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Sigma Series in Pure Mathematics -- Volume 8

   Enlarged Picture

O. Chein, H. O. Pflugfelder, J. D. H. Smith (eds.)

Quasigroups and Loops. Theory and Applications

580 pages, hard cover, ISBN 3-88538-008-0, EUR 75.00, 1990

This book contains 14 chapters written by well-known authors which cover almost all aspects of the algebraic and geometric theory of quasigroups. It is and will remain a source book of lasting value indispensable for any researcher in this area. The bibliography contains ca. 1700 publications.


Contents:

T. Evans: Varieties of loops and quasigroups, 1--26

O. Chein: Examples and methods of construction, 27--94

J. D. H. Smith: Centrality, 95--114

L. Bénéteau: Commutative Moufang loops and related groupoids, 115--142

L. Bénéteau: Cubic hypersurface quasigroups, 143--150

M.-M. Deza, G. Sabidussi: Combinatorial structures, 151--160

E. Goodaire, M. J. Kallaher: Systems with two binary operations and their planes, 161--196

A. Barlotti: Geometry of quasigroups, 197--204

K. H. Hofmann, K. Strambach: Topological and analytical loops, 205--262

V. V. Goldberg: Local differentiable quasigroups and webs, 263--312

T. Grundhoefer, H. Salzmann: Locally compact double loops and ternary fields, 313--356

P. O. Miheev, L. V. Sabinin: Quasigroups and differential geometry, 357--430

P. D. Gerber: Quasigroups arising from differential equations, 431--444

F. B. Kalhoff, S. H. G. Priess-Crampe: Ordered loops and ordered planar ternary rings, 445--466

Bibliography: 467--556


The Contents in detail:

Introduction ix
     
  Chapter I / Zentralblatt-Review  
  T. Evans: Varieties of loops and quasigroups  
I.1 Universal algebraic preliminaries 1
I.2 Free loops and quasigroups 7
I.3 Decision problems and the structure of finitely presented loops 11
I.4 Classifying loop identities and loop varieties 16
I.5 Quasigroup varieties and combinatorics 22
     
  Chapter II / Zentralblatt-Review  
  O. Chein: Examples and methods of construction  
II.1 Introduction 27
II.2 Terminology 28
II.3 Extensions 35
II.4 Other constructions using cartesian products 43
II.5 Isotopy 51
II.6 New operations on algebraic systems 58
II.7 Miscellaneous algebraic constructions 63
II.8 Constructions arising from geometry 74
II.9 Constructions based on designs 83
II.10 Summary 90
     
  Chapter III / Zentralblatt-Review  
  J. D. H. Smith: Centrality  
III.1 Introduction 95
III.2 Equasigroups and their congruences 95
III.3 Central congruences 98
III.4 Central isotopy 102
III.5 Z-quasigroups 105
III.6 Stability congruences 110
     
  Chapter IV / Zentralblatt-Review  
  L. Bénéteau: Commutative Moufang loops and related groupoids  
IV.1 Preliminaries 115
IV.2 Basic facts about central nilpotence and first consequences 122
IV.3 Free objects, related groups, and open problems 128
     
  Chapter V / Zentralblatt-Review  
  L. Bénéteau: Cubic hypersurface quasigroups  
V.1 Definitions and structure theorems 143
V.2 Geometrical motivations 146
     
  Chapter VI / Zentralblatt-Review  
  M. Deza, G. Sabidussi: Combinatorial structures arising from commutative Moufang loops  
VI.1 Introduction 151
VI.2 Matroidal background 153
VI.3 Hall systems as perfect matroid designs 154
VI.4 Erigibility 156
VI.5 Dimensions 159
     
  Chapter VII / Zentralblatt-Review  
  E. G. Goodaire, M. J. Kallaher: Systems with two binary operations, and their planes  
VII.1 Introduction and historical background 161
VII.2 Ternary rings and their planes 162
VII.3 Quasifields 169
VII.4 Semifields 176
VII.5 Derivability 180
VII.6 Construction of cartesian groups 183
VII.7 Neofields 186
VII.8 Loop rings 189
     
