 (1998) · Zbl 0890.15016 · doi:10.1016/S0024-3795(97)00309-1
[28] Rosales, J.C., García-Sánchez, P.A.: Numerical semigroups. In: Developments in Mathematics, vol. 20. Springer, New York (2009) · Zbl 1220.20047
[29] Rosales, J.C., García-Sánchez, P.A., Urbano-Blanco, J.M.: On presentations of commutative monoids. Int. J. Algebra Comput. 9(5), 539-553 (1999) · Zbl 1028.20037 · doi:10.1142/S0218196799000333
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.