Regular CA-Groupoids and Cyclic Associative Neutrosophic Extended Triplet Groupoids (CA-NET-Groupoids) with Green Relations

Wangtao Yuan,Xiaohong Zhang

doi:10.3390/math8020204

Abstract

Based on the theories of AG-groupoid, neutrosophic extended triplet (NET) and semigroup, the characteristics of regular cyclic associative groupoids (CA-groupoids) and cyclic associative neutrosophic extended triplet groupoids (CA-NET-groupoids) are further studied, and some important results are obtained. In particular, the following conclusions are strictly proved: (1) an algebraic system is a regular CA-groupoid if and only if it is a CA-NET-groupoid; (2) if (S, *) is a regular CA-groupoid, then every element of S lies in a subgroup of S, and every ℋ -class in S is a group; and (3) an algebraic system is an inverse CA-groupoid if and only if it is a regular CA-groupoid and its idempotent elements are commutative. Moreover, the Green relations of CA-groupoids are investigated, and some examples are presented for studying the structure of regular CA-groupoids.

Highlights

The theory of group is an essential branch of algebra
Since cyclic associative law is widely used in algebraic systems, we have been focusing on the basic algebraic structure of cyclic associative groupoids (CA-groupoids) and other relevant algebraic structures
Starting from various backgrounds (for examples, non-associative rings with x(yz) = y(zx), cyclic associative Abel-Grassman groupoids, regular semigroup, and regular AG-groupoid), this paper introduces the concept of regular cyclic associative groupoid (CA-groupoid) for the first time

Summary

Introduction

The theory of group is an essential branch of algebra. The research of group has become an important trend in the theory of semigroup. Various algebraic structures are related to groups, such as regular semigroups, generalized groups, and neutrosophic extended triplet groups (see [1,2,3,4,5,6]). This paper focuses on the regularity of non-associative algebraic structures satisfying the cyclic associative law: x(yz) = z(xy). We analyze the structure of cyclic associative neutrosophic extended triplet groupoids (CA-NET-Groupoids). Green’s relations, first studied by Green [21] in 1951, have played a fundamental role in the development of regular semigroup theory. This paper focuses on the Green’s relations of CA-groupoids, in particular regular CA-groupoids. We analyzed these new results and studied them from the perspective of CA-groupoid theory.

Preliminaries

Triplet

Conclusions