##
**Quasigroups and loops: theory and applications.**
*(English)*
Zbl 0719.20036

Sigma Series in Pure Mathematics, 8. Berlin etc.: Heldermann Verlag. xii, 568 p. (1990).

This book contains 14 contributed chapters which cover almost all aspects of the algebraic and geometric theory of quasigroups.

The authors and the titles of the chapters are as follows: T. Evans: Varieties of loops and quasigroups, O. Chein: Examples and methods of construction, J. D. H. Smith: Centrality, L. Bénéteau: Commutative Moufang loops and related groupoids, L. Bénéteau: Cubic hypersurface quasigroups, M. Deza, G. Sabidussi: Combinatorial structures, arising from commutative Moufang loops, E. G. Goodaire, M. J. Kallaher: Systems with two binary operations, A. Barlotti: Geometry of quasigroups, K. H. Hofmann, K. Strambach: Topological and analytic loops, V. V. Goldberg: Local differentiable quasigroups and webs, Th. Grundhöfer, H. Salzmann: Locally compact double loops and ternary fields P. O. Miheev, L. V. Sabinin: Quasigroups and differential geometry, P. D. Gerber: LIP loops and quadratic differential equations, F. B. Kalhoff, S. H. G. Priess- Crampe: Ordered loops and ordered planar ternary rings.

The purely algebraic aspects of the theory of quasigroups and loops are discussed in Chapter I, III, and IV. Hall triple systems and perfect matroid designs are covered by Chapter VI. Some geometrical motivations are discussed in Chapter V and VIII. The classical differential geometry is the setting for Chapter XII which points out the important role that quasigroups, loops and related structures have to play there. A detailed coverage of topological double loops and planes is given in Chapter XI. Local webs and local n-quasigroups that coordinate them form the main topic of Chapter X. An application of certain local analytic loops and corresponding non-associative algebras is examined in Chapter XIII. A survey of topological and analytic loops is given in Chapter IX. Chapter XIV is dedicated to ordered structures. Chapter VII is studying loops for which the loop ring satisfies certain identities. The keystone of the book is Chapter II which presents a wide range of examples of quasigroups and loops as they appear naturally or have been constructed for specific purposes. At the end of the book one can find a very detailed bibliography.

Reviewer’s remarks: 1. In the introduction the Editors give the titles of three pioneering works. The reviewer thinks that three more books should have been mentioned as definitive works. Namely: Algebraic Nets and Quasigroups (1971; Zbl 0245.50005) and N-ary Quasigroups (1972; Zbl 0282.20061) by V. D. Belousov and Finite Geometries (1968; Zbl 0159.500) by P. Dembowski. 2. The reviewer with A. D. Keedwell edited a new book: Latin squares. New developments in the theory and applications (North Holland, Amsterdam 1991; Zbl 0715.00010) on the combinatorial aspects of quasigroups. The reviewer thinks that it is fortunate that there is no overlap between the present volume and the recent joint book of the reviewer and A. D. Keedwell, apart from one exception: Chapter I (T. Evans: Varieties of loops and quasigroups) of the present book and Chapter 7 (T. Evans: Latin squares and universal algebra) of the jointly edited book of the reviewer and A. D. Keedwell, mostly overlap each other.

The authors and the titles of the chapters are as follows: T. Evans: Varieties of loops and quasigroups, O. Chein: Examples and methods of construction, J. D. H. Smith: Centrality, L. Bénéteau: Commutative Moufang loops and related groupoids, L. Bénéteau: Cubic hypersurface quasigroups, M. Deza, G. Sabidussi: Combinatorial structures, arising from commutative Moufang loops, E. G. Goodaire, M. J. Kallaher: Systems with two binary operations, A. Barlotti: Geometry of quasigroups, K. H. Hofmann, K. Strambach: Topological and analytic loops, V. V. Goldberg: Local differentiable quasigroups and webs, Th. Grundhöfer, H. Salzmann: Locally compact double loops and ternary fields P. O. Miheev, L. V. Sabinin: Quasigroups and differential geometry, P. D. Gerber: LIP loops and quadratic differential equations, F. B. Kalhoff, S. H. G. Priess- Crampe: Ordered loops and ordered planar ternary rings.

The purely algebraic aspects of the theory of quasigroups and loops are discussed in Chapter I, III, and IV. Hall triple systems and perfect matroid designs are covered by Chapter VI. Some geometrical motivations are discussed in Chapter V and VIII. The classical differential geometry is the setting for Chapter XII which points out the important role that quasigroups, loops and related structures have to play there. A detailed coverage of topological double loops and planes is given in Chapter XI. Local webs and local n-quasigroups that coordinate them form the main topic of Chapter X. An application of certain local analytic loops and corresponding non-associative algebras is examined in Chapter XIII. A survey of topological and analytic loops is given in Chapter IX. Chapter XIV is dedicated to ordered structures. Chapter VII is studying loops for which the loop ring satisfies certain identities. The keystone of the book is Chapter II which presents a wide range of examples of quasigroups and loops as they appear naturally or have been constructed for specific purposes. At the end of the book one can find a very detailed bibliography.

Reviewer’s remarks: 1. In the introduction the Editors give the titles of three pioneering works. The reviewer thinks that three more books should have been mentioned as definitive works. Namely: Algebraic Nets and Quasigroups (1971; Zbl 0245.50005) and N-ary Quasigroups (1972; Zbl 0282.20061) by V. D. Belousov and Finite Geometries (1968; Zbl 0159.500) by P. Dembowski. 2. The reviewer with A. D. Keedwell edited a new book: Latin squares. New developments in the theory and applications (North Holland, Amsterdam 1991; Zbl 0715.00010) on the combinatorial aspects of quasigroups. The reviewer thinks that it is fortunate that there is no overlap between the present volume and the recent joint book of the reviewer and A. D. Keedwell, apart from one exception: Chapter I (T. Evans: Varieties of loops and quasigroups) of the present book and Chapter 7 (T. Evans: Latin squares and universal algebra) of the jointly edited book of the reviewer and A. D. Keedwell, mostly overlap each other.

Reviewer: J.Dénes (Budapest)

### MSC:

20N05 | Loops, quasigroups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

51A25 | Algebraization in linear incidence geometry |

53A60 | Differential geometry of webs |

22A22 | Topological groupoids (including differentiable and Lie groupoids) |

05B15 | Orthogonal arrays, Latin squares, Room squares |

51A20 | Configuration theorems in linear incidence geometry |

05B07 | Triple systems |

22A30 | Other topological algebraic systems and their representations |

06A99 | Ordered sets |

00B10 | Collections of articles of general interest |