
TL;DR
This paper introduces the concept of sets of lengths in algebraic structures, exploring their properties and significance in understanding non-unique factorizations in rings of integers and transfer Krull monoids.
Contribution
It provides a largely self-contained introduction to the known results about systems of sets of lengths in algebraic number theory and transfer Krull monoids.
Findings
Explains the structure of sets of lengths in algebraic number fields.
Describes the role of transfer Krull monoids in factorization theory.
Summarizes key properties and known results in the area.
Abstract
Oftentimes the elements of a ring or semigroup can be written as finite products of irreducible elements, say , where the number of irreducible factors is distinct. The set of all possible factorization lengths of is called the set of lengths of , and the full system is a well-studied means of describing the non-uniqueness of factorizations of . 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.
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Taxonomy
TopicsOptics and Image Analysis
