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Asymptotic elasticity in atomic monoids. (English) Zbl 1099.20033
Let \(M\) be a (multiplicative) commutative cancellative monoid in which every nonzero nonunit is a product of irreducible elements, and let \(M^*\) be its subset of nonzero nonunits. For an \(x\in M^*\), its elasticity (resp., asymptotic elasticity) is \(\rho(x)=\sup\{m/n\mid x=x_1\cdots x_m=y_1\cdots y_n\) with each \(x_i,y_j\in M\) irreducible} (resp., \(\overline\rho(x)=\lim_{n\to\infty}\rho(x^n)\)). The elasticity (resp., asymptotic elasticity) of \(M\) is \(\rho(M)=\sup\{\rho(x)\mid x\in M^*\}\), (resp., \(\overline\rho(M)=\sup\{\rho(x)\mid x\in M^*\}\)), and the set of elasticities (resp., asymptotic elasticities) of \(M\) is \({\mathcal R}(M)=\{\rho(x)\mid x\in M^*\}\) (resp., \(\overline{\mathcal R}(M)=\{\overline\rho(x)\mid x\in M^*\}\)).
In this paper, the authors investigate the function \(\overline\rho\) when \(M\) is a block monoid over an Abelian group. Their main result is that \({\mathcal R}(M)= \overline{\mathcal R}(M)=[1,\rho(M)]\cap\mathbb{Q}\) for certain block monoids \(M\).

MSC:
20M14 Commutative semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20M05 Free semigroups, generators and relations, word problems
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