Power monoids: A bridge between factorization theory and arithmetic combinatorics

JOURNAL OF ALGEBRA

卷:512

Article

Fan, Yushuang¹  Tringali, Salvatore² 

[1] China Univ Geosci, Math Coll, Beijing, Peoples R China

[2] Karl Franzens Univ Graz, Inst Math & Sci Comp, NAWI Graz, Heinrichstr 36, A-8010 Graz, Austria

关键词: Atoms;    Catenary degree;    Equimorphisms;    Irreducible sets;    Monoids;    Non-unique factorization;    Power monoids;    Sets of lengths;    Set of distances;    Sumsets;    Transfer techniques;   

DOI  :  10.1016/j.jalgebra.2018.07.010

来源: Elsevier

【 摘 要 】

We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of Baeth and Smertnig's work on the factorization theory of non-commutative, but cancellative monoids (Baeth and Smertnig (2015) [5]). Then, we bring in power monoids and, applying the abstract machinery developed in the first part, we undertake the study of their arithmetic. More in particular, let H be a multiplicatively written monoid. The set P-fi(n) (H) of all non-empty finite subsets of H is naturally made into a monoid, which we call the power monoid of H and is non-cancellative unless H is trivial, by endowing it with the operation (X, Y) bar right arrow {xy : (x, y) is an element of X x Y}. Power monoids are, in disguise, one of the primary objects of interest in arithmetic combinatorics, and here for the first time we tackle them from the perspective of factorization theory. Proofs lead to consider various properties of finite subsets of N that can or cannot be split into a sumset in a non-trivial way, giving rise to a rich interplay with additive number theory. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free