Table of Contents

Cover image

Title page

Copyright

Foreword to the First Edition

Preface to the First Edition

Acknowledgements (First Edition)

Preface to the Second Edition

Chapter 1: Elementary properties

1.1 The multiplication table of a quasigroup

1.2 The Cayley table of a group

1.3 Isotopy

1.4 Conjugacy and parastrophy

1.5 Transversals and complete mappings

1.6 Latin subsquares and subquasigroups

Chapter 2: Special types of latin square

2.1 Quasigroup identities and latin squares

2.2 Quasigroups of some special types and the concept of generalized associativity

2.3 Triple systems and quasigroups

2.4 Group-based latin squares and nuclei of loops

2.5 Transversals in group-based latin squares

2.6 Complete latin squares

Chapter 3: Partial latin squares and partial transversals

3.1 Latin rectangles and row latin squares

3.2 Critical sets and Sudoku puzzles

3.3 Fuchs’ problems

3.4 Incomplete latin squares and partial quasigroups

3.5 Partial transversals and generalized transversals

Chapter 4: Classification and enumeration of latin squares and latin rectangles

4.1 The autotopism group of a quasigroup

4.2 Classification of latin squares

4.3 History of the classification and enumeration of latin squares

4.4 Enumeration of latin rectangles

4.5 Enumeration of transversals

4.6 Enumeration of subsquares

Chapter 5: The concept of orthogonality

5.1 Existence questions for incomplete sets of orthogonal latin squares

5.2 Complete sets of orthogonal latin squares and projective planes

5.3 Sets of MOLS of maximum and minimum size

5.4 Orthogonal quasigroups, groupoids and triple systems

5.5 Self-orthogonal and other parastrophic orthogonal latin squares and quasigroups

5.6 Orthogonality in other structures related to latin squares

Chapter 6: Connections between latin squares and magic squares

6.1 Diagonal (or magic) latin squares

6.2 Construction of magic squares with the aid of orthogonal latin squares

6.3 Additional results on magic squares

6.4 Room squares: their construction and uses

Chapter 7: Constructions of orthogonal latin squares which involve rearrangement of rows and columns

7.1 Generalized Bose construction: constructions based on abelian groups

7.2 The automorphism method of H.B. Mann

7.3 The construction of pairs of orthogonal latin squares of order ten

7.4 The column method

7.5 The diagonal method

7.6 Left neofields and orthomorphisms of groups

Chapter 8: Connections with geometry and graph theory

8.1 Quasigroups and 3-nets

8.2 Orthogonal latin squares, k-nets and introduction of co-ordinates

8.3 Latin squares and graphs

Chapter 9: Latin squares with particular properties

9.1 Bachelor squares

9.2 Homogeneous latin squares

9.3 Diagonally cyclic latin squares and Parker squares

9.4 Non-cyclic latin squares with cyclic properties

Chapter 10: Alternative versions of orthogonality

10.1 Variants of orthogonality

10.2 Power sets of latin squares

Chapter 11: Miscellaneous topics

11.1 Orthogonal arrays and latin squares

11.2 The direct product and singular direct product of quasigroups

11.3 The Kézdy-Snevily conjecture

11.4 The chromatic number of a latin square

11.5 Practical applications of latin squares

11.6 Latin triangles

11.7 Latin squares and computers

Comment on the Problems

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

New Problems

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Bibliography

Index