A structure theorem for sets of lengths. (English) Zbl 0926.11082

If \(R\) is a ring of integers in an algebraic number field, which is not half-factorial, then the set of lengths of factorization of any non-zero element has a rather regular structure and a similar result holds also in more general domains as well as in certain monoids [the author, Math. Z. 197, 505-529 (1988; Zbl 0618.12002); J. Algebra 188, 331-362 (1997; Zbl 0882.20039)].
In this paper a combinatorial approach is presented, which allows to obtain a structure theorem for sets of factorization lengths in a large class of monoids. This result permits the description of these sets in the case of weakly Krull domains, and hence, in particular, in non-maximal orders of algebraic number fields.


11R27 Units and factorization
20N02 Sets with a single binary operation (groupoids)
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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