# Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
## 1 Introduction
### Generalized Group
_A neutrosophic triplet loop (NTL) is a neutrosophic triplet group if and only if it is associative._
3. _An NTL is a generalized loop if and only if_ \(neut(x)=e(x)\) _is unique for each_ \(x\)_._
4. _An NTL is a loop if and only if it is a quasigroup and_ \(neut(x)=neut(y)\) _for all_ \(x,y\)  
**Example 2**. : _([PERSON] and [PERSON] [36])_  
_Consider \((\mathbb{Z}_{10},\otimes)\) where \(x\otimes y=3 xy\bmod 10\). \((\mathbb{Z}_{10},\otimes)\) is a commutative neutrosophtic triplet group but neither a classical group nor a generalized group._  
**Example 3**. : _([PERSON] and [PERSON] [36])_  
_Consider \((\mathbb{Z}_{10},\star)\) where \(x\star y=5x+y\bmod 10\). \((\mathbb{Z}_{10},\star)\) is a non-commutative neutrosophtic triplet group, but not a classical group._  
**Theorem 1**. : _([PERSON] and [PERSON] [36])_  
_Let \((X,\ast)\) be a neutrosophtic triplet group. The following are true for all \(x,y,z\in X\)._  
1. \(x\ast y=x\ast z\Leftrightarrow neut(x)\ast y=neut(x)\ast z\)_._
2. \(y\ast x=z\ast x\Leftrightarrow y\ast neut(x)=z\ast neut(x)\)_._
3. \(anti(x)\ast y=anti(x)\ast z\Rightarrow neut(x)\ast y=neut(x)\ast z\)_._
4. \(y\ast anti(x)=z\ast anti(x)\Rightarrow y\ast neut(x)=z\ast neut(x)\)_._
5. \(neut(x)\ast neut(x)=neut(x)\) _i.e.,_ \(neut\big{(}neut(x)\big{)}=neut(x)\)_._
6. \(neut(x)^{n}=neut(x)\) _for any_ \(n\in\mathbb{N}\)_;_ \(anti(neut(x))=neut(x)\)_._
7. \(neut(x)\ast anti(x)=anti(x)\ast neut(x)=anti(x)\) _i.e._ \(neut\big{(}anti(x)\big{)}=neut(x)\)_._  
**Definition 6**. : _(Neutrosophic Triplet Loop--NTL)_  
_Let \((X,\ast)\) be a neutrosophtic triplet set. Then, \((X,\ast)\) is called a neutrosophtic triplet loop if \((X,\ast)\) is a groupoid. If in addition, \((X,\ast)\) obeys the commutativity law, then \((X,\ast)\) is called a commutative neutrosophtic triplet loop._  
_Let \((X,\ast)\) be a neutrosophtic triplet loop. If \(neut(xy)=neut(x)neut(y)\) for all \(x,y\in X\), then \(X\) is called normal._  
**Remark 3**. : _An NTL is a neutrosophtic triplet group if and only if it is associative. Thus, an NTL is a generalization of a neutrosophtic triplet group, and it is interesting to study an NTL that obeys weak associative law. NTL was originally introduced by [PERSON]._  
**Example 4**. : _Let \((\mathbb{Z}_{10},+,\cdot)\) be the field of integers modulo \(10\). Consider \((\mathbb{Z}_{10},\ast)\), where for all \(x,y\in\mathbb{Z}_{10},\ x\ast y=2x+2y\). The following are neutrosophtic triplets:_  
\[(0,0,0),(0,0,5),(2,4,0),(2,4,5),(4,8,0),(4,8,5),(6,2,0),(6,2,5),(8,6,0),(8,6,5)\]  
_in \((\mathbb{Z}_{10},\ast)\). Thus, \(\{0,2,4,5,6,8\}\) is a neutrosophtic triplet set. \((\mathbb{Z}_{10},\ast)\) is non-associative because \((x\ast y)\ast z=4x+4y+2z\
eq x\ast(y\ast z)=2x+4y+4z\). \((\mathbb{Z}_{10},\ast)\) is a non-associative NTL (i.e., not a neutrosophtic triplet group) with \(2\ast neut(x)=9x\) and \(4\ast anti(x)=5x\)._  
**Definition 7**. : _(Inverse Properties and Neutrosophic Triplet Loop)_  
\((X,\ast)\) _will be called a right inverse property neutrosophic triplet loop (RIPNTL) if it obeys the right inverse property (RIP)_  
\[(y\ast x)\ast\ anti(x)=y \tag{1}\]  
\((X,\ast)\) _will be called a left inverse property neutrosophic triplet loop (LIPNTL) if it obeys the left inverse property (LIP)_  
\[anti(x)\ast(x\ast y)=y \tag{2}\]  
\((X,\ast)\) _will be called an inverse property neutrosophic triplet loop if it obeys both (_1_) and (_2_)._  
\((X,\ast)\) _will be called a left cross inverse property neutrosophic triplet loop (LCIPNTL) if it obeys the left cross inverse property (LCIP)_  
\[anti(x)\ast(y\ast x)=y\]\((X,*)\) will be called a right cross inverse property neutrosophic triplet loop (RCIPNTL) if it obeys the right cross inverse property (RCIP)_  
\[(x*y)*anti(x)=y \tag{4}\]  
\((X,*)\) will be called a cross inverse property neutrosophic triplet loop (CIPNTL) if it obeys both (3) and (4).