• Donald Yau
Book

Part of the Lecture Notes in Mathematics book series (LNM, volume 2192)

1. Front Matter
Pages i-xi
2. Donald Yau
Pages 1-9
3. ### Part I

1. Front Matter
Pages 11-12
2. Donald Yau
Pages 13-37
3. Donald Yau
Pages 39-61
4. Donald Yau
Pages 63-87
5. Donald Yau
Pages 89-105
6. Donald Yau
Pages 107-155
4. ### Part II

1. Front Matter
Pages 157-157
2. Donald Yau
Pages 159-174
3. Donald Yau
Pages 175-189
4. Donald Yau
Pages 191-209
5. Donald Yau
Pages 211-219
6. Donald Yau
Pages 221-239
5. ### Part III

1. Front Matter
Pages 241-242
2. Donald Yau
Pages 243-267
3. Donald Yau
Pages 269-285
4. Donald Yau
Pages 287-289
6. Back Matter
Pages 291-308

### Introduction

Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and outputs, and shows how such operations are wired together to form a larger and more complex operation. This monograph presents a comprehensive study of the combinatorial structure of the various operads of wiring diagrams, their algebras, and the relationships between these operads.

The book proves finite presentation theorems for operads of wiring diagrams as well as their algebras. These theorems describe the operad in terms of just a few operadic generators and a small number of generating relations. The author further explores recent trends in the application of operad theory to wiring diagrams and related structures, including finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the relational algebra. A partial verification of David Spivak’s conjecture regarding the quotient-freeness of the relational algebra is also provided. In the final part, the author constructs operad maps between the various operads of wiring diagrams and identifies their images.

Assuming only basic knowledge of algebra, combinatorics, and set theory, this book is aimed at advanced undergraduate and graduate students as well as researchers working in operad theory and its applications. Numerous illustrations, examples, and practice exercises are included, making this a self-contained volume suitable for self-study.

### Keywords

MSC (2010): 18D50, 37A60, 55P48, 81P45, 94C15 wiring diagrams undirected wiring diagrams operads colored operads operad algebras operad maps propagator algebra discrete systems open dynamical systems relational algebra

#### Authors and affiliations

• Donald Yau
• 1
1. 1.Department of MathematicsThe Ohio State University at NewarkOhioUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-95001-3
• Copyright Information Springer Nature Switzerland AG 2018
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-95000-6
• Online ISBN 978-3-319-95001-3
• Series Print ISSN 0075-8434
• Series Online ISSN 1617-9692
• Buy this book on publisher's site