Abstract
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.
Highlights
The importance of group theory is self-evident
Smarandache and Mumtaz Ali in [14,15], for the first time, introduced the new notion of neutrosophic triplet group (NTG), which is another generalization of classical group
In NTG, the unit element is generalized as a neutral element, which is relative and local; that is, each element has its own neutral element; and the original inverse element concept is generalized as an anti element, and it is relative to own neutral element, and it cannot be unique
Summary
The importance of group theory is self-evident. As a more general concept, Molaei introduced the new notion of generalized group in [3,4], and some researchers studied its properties [5,6]. The concept of neutrosophic set is introduced by F Smarandache, it is a generalization of (intuitionistic) fuzzy sets [7]. Smarandache and Mumtaz Ali in [14,15], for the first time, introduced the new notion of neutrosophic triplet group (NTG), which is another generalization of classical group. Symmetry 2018, 10, 289 new concepts, include NT-subgroups, strong NT-subgroups, and weak commutative neutrosophic triplet groups (WCNTGs), and prove a homomorphism theorem of weak commutative neutrosophic triplet groups
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