Kandasamy W. B. V. - Smarandache Loops.pdf

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W. B. VASANTHA KANDASAMY
SMARANDACHE
LOOPS
L 15 (8)
B 1
B 5
A 1
A 15
{e}
AMERICAN RESEARCH PRESS
REHOBOTH
2002
133809222.001.png
Smarandache Loops
W. B. Vasantha Kandasamy
Department of Mathematics
Indian Institute of Technology, Madras
Chennai – 600036, India
L
A 7
A 8
A 10
A 11
A 6
A 1
A 2
A 3
A 4
A 5
American Research Press
Rehoboth, NM
2002
1
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The picture on the cover represents the lattice of subgroups of the Smarandache loop L 15 (8).
The lattice of subgroups of the commutative loop L 15 (8) is a non-modular lattice with 22
elements. This is a Smarandache loop which satisfies the Smarandache Lagrange criteria.
But for the Smarandache concepts one wouldn't have studied the collection of subgroups of
a loop.
This book can be ordered in a paper bound reprint from:
Books on Demand
ProQuest Information & Learning
(University of Microfilm International)
300 N. Zeeb Road
P.O. Box 1346, Ann Arbor
MI 48106-1346, USA
Tel.: 1-800-521-0600 (Customer Service)
http://wwwlib.umi.com/bod/
and online from:
Publishing Online, Co. (Seattle, Washington State)
at: http://PublishingOnline.com
This book has been peer reviewed and recommended for publication by:
Dr. M. Khoshnevisan, Sharif University of Technology, Tehran, Iran.
Dr. J. Dezert, Office National d’Etudes et de Recherches Aeorspatiales (ONERA),
29, Avenue de la Division Leclerc, 92320 Chantillon, France.
Professor C. Corduneanu, Texas State University, Department of Mathematics,
Arlington, Texas 76019, USA.
Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy
Rehoboth, Box 141
NM 87322, USA
Many books can be downloaded from:
http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm
ISBN : 1-931233-63-2
Standard Address Number : 297-5092
P RINTED IN THE U NITED S TATES OF A MERICA
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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
2.3 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 K 2n
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
3
2.2 Substructures in loops
17
4.1 Smarandache loops of level II
93
4.3 Smarandache representation of a finite loop L
99
4.4 Smarandache isotopes of loops
102
4.5 Smarandache hyperloops
103
5. Research probles
107
References
113
Index
119
4
4.2 Properties of S-loop II
98

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