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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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