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Unions of sets of lengths. (English) Zbl 1228.20046

Let \(H\) be a Krull monoid such that every class contains a prime (e.g. the multiplicative monoid of integers of an algebraic number field). S. T. Chapman and W. W. Smith [Isr. J. Math. 71, No. 1, 65-95 (1990; Zbl 0717.13014)] introduced for every positive integer \(k\) the set \(\mathcal V_k\), consisting of all integers \(m\) such that there is an element of \(H\) having factorizations into irreducibles of lengths \(k\) and \(m\).
The authors show that for every \(k\) the set \(\mathcal V_k\) is an interval, pose the question whether the same holds also for non-principal orders in algebraic number fields and answer it positively in the case of the order \(\mathbb Z[\sqrt{-7}]\).

MSC:

20M13 Arithmetic theory of semigroups
11R27 Units and factorization
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
20M14 Commutative semigroups
20M05 Free semigroups, generators and relations, word problems

Citations:

Zbl 0717.13014
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Full Text: DOI Euclid

References:

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