# Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
## 2 Main Results
\(X\) _is a loop if and only if_ \(neut(x)=neut(y)\) _for all_ \(x,y\in X\)_._
5. _If_ \(X\) _is associative, then_ \(X\) _is a loop and group._
6. \(X\) _is a group if and only if_ \(X\) _is associative._  
Proof. :
1. By Lemma 2(2), \(neut(x)x=x=neut(x)^{\prime}\Rightarrow neut(x)=neut(x)^{\prime}\). 2. By Lemma 2(2), \(anti(x)x=x=anti(x)^{\prime}\Rightarrow anti(x)=x=anti(x)^{\prime}\). 3. These follow by 1. and Lemma 2(1). 4. By the definition of NTL and loop, and 2. 5. An associative quasigroup is a loop and a group. 6. A loop is a group if and only it is associative. **Theorem 5**. : _Let \(X\) be an NTL._  
1. _If_ \(X\) _is an IPNTL, then for all_ \(x\in X\)_:_ 1. _[label=()]_ 2. \(X\) _is an AAIPNL._ 3. \(R_{x}^{-1}=R_{anti(x)}\) _and_ \(L_{x}^{-1}=L_{anti(x)}\)_._ 4. \(JR_{x}J=L_{x}^{-1}\) _and_ \(JL_{x}J=R_{x}^{-1}\)_._ 5. \(X\) _is a WIPNTL._
2. _If_ \(X\) _is a CIPNTL, then for all_ \(x\in X\)_:_ 1. \(X\) _is an AIPNTL._ 2. \(L_{x}R_{anti(x)}=I=R_{anti(x)}L_{x}\) _and_ \(R_{x}L_{anti(x)}=I=L_{anti(x)}R_{x}\)_._ 3. \(JR_{x}J=R_{anti(x)}\) _and_ \(JL_{x}J=L_{anti(x)}\)_._ 4. \(X\) _is a WIPNTL._
3. _If_ \(X\) _is an RCIPNTL, then for all_ \(x\in X\)_:_ 1. \(X\) _is an AIPNTL._ 2. \(L_{x}R_{anti(x)}=I=R_{anti(x)}L_{x}\)_._ 3. \(JR_{x}J=R_{anti(x)}\) _and_ \(JL_{x}J=L_{anti(x)}\) _if and only if_ \(anti\big{(}anti(x)\big{)}=x\)_._ 4. \(X\) _is an RWIPNTL._
4. _If_ \(X\) _is an LCIPNTL, then for all_ \(x\in X\)_:_ 1. \(X\) _is an AIPNTL._ 2. \(X\) _is an AIPNTL._ 3. \(JR_{x}J=R_{anti(x)}\) _and_ \(JL_{x}J=L_{anti(x)}\) _if and only if_ \(anti\big{(}anti(x)\big{)}=x\)_._ 4. \(X\) _is an RWIPNTL._
5. _If_ \(X\) _is an LCIPNTL, then for all_ \(x\in X\)_:_ 1. \(X\) _  
_._ 2. Let \(X\) be a CIPNTL. 1. \(xy=z\Rightarrow y=z\;anti(x)\Rightarrow anti(x)=y\;anti(z)\Rightarrow anti(z)= anti(x)anti(y)\Rightarrow anti(x)anti(y)=anti(x)\Rightarrow anti(x)\). So, \(JR_{x}J=R_{anti(x)}\Leftrightarrow anti(anti(y))anti(x)=y\;anti(x)\Leftrightarrow anti \big{(}anti(y)\big{)}=y\). 2. By Theorem 2: LCIP implies that \(R_{x}L_{anti(x)}=I=L_{anti(x)}R_{x}\). 3. \(yJR_{x}J=anti\big{(}anti(y)x)=anti\big{(}anti(y)\big{)}anti(x)\). So, \(JR_{x}J=R_{anti(x)}\Leftrightarrow anti\big{(}anti(y)\big{)}anti(x)=y\;anti(x) \Leftrightarrow anti\big{(}anti(y)\big{)}=y\). 4. \(yJL_{x}J=anti\big{(}x\;anti(y)\big{)}=anti(x)anti\big{(}anti(y)\big{)}\). So, \(JL_{x}J=L_{anti(x)}\Leftrightarrow anti(x)anti\big{(}anti(y)\big{)}=anti(x)y \Leftrightarrow anti\big{(}anti(y)\big{)}=y\). 5. \(anti(xy)x=anti(x)anti(y)*x=anti(y)\Rightarrow\) LWIP. So, \(X\) is an LWIPNTL. \(\Box\)  
**Theorem 6**. : _Let \(X\) be an NTL._  
1. _If_ \(X\) _is an LWIPNTL, then for all_ \(x\in X\)_:_ 1. \(neut(x)=anti(neut(x))\)_._ 2. \(neut\big{(}neut(x)\big{)}=neut(x)neut(x)\)_._ 3. \(anti\big{(}anti(x)\big{)}=x\) _and_ \(J^{2}=I\)_._ 4. \(neut(x)=neut\big{(}anti(x)\big{)}\)_._ 5. _J is a bijection._ 6. \(X\) _is a left quasigroup._ 7. \(L_{x}\) _is a bijection._
2. _If_ \(X\) _is an RWIPNTL, then for all_ \(x\in X\)_:_ 1. \(neut(x)=anti\big{(}neut(x)\big{)}\)_._ 2. \(neut\big{(}neut(x)\big{)}=neut(x)neut(x)\)_._ 3. \(anti\big{(}anti(x)\big{)}=x\) _and_ \(J^{2}=I\)_._ 4. \(neut(x)=neut\big{(}anti(x)\big{)}\)_._ 5. _J is a bijection._ 6. \(X\) _is a right quasigroup._ 7. \(R_{x}\) _is a bijection._
3. _The following are equivalent._ 1. \(X\) _is an LWIPNTL and_ \(R_{x}\) _is bijective._ 2. \(X\) _is an RWIPNTL and_ \(L_{x}\) _is bijective._ 3. \(X\) _is an LWIPNTL and_ \(X\) _is a right quasigroup._ 4. \(X\) _is an RWIPNTL and_ \(X\) _is a left quasigroup._
4. _If_ \(X\) _is a WIPNTL, then_ \(L_{x}^{2}=I\Leftrightarrow R_{x}^{2}=I\)_._ 5. _If_ \(X\) _is an LCIPNTL, then_ \(X\) _is a right quasigroup._
6. _If_ \(X\) _is an RCIPNTL, then_ \(X\) _is a left quasigroup._  
Proof. :
1. Let \(X\) be an LWIPNTL, then \(anti(xy)x=anti(y)\).