# Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
## 1 Introduction
### Generalized Group
This mathematical tool is used to handle problems consisting of uncertainty, imprecision, indeterminacy, inconsistency, incompleteness, and falsity. The development process of neutrosophic sets, fuzzy sets, and intuitionistic fuzzy sets are still growing, with various applications; here are some recent research works in these directions [25; 26; 27; 28; 29; 30; 31; 32]. By utilizing the idea of neutrosophic theory, [PERSON] and [PERSON] studied neutrosophic algebraic structures in [33; 34; 35] by introducing an indeterminate element \"\(I\)\" in the algebraic structure and then combining \"\(I\)\" with each element of the structure with respect to the corresponding binary operation \(*\). This was called a neutrosophic element, and the generated algebraic structure was termed a neutrosophic algebraic structure. They further studied several neutrosophic algebraic structures, such as neutrosophic fields, neutrosophic vector spaces, neutrosophic groups, neutrosophic bigroups, neutrosophic N-groups, neutrosophic semigroups, neutrosophic bisemigroups, neutrosophic N-semigroups, neutrosophic loops, neutrosophic biloops, neutrosophic N-loops, neutrosophic groupoids, neutrosophic bigroupoids, and so on. [PERSON] and [PERSON] [36] for the first time introduced the idea of the neutrosophic triplet, which they had previously discussed in [37]. They used these neutrosophic triplets to introduce the neutrosophic triplet group, which is different from the classical group both in structural and fundamental properties. They gave distinction and comparison of neutrosophic triplet group with the classical generalized group. They also drew a brief sketch of the possible applications of the neutrosophic triplet group in some other research areas. [PERSON] [38] studied new algebraic properties of the neutrosophic triplet group with new applications. Some new applications of neutrosophy were announced in [PERSON] and [PERSON] [39], [PERSON] and [PERSON] [40], [PERSON] et al. [41], and [PERSON] [42]. [PERSON] et al. [43] and [PERSON] et al. [44] are some recent works on neutrosophic triplet groups, neutrosophic quadruple, and neutrosophic duplet of algebraic structures. Definition 4: (Neutrosophic Triplet Set)  
Let \(X\) be a set together with a binary operation \(*\) defined on it. Then, \(X\) is called a neutrosophic triplet set if for any \(x\in X\), there exists a neutral of \"\(x\)\" denoted by \(neut(x)\) (not necessarily the identity element) andan opposite of \"\(x\)\" denoted by \(anti(x)\) or \(x\)], with \(neut(x),anti(x)\in X\) such that:  
\[x*neut(x)=neut(x)*x=x\quad\text{and}\quad x*anti(x)=anti(x)*x=neut(x).\]  
The elements \(x,neut(x)\), and \(anti(x)\) are collectively referred to as a neutrosophic triplet, and denoted by \((x,neut(x),anti(x))\). Remark 1: _For the same \(x\in X\), each \(neut(x)\) and \(anti(x)\) may not be unique. In a neutrosophic triplet set \((X,*)\), an element \(y\) (resp. \(z\)) is the second (resp. third) component of a neutrosophic triplet if there exist \(x,z\in X\) (\(x,y\in X\)) such that \(x*y=y*x=x\) and \(x*z=z*x=y\). Thus, \((x,y,z)\) is the neutrosophic triplet._  
Example 1: ([PERSON] and [PERSON] [36])  
_Consider \((\mathbb{Z}_{6},\times_{6})\) where \(\mathbb{Z}_{6}=\{0,1,2,3,4,5\}\) and \(\times_{6}\) is multiplication in modulo \(6\). \((2,4,2),(4,4,4)\), and \((0,0,0)\) are neutrosophic triplets, but \(3\) does not give rise to a neutrosophic triplet._  
Definition 5: (Neutrosophic Triplet Group)  
_Let \((X,*)\) be a neutrosophic triplet set. Then, \((X,*)\) is called a neutrosophic triplet group if \((X,*)\) is a semigroup. If in addition, \((X,*)\) obeys the commutativity law, then \((X,*)\) is called a commutative neutrosophic triplet group._  
Remark 2: _A neutrosophic triplet group is not a group in general, but a group is a neutrosophic triplet group where \(neut(x)=e\) the general identity element for all \(x\in X\) and \(anti(x)\) is unique for each \(x\in X\)._  
1. _A generalized loop is a generalized group if and only if it is associative._
2.