# Sheaves of Algebras over Boolean Spaces

• Arthur Knoebel
Book

1. Front Matter
Pages i-xii
2. Arthur Knoebel
Pages 1-18
3. Arthur Knoebel
Pages 19-53
4. Arthur Knoebel
Pages 55-77
5. Arthur Knoebel
Pages 79-107
6. Arthur Knoebel
Pages 109-146
7. Arthur Knoebel
Pages 147-178
8. Arthur Knoebel
Pages 179-219
9. Arthur Knoebel
Pages 221-234
10. Arthur Knoebel
Pages 235-260
11. Arthur Knoebel
Pages 261-272
12. Arthur Knoebel
Pages 273-281
13. Arthur Knoebel
Pages 283-294
14. Back Matter
Pages 295-329

### Introduction

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.

### Keywords

Boolean algebras Boolean spaces Stone's theorem categorical formulations central idempotent commutative von Neumann regular rings complex and factor element metric space sheaves of algebras shells, rings, lattices

#### Authors and affiliations

• Arthur Knoebel
• 1
1. 1.AlbuquerqueUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-0-8176-4642-4