# Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
## 1 Introduction  
### Generalized Group  
A generalized group is an algebraic structure which has a deep physical background in the unified gauge theory and has direct relation with isotopies. Mathematicians and physicists have been trying to construct a suitable unified theory for twistor theory, isotopies theory, and so on. It was known that generalized groups are tools for constructions in unified geometric theory and electroweak theory. Electroweak theories are essentially structured on Minkowskian axioms, and gravitational theories areconstructed on Riemannian axioms. According to [PERSON] et. al. [1], the generalized group is equivalent to the notion of a completely simple semigroup. Some of the structures and properties of generalized groups have been studied by [PERSON] [2], [PERSON] [3], [4], [PERSON], [PERSON], and [PERSON] [5], [PERSON] [6], [PERSON] et al. [7], and [PERSON] and [PERSON] [8]. Smooth generalized groups were introduced in [PERSON] [9], and later [PERSON] [10] also presented smooth generalized subgroups while [PERSON] [11], [PERSON] and [PERSON] [12] considered the notion of topological generalized groups, and [PERSON] and [PERSON] [13] studied the quotient space of generalized groups. **Definition 1**. : _(Generalized Group)_  
_A generalized group \(G\) is a non-empty set admitting a binary operation called multiplication, subject to the set of rules given below._  
_(i)_ \((xy)z=x(yz)\) _for all_ \(x,y,z\in G\)_._  
_(ii)_ _For each_ \(x\in G\)_, there exists a unique_ \(e(x)\in G\) _such that_ \(xe(x)=e(x)x=x\) _(existence and uniqueness of identity element)._  
_(iii)_ _For each_ \(x\in G\)_, there exists_ \(x^{-1}\in G\) _such that_ \(xx^{-1}=x^{-1}x=e(x)\) _(existence of inverse element)._  
**Definition 2**. : _Let \(L\) be a non-empty set. Define a binary operation \((\cdot)\) on \(L\). If \(x\cdot y\in L\) for all \(x,y\in L\), \((L,\cdot)\) is called a groupoid._  
_If the equation \(a\cdot x=b\) (resp. \(y\cdot a=b\)) has a unique solution relative to \(x\) (resp. \(y\)) (i.e., obeys the left (resp. right) cancellation law), then \((L,\cdot)\) is called a left (resp. right) quasigroup. If a groupoid \((L,\cdot)\) is both a left quasigroup and right quasigroup, then it is called a quasigroup. If there exists an element \(e\in L\) called the identity element such that for all \(x\in L\), \(x\cdot e=e\cdot x=x\), then a quasigroup \((L,\cdot)\) is called a loop._  
For more on quasigroups and loops, readers should check [14; 15; 16; 17; 18; 19; 20]. **Definition 3**. : _(Generalized loop)_  
_A generalized loop is the pair \((G,\cdot)\) where \(G\) is a non-empty set and \"\(\cdot\)\" a binary operation such that the following are true._  
_(i)_ \((G,\cdot)\) _is a groupoid._  
_(ii)_ _For each_ \(x\in G\)_, there exists a unique_ \(e(x)\in G\) _such that_ \(xe(x)=e(x)x=x\)_._  
_(iii)_ _For each_ \(x\in G\)_, there exists_ \(x^{-1}\in G\) _such that_ \(xx^{-1}=x^{-1}x=e(x)\)_._  
A generalized group \(G\) exhibits the following properties:  
_(i)_ _For each_ \(x\in G\)_, there exists a unique_ \(x^{-1}\in G\)_._  
_(ii)_ \(e(e(x))=e(x)\) _and_ \(e(x^{-1})=e(x)\) _whenever_ \(x\in G\)_._  
_(iii)_ _If \(G\) is commutative, then \(G\) is a group._  
Neutrosophy is a new branch of philosophy which studies the nature, origin, and scope of neutralities as well as their interaction with ideational spectra. In 1995, [PERSON] [21] first introduced the concept of neutrosophic logic and neutrosophic sets where each proposition in neutrosophic logic is approximated to have the percentage of truth in a subset \(T\), the percentage of indeterminacy in a subset \(I\), and the percentage of falsity in a subset \(F\) so that this neutrosophic logic is called an extension of fuzzy logic, especially to intuitionistic fuzzy logic. In fact, the neutrosophic set is the generalization of classical sets [22], fuzzy sets [23], intuitionistic fuzzy sets [22; 24] and interval valued fuzzy sets [22], to mention a few.