On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes
1
School of Arts and Sciences, Shaanxi University of Science & Technology, Xi’an 710021, China
2
College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China
3
Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(7), 289; https://doi.org/10.3390/sym10070289
Received: 20 June 2018 / Revised: 6 July 2018 / Accepted: 13 July 2018 / Published: 17 July 2018
(This article belongs to the Special Issue Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets)
As a new generalization of the notion of the standard group, the notion of the neutrosophic triplet group (NTG) is derived from the basic idea of the neutrosophic set and can be regarded as a mathematical structure describing generalized symmetry. In this paper, the properties and structural features of NTG are studied in depth by using theoretical analysis and software calculations (in fact, some important examples in the paper are calculated and verified by mathematics software, but the related programs are omitted). The main results are obtained as follows: (1) by constructing counterexamples, some mistakes in the some literatures are pointed out; (2) some new properties of NTGs are obtained, and it is proved that every element has unique neutral element in any neutrosophic triplet group; (3) the notions of NT-subgroups, strong NT-subgroups, and weak commutative neutrosophic triplet groups (WCNTGs) are introduced, the quotient structures are constructed by strong NT-subgroups, and a homomorphism theorem is proved in weak commutative neutrosophic triplet groups.
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Keywords:
neutrosophic triplet group (NTG); NT-subgroup; homomorphism theorem; weak commutative neutrosophic triplet group
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
MDPI and ACS Style
Zhang, X.; Hu, Q.; Smarandache, F.; An, X. On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes. Symmetry 2018, 10, 289. https://doi.org/10.3390/sym10070289
AMA Style
Zhang X, Hu Q, Smarandache F, An X. On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes. Symmetry. 2018; 10(7):289. https://doi.org/10.3390/sym10070289
Chicago/Turabian StyleZhang, Xiaohong; Hu, Qingqing; Smarandache, Florentin; An, Xiaogang. 2018. "On Neutrosophic Triplet Groups: Basic Properties, NT-Subgroups, and Some Notes" Symmetry 10, no. 7: 289. https://doi.org/10.3390/sym10070289
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