×

A characterization of arithmetical invariants by the monoid of relations. (English) Zbl 1213.20059

It has been shown by S. T. Chapman, P. A. García-Sánchez, D. Llena, V. Ponomarenko and J. C. Rosales [Manuscr. Math. 120, No. 3, 253-264 (2006; Zbl 1117.20045)] that the knowledge of the monoid of relations in a finitely generated monoid permits to calculate several invariants (catenary degree, tame degree, elasticity, …) related to non-unique factorization. In this paper the author extends this approach to the case when the monoids are not finitely generated.

MSC:

20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems
13A05 Divisibility and factorizations in commutative rings

Citations:

Zbl 1117.20045

References:

[1] Baginski, P., Chapman, S., Rodriguez, R., Schaeffer, G., She, Y.: On the delta set and catenary degree of Krull monoids with infinite cyclic divisor class group. J. Pure Appl. Algebra (2010, to appear) · Zbl 1193.20071
[2] Chapman, S., García-Sánchez, P., 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
[3] 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 · doi:10.1007/s00229-006-0008-8
[4] 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
[5] Geroldinger, A., Grynkiewicz, D., Schmid, W.: The catenary degree of Krull monoids I. arXiv:0911.4882v1 [math.NT] · Zbl 1253.11101
[6] Geroldinger, A., Halter-Koch, F.: Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics. Chapman & Hall, New York (2006) · Zbl 1113.11002
[7] Omidali, M.: The catenary and tame degree of certain numerical semigroups. Forum Math. (to appear). arXiv:1001.1646v2 [math.NT] · Zbl 1252.20057
[8] Philipp, A.: Arithmetic of non-principal orders of algebraic number fields. Manuscript (2011)
[9] Rosales, J., García-Sánchez, P.: Numerical semigroups. Developments in Mathematics, vol. 20. Springer, Dordrecht (2009) · Zbl 1220.20047
[10] 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
[11] Skula, L.: Divisorentheorie einer Halbgruppe. Math. Z. 114, 113–120 (1970) · doi:10.1007/BF01110320
[12] Skula, L.: On {\(\delta\)} n-semigroups. Arch. Math. 1, 43–52 (1981) · Zbl 0494.20036
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.