Kandasamy W. B. V. - Smarandache Near-Rings.pdf

(1184 KB) Pobierz
Microsoft Word - Vasantha-Book6.doc
w. b. vasantha kandasamy
smarandache
near-rings
american research press
rehoboth
2002
133809267.001.png
Smarandache Near-rings
W. B. Vasantha Kandasamy
Department of Mathematics
Indian Institute of Technology, Madras
Chennai – 600036, India
American Research Press
Rehoboth, NM
2002
1
The picture on the cover is a representation of the Smarandache near-ring (S-near-ring). The
international near-ring community represents the near-ring by a structure which is nearly a ring, but
not exactly a ring, i.e. a near-ring. For the S-near-ring, of course, we have a "S" within the near-ring
structure!
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
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:
Prof. Geoffrey L. Booth, University of Port Elizabeth, P.O. 1600, South Africa;
Prof. V. Tharmarathnam, Dept. of Mathematics, Univ. of Jaffna, Sri Lanka.
Prof. L. TuÛescu, Dept. of Mathematics, FraÛii BuzeÕti College, Romania.
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 : 1-931233-66-7
Standard Address Number : 297-5092
Printed in the United States of America
2
MI 48106-1346, USA
CONTENTS
Preface
5
Chapter One: PREREQUISITES
1.1 Groups with examples
7
1.2 Definition of groupoids and loops with examples
8
1.3 Semigroups
13
1.4 Semirings
14
1.5 Lattices and its properties
15
Chapter Two: NEAR-RINGS AND ITS PROPERTIES
2.1 Definition of near-ring and some of its basic properties
19
2.2 N-groups, homomorphism and ideal-like subsets
21
2.3 Products, direct sums and subdirect product in near-rings
22
2.4 Ideals in near-rings
23
2.5 Modularity in near-rings
25
2.6 Near polynomial rings
26
2.7 Near matrix rings
27
Chapter Three: SPECIAL CLASSES OF NEAR-RINGS
AND THEIR GENERALIZATIONS
3.1 IFP near-rings
29
3.2 Group near-rings and its generalizations
35
3.3 Loop near-rings and its generalizations
41
3.4 Groupoid near-rings and its properties
54
3.5 Special properties of near-rings
58
Chapter Four: SMARANDACHE NEAR-RINGS
4.1 Definition of S-near-ring with examples
67
4.2 Smarandache N-groups
68
4.3 Smarandache direct product and Smarandache free near-rings
70
4.4 Smarandache ideals in near-rings
72
4.5 Smarandache modularity in near-rings
74
Chapter Five: SPECIAL PROPERTIES OF CLASSES OF
SMARANDACHE NEAR-RINGS AND ITS GENERALIZATIONS
5.1 Smarandache mixed direct product of near-rings and seminear-rings
77
5.2 Special classes of Smarandache near-rings
80
3
5.3 Smarandache group near-rings and their generalizations
83
5.4 On a special class of Smarandache seminear-rings
and their genearlizations
87
5.5 Some special properties in S-near-rings
92
Chapter Six: SMARANDACHE SEMINEAR-RINGS
6.1 Definition and properties of S-seminear-ring
97
6.2 Homomorphism and ideals of a S-seminear-ring
100
6.3 Smarandache seminear-rings of level II
102
6.4 Smarandache pseudo seminear-ring
104
6.5 Miscellaneous properties of some new classes of S-seminear-rings
106
Chapter Seven: SOME APPLICATIONS OF SMARANDACHE
NEAR-RINGS AND NEAR-RINGS
7.1 Basics on automaton and on semi-automaton
111
7.2 Smarandache S-semigroup semi-automaton and
the associated Smarandache syntactic near-ring
119
7.3 Applications of near-rings to error correcting codes
and their Smarandache analogue
123
Chapter Eight: SMARANDACHE NON-ASSOCIATIVE
NEAR-RINGS AND SEMINEAR-RINGS
8.1 Smarandache non-associative seminear-ring and its properties
125
8.2 Some special Smarandache non-associative seminear-rings of type II
129
8.3 Smarandache non-associative near-rings
132
8.4 Smarandache loop seminear-rings and
Smarandache groupoid seminear-rings
141
8.5 New notions of S-NA near-rings and S-NA seminear-rings
150
Chapter Nine: FUZZY NEAR-RINGS AND SMARANDACHE
FUZZY NEAR-RINGS
9.1 Basic notions on fuzzy near-rings
157
9.2 Some special classes of fuzzy near-rings
161
9.3 Smarandache fuzzy near-rings
168
Chapter Ten: SUGGESTED PROBLEMS
171
Bibliography
183
Index
191
4

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin