manuscripta math. 120, 253–264 (2006) © Springer-Verlag 2006 S. T. Chapman · P. A. García-Sánchez · D. Llena · V. Ponomarenko · J. C. Rosales The catenary and tame degree in finitely generated commutative cancellative monoids Received: 14 March 2006 / Revised: 30 March 2006 Published online: 20 May 2006 Abstract Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancel- lative atomic monoid, then the catenary degree of S (denoted c( S)) and the tame degree of S (denoted t( S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c( S) and t( S) when M is a finitely generated commutative cancellative monoid. 1. Introduction The study of combinatorial properties of non-unique factorizations in integral domains and monoids has become an active area of interest (see [9] and its ref- erences). Early work in this area focused on study of the elasticity of factorization which describes non-unique factorizations in a “coarse” sense (see for instance [2] where the first, second and fifth authors of the current paper construct an algorithm to compute the elasticity of a Krull monoid with finite divisor class group). Re- cently, the study of more precise invariants associated to non-unique factorizations has become popular (see for instance the papers [5–11]. The two principal such invariants are known as the catenary degree and the tame degree. A summary of the up to date status of research concerning these constants can be found in [9, Chapters 6.4 and 6.5], but needless to say, exact computations of these constants (especially in the case of the tame degree) are not abundant. In Section 3 of this paper, we will describe two methods to compute the catenary degree of a finitely generated commutative cancellative monoid S. These methods are based on the computation of a minimal presentation of S, and we review these computations in Section 2. The material on presentations draws heavily on results from [3] and [13]. Our computations with the catenary degree will lead to a similar method S. T. Chapman (B ): Department of Mathematics, Trinity University, One Trinity Place, San Antonio, TX 78212-7200, USA. e-mail: schapman@trinity.edu P. A. García-Sánchez · J. C. Rosales: Departamento de Álgebra, Universidad de Granada, Granada 18071, España. e-mail: pedro@ugr.es D. Llena: Departamento de Geometría, Topología y Química Orgánica, Universidad de Almería, Almería 04120, España. e-mail: dllena@ual.es V. Ponomarenko: Department of Mathematics, SanDiego State University, 5500 Campanile Dr., San Diego, CA 92182-7720, USA DOI: 10.1007/s00229-006-0008-8