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On the Delta set and catenary degree of Krull monoids with infinite cyclic divisor class group. (English) Zbl 1193.20071

Let \(M\) be a Krull monoid. For a non-unit \(x\in M\) let \(l(x),L(x)\) denote the smallest, resp. the largest length of a factorization of \(x\) into irreducibles. The ratio \(\rho(x)=L(x)/l(x)\) is called the elasticity of \(x\), and the elasticity of \(M\) is defined by \(\rho(M))=\sup_x\{\rho(x)\}\).
The authors consider Krull monoids with infinite cyclic class group and prove (Theorem 1.1) that the condition \(\rho(M)<\infty\) is equivalent to each of three other finiteness conditions, involving the Delta set of \(M\) and its catenary degree.

MSC:

20M14 Commutative semigroups
11B75 Other combinatorial number theory
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20M05 Free semigroups, generators and relations, word problems
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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