The catenary and tame degree in finitely generated commutative cancellative monoids. (English) Zbl 1117.20045

Let \(S\) be a commutative cancellative atomic monoid. The catenary degree and the tame degree of \(S\) are (somewhat technical) combinatorial invariants of \(S\) which describe the behavior of chains of factorizations in \(S\) (see for example, the book Non-unique factorizations. Algebraic, combinatorial and analytic theory by A. Geroldinger and F. Halter-Koch [Pure Appl. Math. 278. Boca Raton, FL: Chapman & Hall/CRC (2006; Zbl 1113.11002)]).
In this paper, the authors give methods to compute both invariants when \(S\) is finitely generated. These methods are based on the computation of a minimal presentation of \(S\), and there are algorithms to compute this. Several examples are given to illustrate the theory.


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


Zbl 1113.11002


Full Text: DOI


[1] Bowles, C., Chapman, S.T., Kaplan, N., Reiser, D. On delta sets of numerical monoids. J. Algebra Appl (2006)(in press) · Zbl 1115.20052
[2] Chapman S.T., García-García J.I., García-Sánchez P.A., Rosales J.C. (2001) Computing the elasticity of a Krull monoid. Linear Algebra Appl. 336, 201–210 · Zbl 0995.20040
[3] 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. (2006) (in press) · Zbl 1106.20046
[4] Chapman S.T., Krause U., Oeljeklaus E. (2002) On diophantine monoids and their class groups. Pacific J. Math. 207, 125–147 · Zbl 1060.20050
[5] Geroldinger A. (1996) On the structure and arithmetic of finitely primary monoids. Czechoslovak Math. J. 121, 677–695 · Zbl 0879.20032
[6] Geroldinger A. (1997) The catenary degree and tameness of factorizations in weakly Krull domains, Factorization in Integral Domains. Lect. Notes Pure Appl. Math. 180, 113–153 · Zbl 0897.13002
[7] Geroldinger A. (1997) Chains of factorizations and sets of lengths. J. Algebra 188, 331–363 · Zbl 0882.20039
[8] Geroldinger A. (1997) Chains of factorizations in orders of global fields. Colloq. Math. 72, 83–102 · Zbl 0874.11073
[9] Geroldinger A., Halter-Koch F. (2006) Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, vol 278. Chapman & Hall/CRC, Boca Raton · Zbl 1113.11002
[10] Lambert J.-L. (1987) Une borne pour les générateurs des solutions entières positives d ’ une équation diophantienne linéaire. C. R. Acad. Sci. Paris Sér. I Math. 305(2): 39–40
[11] Hassler W. (2004) Factorization in finitely generated domains. J. Pure Appl. Algebra 186, 151–168 · Zbl 1061.13002
[12] Rédei L. (1965) The theory of finitely generated commutative semigroups. Pergamon, Oxford-Edinburgh-New York
[13] Rosales J.C., García-Sánchez P.A., Urbano-Blanco J.M. (1999) On presentations of commutative monoids. Int. J. Algebra Comput. 9(5): 539–553 · Zbl 1028.20037
[14] Rosales J.C., García-Sánchez P.A. (1999) Finitely Generated Commutative Monoids. Nova Science Publishers, New York · Zbl 0966.20028
[15] Schmid W.A. (2005) On invariants related to non-unique factorizations in block monoids and rings of algebraic integers. Math. Slovaca 55, 21–37 · Zbl 1108.11074
[16] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4, 2004. (http://www.gap-system.org)
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.