# Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
## 2 Main Results
_LCIP._Let \(X\) be an RIPNL. Then, \(yx*anti(x)=y\). RCIP implies \(xy*anti(x)=y\Rightarrow xy*anti(x)=yx*anti(x)\Rightarrow xy=yx\). Conversely, RIP and commutativity imply \(xy*anti(x)=y\) and \(anti(x)*yx=y\) imply RCIP and LCIP. 2. Let \(X\) be an LIPNL. Then, \(anti(x)*xy=y\). LCIP implies \(anti(x)*yx=y\Rightarrow anti(x)*yx=anti(x)*xy=y\Rightarrow xy\). Conversely, LIP and commutativity imply \(xy*anti(x)=y\) and \(anti(x)*yx*=y\) imply RCIP and LCIP. 3. This follows from 1 and 2. 4. Let \(X\) be commutative. \(X\) has the RIP iff \(yx*anti(x)=y\Leftrightarrow anti(x)*xy=y\) iff \(X\) has the LIP. \(X\) has the RIP iff \(yx*anti(x)=y\Leftrightarrow anti(x)*yx*=y\) iff \(X\) has the LCIP. \(\Box\)  
**Theorem 4**. : _Let \(X\) be an IPNTL. For all \(x,y\in X\),_  
1. \(\big{(}x,y,anti(y)\big{)}=anti\big{(}x\text{ }neut(y)\big{)}x\)_,_ \(\big{(}x,x,anti(x)\big{)}=neut(x)\)_._
2. \(\big{(}anti(y),y,x\big{)}=anti(x)*neut(y)x\)_,_ \(\big{(}anti(x),x,x\big{)}=neut(x)\)_._
3. \(\big{[}x,y,anti(y)\big{)}=x\text{ }anti\big{(}x\text{ }neut(y)\big{)}\)_,_ \(\big{[}x,x,anti(x)\big{]}=neut(x)\)_._
4. \(\big{[}anti(y),y,x\big{]}=neut(y)\)_,_ \(\big{[}anti(x),x,x\big{]}=neut(x)\)_._
5. \((x,y)=anti(yx)(xy)\) _and_ \([x,y]=(xy)anti(yx)\)_._
6. \(\big{(}x,y,anti(y)\big{)}=\big{[}x,y,anti(y)\big{]}\Leftrightarrow x*anti( neut(y))anti(x)=anti(neut(y))\)_._
7. \(\big{(}anti(y),y,x\big{)}=\big{[}anti(y),y,x\big{]}\Leftrightarrow x\text{ }neut(y)=neut(y)x\)_._
8. \(anti\Big{(}x\big{[}anti(y),y,x\big{]}\Big{)}x=\big{(}x,y,anti(y)\big{)}\)_._
9. \(anti(x)*\big{[}anti(y),y,x\big{]}x=\big{(}anti(y),y,x\big{)}\)_._
10. \(x\text{ }anti\Big{(}x\big{[}anti(y),y,x\big{]}\Big{)}x=\big{[}x,y,anti(y)\big{]}\)_._
11. \(\big{(}neut(x),x\big{)}=neut(x)=\big{[}neut(x),x\big{]}\) _and_ \((x,x)=neut(xx)=[x,x]\)_._
12. \(\big{(}x,neut(x),anti(x)\big{)}=neut(neut(x))=\big{(}anti(x),neut(x),x\big{)}\)_._
13. \(\big{(}neut(x),x,anti(x)\big{)}=\big{(}anti(x),x,neut(x)\big{)}=\big{(}x,anti(x), neut(x)\big{)}=\big{(}neut(x),anti(x),x\big{)}=neut(x)\)_._  
**Proof. **  
1. From the right associator, \(xy*anti(y)=(x*y\text{ }anti(y))\big{(}x,y,anti(y)\big{)}\Rightarrow x=x\text{ }neut(y)* \big{(}x,y,anti(y)\big{)}\Rightarrow anti(x\text{ }neut(y))x=anti(x\text{ }neut(y))\big{[}x\text{ }neut(y)*\big{(}x,y,anti(y)\big{)}\big{]} \Rightarrow\big{(}x,y,anti(y)\big{)}=anti(x\text{ }neut(y))x\). Hence, \(\big{(}x,x,anti(x)\big{)}=neut(x)\). 2. From the right associator, \(anti(y)y*x=(anti(y)*yx)\big{(}anti(y),y,x\big{)}\Rightarrow neut(y)x=x\big{(}anti( y),y,x\big{)}\Rightarrow anti(x)*x\big{(}anti(y),y,x\big{)}\Rightarrow\big{(}anti(y),y,x \big{)}=anti(x)*neut(y)x\). Hence, \(\big{(}anti(x),x,x\big{)}=neut(x)\). 3. From the left associator, \(xy*anti(y)=\big{[}x,y,anti(y)\big{)}(x*y\text{ }anti(y))\Rightarrow x=[x,y,anti(y)](x\text{ }neut(y))\Rightarrow x\text{ }anti(x\text{ }neut(y))=[x,y,anti(y)](x\text{ }neut(y))*anti(x\text{ }neut(y))\Rightarrow[x,y,anti(y)]=x\text{ }anti(x\text{ }neut(y))\). Hence, \(\big{[}x,x,anti(x)\big{]}=neut(x)\). 4. From the left associator, \(anti(y)y*x=\big{[}anti(y),y,x\big{]}\big{(}anti(y)*yx)\Rightarrow neut(y)x=[anti(y),y, x]x\xrightarrow{Lump\ 2}[anti(y),y,x]=neut(y)\). Hence, \(\big{[}anti(x),x,x\big{]}=neut(x)\). 5. From the right commutator, \(x*y=(y*x)(x,y)\Rightarrow anti(yx)*xy=anti(yx)*(yx)(x,y)\Rightarrow(x,y)=anti( yx)(xy)\). From the left commutator, \(x*y=[x,y](y*x)\Rightarrow xy*anti(yx)=[x,y](yx)*anti(yx)\Rightarrow[x,y]=(xy)anti(yx)\). 6. By 1 and 3, \(\big{(}x,y,anti(y)\big{)}=\big{[}x,y,anti(y)\big{]}\Leftrightarrow anti(x\text{ }neut(y))x=x\text{ }anti(x\text{ }neut(y))\xrightarrow{\ A\ A\ A\ P}{This follows combining by 1 and 4. * This follows combining by 2 and 4. * This follows combining by 3 and 4. * Apply 5. * Apply the right and left associators. **Lemma 5**. : _Let \(X\) be a CIPNTL or an IPLNTL. Then:_  
1. \(neut(x)\) _is unique for each_ \(x\in X\)_._
2. \(anti(x)\) _is unique for each_ \(x\in X\)_._
3. \(X\) _is a generalized loop and a quasigroup._
4.