  Chapter VIII / Zentralblatt-Review  
  A. Barlotti: Geometry of quasigroups  
VIII.1 Introduction 197
VIII.2 Staudt's point of view 197
VIII.3 k-transitive and ω-transitive permutation groups 201
VIII.4 Klein's point of view in web geometry 202
     
  Chapter IX / Zentralblatt-Review  
  K. H. Hofmann, K. Strambach: Topological and analytic loops  
IX.1 The general theory, non-specific properties 205
IX.2 The multiplication group, uniformities 217
IX.3 Special hypotheses: weak forms of associativity 223
IX.4 Geometry of topological loops 227
IX.5 Topological distributive quasigroups 229
IX.6 The Lie theory of analytic loops 234
IX.7 Topological double loops 255
     
  Chapter X / Zentralblatt-Review  
  V. V. Goldberg: Local differentiable quasigroups and webs  
X.1 Fibrations and d-webs W(d,n,r) of codimension r on a differentiable manifold Xnr 263
X.2 Local differentiable n-quasigroups connected with webs W(n+1,n,r) 273
X.3 Local Akivis algebras of three-webs W(3,2,r) 280
X.4 Canonical expansions of the equations of a local differentiable n-quasigroup 283
X.5 One-parameter n-subquasigroups and n-subgroups of a local differentiable n-quasigroup 287
X.6 Special classes of webs and local differentiable quasigroups 289
X.7 Realizations of webs and local differentiable quasigroups 308
     
  Chapter XI / Zentralblatt-Review  
  T. Grundhoefer, H. Salzmann: Locally compact double loops and ternary fields  
XI.1 Double loops 313
XI.2 Ternary fields 316
XI.3 Automorphism groups of ternary fields 318
XI.4 Linear ternary fields 319
XI.5 Quasifields 321
XI.6 Nearfields 327
XI.7 Division algebras, fields and alternative fields 329
XI.8 Double loops 333
XI.9 Automorphism groups 335
XI.10 Homogeneity of planes 338
XI.11 Dimension 1 345
XI.12 Dimension 2 347
XI.13 Dimension 4 350
XI.14 Dimension 8 353
     
  Chapter XII / Zentralblatt-Review  
  P. O. Miheev, L. V. Sabinin: Quasigroups and differential geometry  
XII.1 Smooth universal algebras 357
XII.2 Canonical affine connections of loopuscular and geodular structures 366
XII.3 Reductive and symmetric geoodular spaces 374
XII.4 s-Spaces 389
XII.5 Lie triple algebras 395
XII.6 Semidirect products of a quasigroup by its transassociants 398
XII.7 The infinitesimal theory of local analytic loops 405
XII.8 Smooth Bol loops 418
     
  Chapter XIII / Zentralblatt-Review  
  P. D. Gerber: LIP loops and quadratic differential equations  
XIII.1 Introduction 431
XIII.2 Notation and general background 431
XIII.3 Quadratic systems 435
XIII.4 First degree groups 437
XIII.5 First degree LIP loops 440
     
  Chapter XIV / Zentralblatt-Review  
  F. B. Kalhoff, S. H. G. Priess-Crampe: Ordered loops and ordered planar ternary rings  
XIV.1 Some basic facts about ordered loops 445
XIV.2 Archimedean ordering 448
XIV.3 Chain of convex subloops and orderability of loops 451
XIV.4 Ordered double loops and ordered planar ternary rings 454
XIV.5 Archimedean planar ternary rings and double loops 458
XIV.6 Preorderings and Artin-Schreier characterization in PTRs 461
XIV.7 Spaces of orderings and Witt rings of planar ternary rings 463
     
  Bibliography 467
  Subject index 557