Sets of Lengths

2015/09/24 by Alfred Geroldinger, Geroldinger, Alfred · 4 citations
Business, Management and Accounting · #11B30 #11R27 #13A05 #13F05 #16H10 #16U30 #20M13 #FOS: Mathematics #Group Theory (math.GR) #Optics and Image Analysis #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1509.07462

openalex publication_date 2015/09/24 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Oftentimes the elements of a ring or semigroup H can be written as finite products of irreducible elements, say a=u₁ ⋅ … ⋅ u_k = v₁ ⋅ … ⋅ v_ℓ, where the number of irreducible factors is distinct. The set \mathsf L (a) ⊂ \mathbb N of all possible factorization lengths of a is called the set of lengths of a, and the full system \mathcal L (H) = \ \mathsf L (a) | a ∈ H \ is a well-studied means of describing the non-uniqueness of factorizations of H. 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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