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Relative block semigroups and their arithmetical applications. (English) Zbl 0769.11038

From the author’s introduction: In a series of papers A. Geroldinger, W. Narkiewicz and the author investigated phenomena of non-unique factorizations in an abstract context but mainly with emphasis to the rings of integers of algebraic number fields. If we are merely interested in the different lengths of factorizations of a given integer, the concept of block semigroups turned out to be the appropriate combinatorial tool for this question. (...) In this paper we shall refine this tool: we introduce relative blocks; with the aid of them we shall study lengths of factorizations of elements in given residue classes.
In §1 we introduce relative block semigroups and determine their algebraic structure; in §2 we apply them to the arithmetic of arbitrary Krull semigroups. In §3 we recall some abstract analytic formula for the number of elements with a given block. Finally in §4 we give arithmetical applications for algebraic number fields.

MSC:

11R27 Units and factorization
11R47 Other analytic theory
20M14 Commutative semigroups
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