Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures

TL;DR

This work introduces and explores neutrosophic algebraic structures across various classical algebraic systems, extending traditional theorems to these new contexts and constructing novel examples using number theory.

Contribution

It is the first comprehensive study of neutrosophic algebraic structures, including their N-variants, and applies classical theorems to these new frameworks.

Findings

Defined neutrosophic groups, loops, semigroups, and groupoids with examples.

Extended classical theorems like Lagrange and Sylow to neutrosophic structures.

Constructed new classes of neutrosophic loops using modulo integers.

Abstract

In this book, for the first time we introduce the notion of neutrosophic algebraic structures for groups, loops, semigroups and groupoids; and also their neutrosophic N-algebraic structures. One is fully aware of the fact that many classical theorems like Lagrange, Sylow and Cauchy have been studied only in the context of finite groups. Here we try to shift the paradigm by studying and introducing these theorems to neutrosophic semigroups, neutrosophic groupoids, and neutrosophic loops. This book has seven chapters. Chapter one provides several basic notions to make this book self-contained. Chapter two introduces neutrosophic groups and neutrosophic N-groups and gives several examples. The third chapter deals with neutrosophic semigroups and neutrosophic N-semigroups, giving several interesting results. Chapter four introduces neutrosophic loops and neutrosophic N-loops. We introduce several new, related definitions. In fact we construct a new class of neutrosophic loops using modulo integer Z_n, n > 3, where n is odd. Several properties of these structures are proved using number theoretic techniques. Chapter five just introduces the concept of neutrosophic groupoids and neutrosophic N-groupoids. Sixth chapter innovatively gives mixed neutrosophic structures and their duals. The final chapter gives problems for the interested reader to solve.