The Characterization of Finite Elasticities

Factorization Theory in Krull Monoids via Convex Geometry

David J. Grynkiewicz (Autor)

Buch | Softcover

XII, 282 Seiten

2022 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-14868-2 (ISBN)

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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.

- 1. Introduction. - 2. Preliminaries and General Notation. - 3. Asymptotically Filtered Sequences, Encasement and Boundedness. - 4. Elementary Atoms, Positive Bases and Reay Systems. - 5. Oriented Reay Systems. - 6. Virtual Reay Systems. - 7. Finitary Sets. - 8. Factorization Theory.

"This work is a well-written and well-organized research monograph which is accessible for the non-specialist. It is a masterpiece of research in factorization theory and convex geometry." (Alfred Geroldinger, Mathematical Reviews, July, 2024)

Erscheinungsdatum	28.10.2022
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	XII, 282 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	456 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Carathéordory's Theorem • Carathéordory’s Theorem • catenary degree • Convex cone • Delta Set • Elasticity • Factorization • Infinite Subsets of Lattice Points • Krull domain • Krull Monoid • lattice • Minimal Positive Basis • Positive Basis • Primitive Partition Identities • sets of lengths • Simplicial Fan • Structure Theorem for Unions • Transfer Krull Domain • Well-quasi-ordering • zero-sum • Zero-sum Sequence
ISBN-10	3-031-14868-1 / 3031148681
ISBN-13	978-3-031-14868-2 / 9783031148682
Zustand	Neuware