Preface 5
1. General Fundamentals
1.1 Basic Concepts 7
1.2 A few properties of groups and graphs 8
1.3 Lattices and its properties 11
2. Loops and its properties
2.1 Definition of loop and examples 15
2.2 Substructures in loops 17
2.3 Special identities in loops 22
2.4 Special types of loops 23
2.5 Representation and isotopes of loops 29
2.6 On a new class of loops and its properties 31
2.7 The new class of loops and its
application to proper edge colouring of the graph K2n 40
3. Smarandache Loops
3.1 Definition of Smarandache loops with examples 47
3.2 Smarandache substructures in loops 51
3.3 Some new classical S-loops 56
3.4 Smarandache commutative and commutator subloops 61
3.5 Smarandache associativite and associator subloops 67
3.6 Smarandache identities in loops 71
3.7 Some special structures in S-loops 74
3.8 Smarandache mixed direct product loops 78
3.9 Smarandache cosets in loops 84
3.10 Some special type of Smarandache loops 91
4. Properties about S-loops
4
4.1 Smarandache loops of level II 93
4.2 Properties of S-loop II 98
4.3 Smarandache representation of a finite loop L 99
4.4 Smarandache isotopes of loops 102
4.5 Smarandache hyperloops 103
5. Research problems 107
References 113
Index 119