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Sets of lengths. (English) Zbl 1391.13004

Summary: Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements. An element \(a\) can be a product of \(k\) irreducibles and a product of \(l\) irreducibles. The set \(\mathrm L(a)\) of all possible factorization lengths of \(a\) is called the set of lengths of \(a\), and the system consisting of all these sets \(\mathrm L(a)\) is a well-studied means of describing the nonuniqueness of factorizations of a ring or semigroup. We provide a friendly introduction, which is largely self-contained, to what is known about systems of sets of lengths for rings of integers of algebraic number fields and for transfer Krull monoids of finite type as their generalization.

MSC:

13A05 Divisibility and factorizations in commutative rings
20M13 Arithmetic theory of semigroups
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11R37 Class field theory

Software:

numericalsgps
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References:

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