LEADER 04895nam 22007095 450 001 9910624377103321 005 20251113182510.0 010 $a9783031148699$b(electronic bk.) 010 $z9783031148682 024 7 $a10.1007/978-3-031-14869-9 035 $a(MiAaPQ)EBC7123169 035 $a(Au-PeEL)EBL7123169 035 $a(CKB)25201561300041 035 $a(PPN)265855349 035 $a(OCoLC)1349281735 035 $a(DE-He213)978-3-031-14869-9 035 $a(EXLCZ)9925201561300041 100 $a20221026d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe Characterization of Finite Elasticities $eFactorization Theory in Krull Monoids via Convex Geometry /$fby David J. Grynkiewicz 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (291 pages) 225 1 $aLecture Notes in Mathematics,$x1617-9692 ;$v2316 311 08$aPrint version: Grynkiewicz, David J. The Characterization of Finite Elasticities Cham : Springer International Publishing AG,c2022 9783031148682 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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. 330 $aThis 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. 410 0$aLecture Notes in Mathematics,$x1617-9692 ;$v2316 606 $aNumber theory 606 $aCommutative algebra 606 $aCommutative rings 606 $aGroup theory 606 $aConvex geometry 606 $aDiscrete geometry 606 $aNumber Theory 606 $aCommutative Rings and Algebras 606 $aGroup Theory and Generalizations 606 $aConvex and Discrete Geometry 615 0$aNumber theory. 615 0$aCommutative algebra. 615 0$aCommutative rings. 615 0$aGroup theory. 615 0$aConvex geometry. 615 0$aDiscrete geometry. 615 14$aNumber Theory. 615 24$aCommutative Rings and Algebras. 615 24$aGroup Theory and Generalizations. 615 24$aConvex and Discrete Geometry. 676 $a516.08 676 $a516.08 700 $aGrynkiewicz$b David J.$f1978-$01337905 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910624377103321 996 $aThe characterization of finite elasticities$93057590 997 $aUNINA