## 2 The Norm Systems
### The Repeatability Norms
_The operation_ \(\circ_{s}:\textbf{n-ScIFN}\times\textbf{n-ScIFN}\rightarrow\textbf{n-ScIFN}\) _is called the_ _s-norm in the set_ _n-ScIFN_ _ordered by the relation_ \(\leq_{n}\)_, when for any_ \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle,\langle\mu_{3},v_{3} \rangle\in\textbf{n-ScIFN}\)_:_ _(a)_ \(\langle 1,0\rangle\circ_{s}\langle\mu_{1},v_{1}\rangle=\langle 1,0\rangle, \langle\mu_{1},v_{1}\rangle\circ_{s}\langle 0,1\rangle=\langle\mu_{1},v_{1}\rangle\)__--boundary conditions, _(b)_ \(\langle\mu_{1},v_{1}\rangle\circ_{s}\langle\mu_{2},v_{2}\rangle\leq_{n}\langle \mu_{3},v_{3}\rangle\circ_{s}\langle\mu_{2},v_{2}\rangle\)_, when_ \(\langle\mu_{1},v_{1}\rangle\leq_{n}\langle\mu_{3},v_{3}\rangle\)__--_monotonicity, _(c)_ \(\langle\mu_{1},v_{1}\rangle\circ_{s}\langle\mu_{2},v_{2}\rangle=\langle\mu_{2}, v_{2}\rangle\circ_{s}\langle\mu_{1},v_{1}\rangle\)__--commutativity, _(d)_ \(\langle\mu_{1},v_{1}\rangle\circ_{s}(\langle\mu_{2},v_{2}\rangle\circ_{s} \langle\mu_{3},v_{3}\rangle)=(\langle\mu_{1},v_{1}\rangle\circ_{s}\langle\mu_{2 },v_{2}\rangle)\circ_{s}\langle\mu_{3},v_{3}\rangle\)__--associativity. **Theorem 9**. : _The Yager operator \(\otimes\) on the **n-ScIFN** is the \(t\)-norm in the set **n-ScIFN**. The Yager operator \(\oplus\) in the **n-ScIFN** is the s-norm._  
Proof. : Conditions (4)-(6) of Theorem 8 prove that the conditions (a), (c), (d) of Definition 7 (1) are satisfied with the t-norm in the set _n-ScIFN_ by the operator \(\otimes\). It is enough to prove that monotonicity is satisfied by this operation. Let for any \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle,\langle\mu_{3},v_{3} \rangle\in\textbf{n-ScIFN}\):  
* \(\langle\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle=\langle Sc_{n}^{-1}(Sc _{n}(\mu_{1})\bullet_{t}Sc_{n}(\mu_{2})),Sc_{n}^{-1}(Sc_{n}(v_{1})\bullet_{s}Sc_{n }(\
u_{2}))\rangle\),
* \(\langle\mu_{3},v_{3}\rangle\otimes\langle\mu_{2},v_{2}\rangle=\langle Sc_{n}^{-1}(Sc _{n}(\mu_{3})\bullet_{t}Sc_{n}(\mu_{2})),Sc_{n}^{-1}(Sc_{n}(v_{3})\bullet_{s}Sc_{n }(\
u_{2}))\rangle\),  
and let \(\langle\mu_{1},v_{1}\rangle\leq_{n}\langle\mu_{3},v_{3}\rangle\). Then \(\mu_{1}\leq\mu_{3},v_{1}\geq v_{3}\). Hence and form the monotonicity of t-norm and s-norm as well as the scale function \(Sc_{n}\), there are inequalities:  
\(Sc_{n}(\mu_{1})\bullet_{t}Sc_{n}(\mu_{2})\leq Sc_{n}(\mu_{3})\bullet_{t}Sc_{n }(\mu_{2})\),  
\(Sc_{n}(v_{3})\bullet_{s}Sc_{n}(v_{2})\leq Sc_{n}(v_{1})\bullet_{s}Sc_{n}(\
u_{2}))\) iff  
\(Sc_{n}^{-1}(Sc_{n}(\mu_{1})\bullet_{t}Sc_{n}(\mu_{2}))\leq Sc_{n}^{-1}(Sc_{n}( \mu_{3})\bullet_{t}Sc_{n}(\mu_{2}))\),  
\(Sc_{n}^{-1}(Sc_{n}(\mu_{3})\bullet_{s}Sc_{n}(\
u_{2}))\leq Sc_{n}^{-1}(Sc_{n}(v_{3}) \bullet_{s}Sc_{n}(\
u_{2}))\) iff  
\((Sc_{n}^{-1}(Sc_{n}(\mu_{1})\bullet_{t}Sc_{n}(\mu_{2})),Sc_{n}^{-1}(Sc_{n}(v_{1}) \bullet_{s}Sc_{n}(\
u_{2})))\leq_{n}\)\(\langle\mathcal{S}c_{n}^{-1}(Sc_{n}(\mu_{3})\bullet_{t}Sc_{n}(\mu_{2})),Sc_{n}^{-1 }(Sc_{n}(v_{3})\bullet_{s}Sc_{n}(v_{2}))\rangle\) iff \(\langle\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle\leq_{n}\langle \mu_{3},v_{3}\rangle\otimes\langle\mu_{2},v_{2}\rangle\). The fulfillment by the operator \(\oplus\) of the conditions of Definition 7 (2) of the s-norm in the _n-ScIFN_ set is proved analogously. The monotonicity of the operators \(\oplus,\otimes\) describes important Theorem:  
Theorem 3.1: _For any \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle,\langle\mu_{3},v_{3} \rangle\in\textbf{n-ScIFN}\):_  
1. \(\langle\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle\leq_{n}\langle \mu_{1},v_{1}\rangle\) _and_ \(\langle\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle\leq_{n}\langle \mu_{2},v_{2}\rangle\)_,_
2. \(\langle\mu_{1},v_{1}\rangle\leq_{n}\langle\mu_{1},v_{1}\rangle\oplus\langle \mu_{2},v_{2}\rangle\) _and_ \(\langle\mu_{2},v_{2}\rangle\leq_{n}\langle\mu_{1},v_{1}\rangle\oplus\langle \mu_{2},v_{2}\rangle\)_,_
3. \(\langle\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle\leq_{n}\langle \mu_{1},v_{1}\rangle\oplus\langle\mu_{2},v_{2}\rangle\)_,_
4.