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Non Associative Algebraic Structures Using Finite Complex Numbers

By Smarandache, Florentin

Book Id:WPLBN0002828486 Format Type:PDF (eBook) File Size:1.53 mb Reproduction Date:8/2/2013

Smarandache, F., & Vasantha Kandasamy, W. B. (2013). Non Associative Algebraic Structures Using Finite Complex Numbers. Retrieved from http://gutenberg.us/

Description
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(Zn). This chapter gives 77 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.
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.

Summary
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.

Table of Contents
THEOREM 2.1: Let G = {C(Zn), *, (t, u); t, u ∈ Zn} be a complex modulo integer groupoid. If H ⊆ G is such that H is a Smarandache modulo integer subgroupoid, then G is a Smarandache complex modulo integer groupoid. But every subgroupoid of G need not be a Smarandache complex modulo interger subgroupoid even if G is a Smarandache groupoid.
Proof is direct and hence is left as an exercise to the reader.
Example 2.28: Consider G = {C(Z8), *, (2, 4)}, a complex modulo integer groupoid.