Generalized Abel-Grassmann’s Neutrosophic Extended Triplet Loop

Xiaogang An,Xiaohong Zhang,Yingcang Ma

doi:10.3390/math7121206

Xiaogang An, Xiaohong Zhang + Show 1 more

Open Access

https://doi.org/10.3390/math7121206

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Abstract

A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet loop (GAG-NET-Loop) is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is an AG-NET-Loop if and only if it is a strong inverse AG-groupoid; (2) an algebraic system is a GAG-NET-Loop if and only if it is a quasi strong inverse AG-groupoid; (3) an algebraic system is a weak commutative GAG-NET-Loop if and only if it is a quasi Clifford AG-groupoid; and (4) a finite interlaced AG-(l,l)-Loop is a strong AG-(l,l)-Loop.

Highlights

The concept of an Abel-Grassmann’s groupoid (AG-groupoid) was first given by Kazim and Naseeruddin [1] in 1972 and they have called it a left almost semigroup (LA-semigroup)
Smarandache proposed the new concept of neutrosophic set, which is an extension of fuzzy set and intuitionistic fuzzy set [10]
Neutrosophic sets have been applied to many fields such as decision making [11,12,13], forecasting [14], best product selection [15], the shortest path problem [16], minimum spanning tree [17], neutrosophic portfolios of financial assets [18], etc

Summary

Introduction

The concept of an Abel-Grassmann’s groupoid (AG-groupoid) was first given by Kazim and Naseeruddin [1] in 1972 and they have called it a left almost semigroup (LA-semigroup). In [25], Xiaohong Zhang introduced the concept of Abel-Grassmann’s neutrosophic extended triplet loop (AG-NET-loop), and some properties and structure about AG-NET-loop are discussed. A new algebraic system, generalized neutrosophic extended triplet set, is proposed in [26]. We combine the notions of generalized neutrosophic extended triplet set and AG-groupoid, introduce the new concept of generalized Abel-Grassmann’s neutrosophic extended triplet loop (GAG-NET-loop); that is, GAG-NET-loop is both AG-groupoid and generalized neutrosophic extended triplet set. Mathematics 2019, 7, 1206 into smaller ones This is the embodiment of the research method of regular semigroups to quasi-regular semigroups in non-associative groupoid.

Basic Definitions

Strong Inverse AG-Groupoid and Finite Interlaced AG-Groupoid

GAG-NET-Loop

Conclusions