Wagner’s Theory of Generalised Heaps

1. Front Matter

   Pages i-xv

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2. Introduction

   • Christopher D. Hollings, Mark V. Lawson

   Pages 1-3

3. Viktor Vladimirovich Wagner (1908–1981)

   • Christopher D. Hollings, Mark V. Lawson

   Pages 5-6

4. Wagner’s Work in Historical Context

   • Christopher D. Hollings, Mark V. Lawson

   Pages 7-23

5. Notes on the Translations

   • Christopher D. Hollings, Mark V. Lawson

   Pages 25-29

6. A Ternary Algebraic Operation in the Theory of Coordinate Structures

   • Christopher D. Hollings, Mark V. Lawson

   Pages 31-35

7. On the Theory of Partial Transformations

   • Christopher D. Hollings, Mark V. Lawson

   Pages 37-41

8. Generalised Groups

   • Christopher D. Hollings, Mark V. Lawson

   Pages 43-47

9. Theory of Generalised Heaps and Generalised Groups

   • Christopher D. Hollings, Mark V. Lawson

   Pages 49-145

10. Generalised Heaps as Affine Structures

    • Christopher D. Hollings, Mark V. Lawson

    Pages 147-169

11. Back Matter

    Pages 171-189

    PDF

The theories of V. V. Wagner (1908-1981) on abstractions of systems of binary relations are presented here within their historical and mathematical contexts.  This book contains the first translation from Russian into English of a selection of Wagner’s papers, the ideas of which are connected to present-day mathematical research.  Along with a translation of Wagner’s main work in this area,  his 1953 paper ‘Theory of generalised heaps and generalised groups,’ the book also includes translations of three short precursor articles that provide additional context for his major work.

Researchers and students interested in both algebra (in particular, heaps, semiheaps, generalised heaps, semigroups, and groups) and differential geometry will benefit from the techniques offered by these translations, owing to the natural connections between generalised heaps and generalised groups, and the role played by these concepts in differential geometry.  This book gives examples from present-day mathematics where ideas related to Wagner’s have found fruitful applications.

• V. V. Wagner
• generalised groups
• differential geometry
• inverse semigroups
• Historic mathematics
• binary relations
• mappings between sets
• modern algebra
• Pseudogroups
• bitorsors
• Generalised heaps
• partial transformations
• coordinate structures
• Ternary operations
• Semigroup theory
• Viktor VladimirovichWagner
• mathematicians in USSR
• 20th century mathematics history

“This book will be invaluable to whoever is interested in the algebraic structures afforded by the description of global objects through local charts, the history of the surrounding ideas, and the sources. Justifying its title, it focusses on Wagner's contributions to the theory of heaps and their generalisations … . As a consequence, even readers who are not interested primarily in Wagner or heaps will find rewards for their time spent with this book.” (Markus Szymik, Mathematical Reviews, July, 2018)

“This book presents the in the West little-known work of the Russian differential geometer Wagner … . the book is easy to read and should be of interest to mathematicians and PhD students interested in differential geometry and contemporary theories of physics.” (Jaak Henno, zbMATH 1403.20001, 2019)

• Mathematical Institute, University of Oxford, Oxford, United Kingdom

  Christopher D. Hollings

• Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom

  Mark V. Lawson

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