Journal of Intelligent & Fuzzy Systems 37 (2019) 5743–5753 DOI:10.3233/JIFS-181742 IOS Press 5743 On neutrosophic extended triplet groups (loops) and Abel-Grassmann’s groupoids (AG-groupoids) Xiaohong Zhang a,b,∗ , Xiaoying Wu a , Xiaoyan Mao c , Florentin Smarandache d and Choonkil Park e a Department of Mathematics, Shaanxi University of Science & Technology, Xi’an, China b Department of Mathematics, Shanghai Maritime University, Shanghai, China c College of Science and Technology, Ningbo University, Ningbo, China d Department of Mathematics, University of New Mexico, Gallup, NM, USA e Department of Mathematics, Hanyang University, Seoul, Korea Abstract. From the perspective of semigroup theory, the characterizations of a neutrosophic extended triplet group (NETG) and AG-NET-loop (which is both an Abel-Grassmann groupoid and a neutrosophic extended triplet loop) are systematically analyzed and some important results are obtained. In particular, the following conclusions are strictly proved: (1) an algebraic system is neutrosophic extended triplet group if and only if it is a completely regular semigroup; (2) an algebraic system is weak commutative neutrosophic extended triplet group if and only if it is a Clifford semigroup; (3) for any element in an AG-NET-loop, its neutral element is unique and idempotent; (4) every AG-NET-loop is a completely regular and fully regular Abel-Grassmann groupoid (AG-groupoid), but the inverse is not true. Moreover, the constructing methods of NETGs (completely regular semigroups) are investigated, and the lists of some finite NETGs and AG-NET-loops are given. Keywords: Semigroup, neutrosophic extended triplet group (NETG), completely regular semigroup, Clifford semigroup, Abel-Grassmann’s groupoid (AG-groupoid) 1. Introduction Smarandache proposed the new concept of neu- trosophic set, which is an extension of fuzzy set and intuitionistic fuzzy set [1]. Until now, neutrosophic sets have been applied to many fields [2–4], and some new theoretical studies are developed [5, 6]. As an application of the basic idea of neutrosophic sets (more general, neutrosophy), the new notion of neutrosophic triplet group (NTG) is introduced by Smarandache and Ali in [7, 8]. As a new algebraic ∗ Corresponding author. Xiaohong Zhang. E-mails: zhangxiao hong@sust.edu.cn; zhangxh@shmtu.edu.cn. structure, NTG is a generalization of classical group, but it has different properties from classical group. For NTG, the neutral element is relative and local, that is, for a neutrosophic triplet group (N, ∗ ), every element a in N has its own neutral ele- ment (denote by neut (a)) satisfying condition a ∗ neut (a) = neut (a) ∗ a = a, and there exits at least one opposite element (denote by anti (a)) in N relative to neut (a) such condition a ∗ anti (a) = anti (a) ∗ a = neut (a). In the original definition of NTG in [8], neut (a) is different from the traditional unit element. Later, the concept of neutrosophic extended triplet group (NETG) was introduced (see [7]), in which the neutral element may be traditional unit element, it is just a special case. ISSN 1064-1246/19/$35.00 © 2018 – IOS Press and the authors. All rights reserved