04895nam 22007095 450 991062437710332120251113182510.09783031148699(electronic bk.)978303114868210.1007/978-3-031-14869-9(MiAaPQ)EBC7123169(Au-PeEL)EBL7123169(CKB)25201561300041(PPN)265855349(OCoLC)1349281735(DE-He213)978-3-031-14869-9(EXLCZ)992520156130004120221026d2022 u| 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierThe Characterization of Finite Elasticities Factorization Theory in Krull Monoids via Convex Geometry /by David J. Grynkiewicz1st ed. 2022.Cham :Springer International Publishing :Imprint: Springer,2022.1 online resource (291 pages)Lecture Notes in Mathematics,1617-9692 ;2316Print version: Grynkiewicz, David J. The Characterization of Finite Elasticities Cham : Springer International Publishing AG,c2022 9783031148682 Includes bibliographical references and index.Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Convex Geometry -- 1.2 Krull Domains, Transfer Krull Monoids and Factorization -- 1.3 Zero-Sum Sequences -- 1.4 Overview of Main Results -- 2 Preliminaries and General Notation -- 2.1 Convex Geometry -- 2.2 Lattices and Partially Ordered Sets -- 2.3 Sequences and Rational Sequences -- 2.4 Arithmetic Invariants for Transfer Krull Monoids -- 2.5 Asymptotic Notation -- 3 Asymptotically Filtered Sequences, Encasement and Boundedness -- 3.1 Asymptotically Filtered Sequences -- 3.2 Encasement and Boundedness -- 4 Elementary Atoms, Positive Bases and Reay Systems -- 4.1 Basic Non-degeneracy Characterizations -- 4.2 Elementary Atoms and Positive Bases -- 4.3 Reay Systems -- 4.4 -Filtered Sequences, Minimal Encasement and Reay Systems -- 5 Oriented Reay Systems -- 6 Virtual Reay Systems -- 7 Finitary Sets -- 7.1 Core Definitions and Properties -- 7.2 Series Decompositions and Virtualizations -- 7.3 Finiteness Properties of Finitary Sets -- 7.4 Interchangeability and the Structure of X(G0) -- 8 Factorization Theory -- 8.1 Lambert Subsets and Elasticity -- 8.2 The Structure of Atoms and Arithmetic Invariants -- Summary -- 8.3 Transfer Krull Monoids Over Subsets of Finitely Generated Abelian Groups -- Summary -- References -- Index.This book develops a new theory in convex geometry, generalizing positive bases and related to Carathéordory’s Theorem by combining convex geometry, the combinatorics of infinite subsets of lattice points, and the arithmetic of transfer Krull monoids (the latter broadly generalizing the ubiquitous class of Krull domains in commutative algebra) This new theory is developed in a self-contained way with the main motivation of its later applications regarding factorization. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. Among the most important is the elasticity, which measures the ratio between the maximum and minimum number of atoms in any factorization. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. Via the developed material in convex geometry, we characterize when finite elasticity holds for any Krull domain with finitely generated class group $G$, with the results extending more generally to transfer Krull monoids. This book is aimed at researchers in the field but is written to also be accessible for graduate students and general mathematicians.Lecture Notes in Mathematics,1617-9692 ;2316Number theoryCommutative algebraCommutative ringsGroup theoryConvex geometryDiscrete geometryNumber TheoryCommutative Rings and AlgebrasGroup Theory and GeneralizationsConvex and Discrete GeometryNumber theory.Commutative algebra.Commutative rings.Group theory.Convex geometry.Discrete geometry.Number Theory.Commutative Rings and Algebras.Group Theory and Generalizations.Convex and Discrete Geometry.516.08516.08Grynkiewicz David J.1978-1337905MiAaPQMiAaPQMiAaPQ9910624377103321The characterization of finite elasticities3057590UNINA