Table Of Content

� � �� W. B. Vasantha Kandasamy Florentin Smarandache � ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2012 by Zip Publishing and the Authors Peer reviewers: Florentin Popescu, Facultatea de Mecanica, University of Craiova, Romania. Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA. Prof. Mihàly Bencze, Department of Mathematics, Áprily Lajos College, Bra�ov, Romania Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-169-8 EAN: 9781599731698 Printed in the United States of America 2 � � �� 5� � �� 7� �� 13� �� 45� � 3.1 Groupoids and Loops over the Complex Modulo Integer Rings (Complex Rings) 45 3.2 Complex Loops and Complex Groupoids over Real Rings 77 �� 119� 3 �� 155 � � �� 157 �� 209 �� 211 �� 213 4 � � � � �� Authors in this book for the first time have constructed nonassociative structures like groupoids, quasi loops, non associative semirings and rings using finite complex modulo integers. The Smarandache analogue is also carried out. We see the nonassociative complex modulo integers groupoids satisfy several special identities like Moufang identity, Bol identity, right alternative and left alternative identities. P-complex modulo integer groupoids and idempotent complex modulo integer groupoids are introduced and characterized. This book has six chapters. The first one is introductory in nature. Second chapter introduces complex modulo integer groupoids and complex modulo integer loops using C(Z ). This chapter gives 77 n examples and forty theorems. Chapter three introduces the notion of nonassociative complex rings both finite and infinite using complex groupoids and complex loops. This chapter gives over 120 examples and thirty theorems. 5 Forth chapter introduces nonassociative structures using complex modulo integer groupoids and quasi loops. This new notion is well illustrated by 140 examples. These can find applications only in due course of time, when these new concepts become familiar. The final chapter suggests over 300 problems some of which are research problems. We thank Dr. K.Kandasamy for proof reading and being extremely supportive. � �� 6 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:2)(cid:10)(cid:11)(cid:8)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:12) (cid:2) (cid:13)(cid:14)(cid:15)(cid:10)(cid:16)(cid:17)(cid:3)(cid:14)(cid:12)(cid:10)(cid:13)(cid:2) (cid:2) (cid:2) (cid:2) In this chapter we for just recall some definitions and give the notations to make this book a self contained one. Z , Z, Q, R n and C denote the modulo integers, integers, rationals, reals and complex numbers respectively. C(Z ) = {a + bi | a, b ∈ Z , i2 = n – 1} denotes the ring of n F n F complex modulo integers. C(Z) = {a + bi | a, b ∈ Z, i2 = –1} denotes the ring of complex integers. C(Q) = {a + bi | a, b ∈ Q, i2 = –1} is the field of rational complex numbers. C(R) = C = {a + bi | a, b ∈ R, i2 = –1} is the field of complex numbers. R+ ∪ {0}, Q+ ∪ {0}, Z+ ∪ {0} denote the positive numbers and they form the semifield. However, since i2 = –1 we cannot have complex semifield. Now we proceed onto define groupoids and loops and for more about these concepts refer [14-5]. 7 DEFINITION 1.1: Let G = {Z , *, (t, u); t, u ∈ Z \ {0, 1}, (t, u) = 1; n n t and u are primes} be a groupoid of type I. If in this definition (t, u) = 1; t and u are not primes but relatively prime we get a groupoid of type II. If in this definition 1.1 if we replace (t, u) = d ≠ 0; (t, u ∈ Z \ {0}) we get groupoid n of type III. (Also (t, t) gives a groupoid of type V), if t or u = 0 then also (t, 0) or (0, t) will give a groupoid of type IV. We give examples of them. Example 1.1: Let G = {Z , (7, 13), *} be groupoid of type I. 45 Example 1.2: Let G = {Z , (15, 8), *} be groupoid of type II. 36 Example 1.3: Let G = {Z , (9, 24), *} be groupoid of type III. 40 Example 1.4: Let G = {Z , (12, 12), *} be groupoid of type IV. 29 Example 1.5: Let G = {Z , (13, 0), *} be groupoid of type V. 20 Now we have several associated properties with them. For more please refer [14]. DEFINITION 1.2: Let (G, *) be a groupoid. If H ⊆ G and (H, *) is a groupoid then we call (H, *) to be a subgroupoid of G. If (H, *) is a semigroup we define (G, *) to be a Smarandache groupoid. All groupoids in general are not Smarandache groupoids. We can define special identities on groupoids [14]. Now we proceed onto recall the definition of the new class of loops. DEFINITION 1.3: Let L (m) = {e, 1, 2, …, n} be the set where n n > 3, n is odd and m is a positive integer such that (m, n) = 1 and (m–1, n) = 1 with m < n. 8 Define on L (m) a binary operation ‘o’ as follows. n (i) e o i = i o e = i for all i ∈ L (m) n (ii) i2 = i o i = e for all i ∈ L (m) n (iii) i o j = t where t = (mj – (m – 1)i) for all i, j ∈ L (m), i ≠ j, i ≠ e and j ≠ e; then L (m) is a loop n n under the binary operation. We just give one or two examples. Example 1.6: Let L (2) = {e, 1, 2, 3, 4, 5}. The table for L (2) is 5 5 as follows: 0 e 1 2 3 4 5 e e 1 2 3 4 5 1 1 e 3 5 2 4 2 2 5 e 4 1 3 3 3 4 1 e 5 2 4 4 3 5 2 e 1 5 5 2 4 1 3 e L (2) is a loop of order six. Clearly L (2) is non associative and 5 5 non commutative. Example 1.7: L (8) be the loop given by the following table. 9 0 e 1 2 3 4 5 6 7 8 9 e e 1 2 3 4 5 6 7 8 9 1 1 e 9 8 7 6 5 4 3 2 2 2 3 e 1 9 8 7 6 5 4 3 3 5 4 e 2 1 9 8 7 6 4 4 7 6 5 e 3 2 1 9 8 5 5 9 8 7 6 e 4 3 2 1 6 6 2 1 9 8 7 e 5 4 3 7 7 4 3 2 1 9 8 e 6 5 8 8 6 5 4 3 2 1 9 e 7 9 9 8 7 6 5 4 3 2 1 e L (8) is a Smarandache loop [15]. 9 9