Umfang:
Online-Ressource (XII, 331p. 63 illus, digital)
ISBN:
9780817646424
,
0817646426
,
9781283444408
,
1283444402
Serie:
SpringerLink
Inhalt:
Preface -- Introduction -- Algebra -- Tools -- Complexes and their Sheaves -- Boolean Subsemilattices -- Sheaves from Factor Congruences -- Shells -- Baer-Stone Shells -- Strict Shells -- Varieties Generated by Preprimal Algebras -- Return to General Algebras -- Further Examples Pointing to Future Research -- List of Symbols -- References -- Index.
Inhalt:
Sheaves of Algebras over Boolean Spaces comprehensively covers sheaf theory as applied to universal algebra. Sheaves decompose general algebras into simpler pieces called the stalks. A classical case is commutative von Neumann regular rings, whose stalks are fields. Other classical theorems also extend to shells, a common generalization of rings and lattices. This text presents intuitive ideas from topology such as the notion of metric space and the concept of central idempotent from ring theory. These lead to the abstract notions of complex and factor element, respectively. Factor elements are defined by identities, discovered for shells for the first time, explaining why central elements in rings and lattices have their particular form. Categorical formulations of the many representations by sheaves begin with adjunctions and move to equivalences as the book progresses, generalizing Stone’s theorem for Boolean algebras. Half of the theorems provided in the text are new; the rest are presented in a coherent framework, starting with the most general, and proceeding to specific applications. Many open problems and research areas are outlined, including a final chapter summarizing applications of sheaves in diverse fields that were not covered earlier in the book. This monograph is suitable for graduate students and researchers, and it will serve as an excellent reference text for those who wish to learn about sheaves of algebras.
Anmerkung:
Includes bibliographical references and index
,
Sheaves of Algebras over Boolean Spaces; Preface; Contents; I: Introduction; 1. History; 2. Survey of Results; II: Algebra; 1. Universal Algebra; 2. Products and Factor Objects; III: Tools; 1. Model Theory; 2. Category Theory; 3. Topology; 4. Boolean Algebras; IV: Complexes and their Sheaves; 1. Concepts; 2. Constructions; 3. Categorical Reformulation; V: Boolean Subsemilattices; 1. Identifying the Congruences; 2. Constructing the Complex; 3. Special Sheaves; 4. Categorical Recapitulation; VI: Sheaves from Factor Congruences; 1. Factorial Braces; 2. Boolean Algebras of Factor Objects
,
3. Algebras Having Boolean Factor Congruences4. Their Categories; VII: Shells; 1. Algebras with a Multiplication; 2. Half-shells; 3. Shells; 4. Reprise; 5. Separator Algebras; 6. Categories of Shells; VIII: Baer-Stone Shells; 1. Integrality; 2. Regularity; IX: Strict Shells; 1. Nilpotents and Null-symmetry; 2. Converses and Axiomatics; 3. Adding a Unity or a Loop; X: Varieties Generated by Preprimal Algebras; 1. Overview; 2. From Permutations; 3. From Groups; 4. From Subsets; 5. Remaining Preprimal Varieties; XI: Return to General Algebras; 1. Iteration; 2. Self Help
,
XII: Further Examples Pointing to Future Research1. From Classical Algebra; 2. Algebras from Logic; 3. From Model Theory; 4. Beyond Sheaves over Boolean Spaces; 5. Many Choices; List of Symbols; References; Index
Weitere Ausg.:
ISBN 9780817642181
Weitere Ausg.:
Buchausg. u.d.T. ISBN 9780817642181
Sprache:
Englisch
Schlagwort(e):
Bündel
;
Boolescher Raum
;
Boolesche Garbe
DOI:
10.1007/978-0-8176-4642-4
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
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