Some Results on Neutrosophic Triplet Group and Their Applications

Abstract

This article is based on new developments on a neutrosophic triplet group (NTG) and applications earlier introduced in 2016 by Smarandache and Ali. NTG sprang up from neutrosophic triplet set X: a collection of triplets (b, neut(b), anti(b)) for an b ∈ X that obeys certain axioms (existence of neutral(s) and opposite(s)). Some results that are true in classical groups were investigated in NTG and were shown to be either universally true in NTG or true in some peculiar types of NTG. Distinguishing features between an NTG and some other algebraic structures such as: generalized group (GG), quasigroup, loop and group were investigated. Some neutrosophic triplet subgroups (NTSGs) of a neutrosophic triplet group were studied. In particular, for any arbitrarily fixed a ∈ X, the subsets X a = {b ∈ X : neut(b) = neut(a)} and ker f a = {b ∈ X| f (b) = neut(f (a))} of X, where f : X → Y is a neutrosophic triplet group homomorphism, were shown to be NTSG and normal NTSG, respectively. Both X a and ker f a were shown to be a-normal NTSGs and found to partition X. Consequently, a Lagrange-like formula was found for a finite NTG X; |X| = ∑ a∈X [X a : ker f a ]| ker f a | based on the fact that | ker f a | |X a |. The first isomorphism theorem X/ ker f ∼ = Im f was established for NTGs. Using an arbitrary non-abelian NTG X and its NTSG X a , a Bol structure was constructed. Applications of the neutrosophic triplet set, and our results on NTG in relation to management and sports, are highlighted and discussed.

Singular neutrosophic extended triplet groups and generalized groups

“…a2X ½X a : ker f a Á jker f a j It follows that Thereom 6 (6) and (8) in (Jaiyeola & Smarandache, 2018) are not true.…”

“…For a neutrosophic extended triplet group, the Ref. Jaiyeola and Smarandache (2018) gives some important research directions, but there were some errors. In this section, some counterexamples will be constructed.…”

“…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 ßahin, & Kargin, 2018;Ali, Smarandache, & Khan, 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).…”

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

“…And in [30], sorts of general neutrosophic triplet structures were pointed out, and their basic properties were investigated. For the structure of NETG, some research papers are published with a series of results [31][32][33][34][35].…”

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

“…Finally, a practical case study along with a comparative study are carried out to reveal the accuracy and reliability of the constructed three-way decisions model. The next article [39] is based on new developments on a NTG and applications earlier introduced in 2016 by Smarandache and Ali. NTG sprang up from neutrosophic triplet set X: a collection of triplets (b,neut(b),anti(b)) for an b∈X that obeys certain axioms (existence of neutral(s) and opposite(s)).…”