Singular neutrosophic extended triplet groups and generalized groups Xiaohong Zhang a,b,⇑ , Xuejiao Wang a , Florentin Smarandache c , Te `mı ´to ´pe ´ Gbo ´la ´ha `n Jaı ´ye ´ola ´ d , Tieyan Lian a a Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China b Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China c Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA d Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria Received 9 August 2018; received in revised form 27 September 2018; accepted 14 October 2018 Abstract Neutrosophic extended triplet group (NETG) is an interesting extension of the concept of classical group, which can be used to express general symmetry. This paper further studies the structural characterizations of NETG. First, some examples are given to show that some results in literature are false. Second, the differences between generalized groups and neutrosophic extended triplet groups are investigated in detail. Third, the notion of singular neutrosophic extended triplet group (SNETG) is introduced, and some homomor- phism properties are discussed and a Lagrange-like theorem for finite SNETG is proved. Finally, the following important result is proved: a semigroup is a singular neutrosophic extended triplet group (SNETG) if and only if it is a generalized group. Ó 2018 Elsevier B.V. All rights reserved. Keywords: Neutrosophic extended triplet group; Generalized group; Semigroup; Singular neutrosophic extended triplet group; Kernel of homomorphism MSC: 20N02; 20N05 1. Introduction and basic concepts The theory of neutrosophic set was introduced by Smarandache, and it is applied to many fields (see Smarandache, 2005; Ye, 2014; Liu, Khan, Ye, & Mahmood, 2018; Zhang, Bo, Smarandache, & Dai, 2018; Zhang, Bo, Smarandache, & Park, 2018). In recent years, the ideology of neutrosophic set has been applicable in related algebraic structures. In particular, Smarandache and Ali (2018) introduced the notion of neutrosophic triplet group, which is a new extension of the concept of classical group. Now, this new algebraic structure has aroused scholars’ interest, and some new research papers have been published one after another (see Smarandache, 2017; Zhang, Smarandache, & Liang, 2017; Jaiyeola and Smarandache, 2018; Smarandache, S ß ahin, & Kargin, 2018; Ali, Smarandache, & Khan, 2018; Zhang, Hu, Smarandache, & An, 2018). In fact, neutrosophic triplet structures are closely connected with related non-classical logic algebras (see Zhang, Wu, Smarandache, & Hu, 2018; Zhang, 2017; Zhang, Park, & Wu, 2018). In (Smarandache, 2017), the notion of neutrosophic extended triplet group (NETG) was introduced as a generalization of neutrosophic triplet group. On the other hand, Molaei (Molaei, 1999) introduced the notion of generalized group, as a class of algebras of https://doi.org/10.1016/j.cogsys.2018.10.009 1389-0417/Ó 2018 Elsevier B.V. All rights reserved. ⇑ Corresponding author at: Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China. E-mail addresses: zhangxiaohong@sust.edu.cn, zhangxh@shmtu.edu. cn (X. Zhang), smarand@unm.edu (F. Smarandache), tjayeola@oauife. edu.ng (T.G. Jaı ´ye ´ola ´), liantieyan@sust.edu.cn (T. Lian). www.elsevier.com/locate/cogsys Available online at www.sciencedirect.com ScienceDirect Cognitive Systems Research xxx (xxxx) xxx Please cite this article as: X. Zhang, X. Wang, F. Smarandache et al., Singular neutrosophic extended triplet groups and generalized groups, Cognitive Systems Research, https://doi.org/10.1016/j.cogsys.2018.10.009