## Differences in sets of lengths of Krull monoids with finite class group.(English)Zbl 1090.20034

Let $$G$$ be a finite Abelian group written additively, and for any subset $$G_0$$ of $$G$$ let $$F(G_0)$$ be the free semigroup with unit, generated by elements of $$G_0$$. Moreover let $$B(G_0)=\{\prod_ig_i\in G_0:\sum_ig_i=0\}$$. Elements $$a\in B(G_0)$$ which do not have proper divisors are called minimal, and every $$a$$ is a product of minimal elements. If every factorization of each $$a\in B(G_0)$$ is of the same length, then $$G_0$$ is called half-factorial. Let $$L(a)=\{l_1<l_2<\cdots<l_k\}$$ be the set of all possible lengths of factorizations of $$a$$, denote by $$\Delta(a)$$ the set of consecutive differences $$\{l_2-l_1,l_3-l_2,\dots\}$$, and put $$\Delta(G_0)=\bigcup_{a\in G_0}\Delta(a)$$.
The author is interested in the set $$\Delta^*(G)$$, consisting of all values of $$\min\Delta(G_0)$$, when $$G_0$$ runs over all non-empty subsets of $$G$$ which are not half-factorial. He obtains bounds for maximal and minimal elements of $$\Delta^*(G)$$, and shows that if $$G$$ is an elementary $$p$$-group, $$G=C_p^r$$, then $$\max\Delta^*(G)=\max\{p-2,r-1\}$$. He shows finally that if $$p,q$$ are primes, $$G$$ is an elementary $$p$$-group, $$H$$ is an elementary $$q$$-group, and the sets $$\{L(a):a\in B(G)\}$$ and $$\{L(a):a\in B(H)\}$$ are equal, then $$G$$ and $$H$$ are isomorphic, apart of the case $$G=C_3$$, $$H=C_2^2$$.

### MSC:

 20M14 Commutative semigroups 11R27 Units and factorization 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20K01 Finite abelian groups
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### References:

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