Long Sets of Lengths With Maximal Elasticity. (1st December 2018)

• Record Type: Journal Article
• Authors: Geroldinger, Alfred; Zhong, Qinghai
• Is Part Of: Canadian journal of mathematics. Volume 70:Number 6(2018)
• Page Start: 1284
• Page End: 1318
• Publication Date: 2018-12-01
• Subjects: 13A05 -- 13F05 -- 16H10 -- 16U30 -- 20M13; transfer Krull monoid -- weakly Krull monoid -- set of length -- elasticity; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2017-043-4
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: We introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid $H$, let ${{\Delta }_{\rho }}\left( H \right)$ be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity $\rho \left( H \right)$ . We study ${{\Delta }_{\rho }}\left( H \right)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.