## 2 The Norm Systems
### The Repeatability Norms
\((\langle\mu_{1},v_{1}\rangle\oplus\langle\mu_{2},v_{2}\rangle)\oplus\langle\mu_{3},v_{3} \rangle=\langle\mu_{1},v_{1}\rangle\oplus(\langle\mu_{2},v_{2}\rangle\oplus \langle\mu_{3},v_{3}\rangle)\)_,_\(\langle\mu_{1},v_{1}\rangle\otimes\langle 1,0\rangle=\langle\mu_{1},v_{1}\rangle, \langle\mu_{1},v_{1}\rangle\oplus\langle 0,1\rangle=\langle 1,0\rangle\),  
\(\langle\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle=\langle\mu_{2}, v_{2}\rangle\otimes\langle\mu_{1},v_{1}\rangle\),  
\(\langle(\mu_{1},v_{1})\otimes\langle\mu_{2},v_{2}\rangle\rangle\otimes \langle\mu_{3},v_{3}\rangle=\langle\mu_{1},v_{1}\rangle\otimes(\langle\mu_{2},v_{2}\rangle\otimes\langle\mu_{3},v_{3}\rangle)\),  
\(\lambda(\langle\mu_{1},v_{1}\rangle\oplus\langle\mu_{2},v_{2}\rangle)= \lambda\langle\mu_{1},v_{1}\rangle\oplus\lambda\langle\mu_{2},v_{2}\rangle\),  
\(\langle(\mu_{1},v_{1}\rangle\otimes\langle\mu_{2},v_{2}\rangle)\lambda= \langle\mu_{1},v_{1}\rangle\lambda\otimes\langle\mu_{2},v_{2}\rangle\lambda\),  
\(\langle(\lambda_{1}+\lambda_{2})\langle\mu_{1},v_{1}\rangle=\lambda_{1} \langle\mu_{1},v_{1}\rangle\oplus\lambda_{2}\langle\mu_{1},v_{1}\rangle\),  
\(\langle\mu_{1},v_{1}\rangle^{\lambda_{1}+\lambda_{2}}=\langle\mu_{1},v_{1} \rangle^{\lambda_{1}}\otimes\langle\mu_{1},v_{1}\rangle^{\lambda_{2}}\). **4. Triangular Norms in n-ScIFN and n-ScIFN Algebra**  
Any set _n-ScIFN_ can be ordered by the relation\(\leq_{n}\) defined as follows:  
**Definition 7**. : _For any \(\langle\mu_{1},v_{1}\rangle,\langle\mu_{2},v_{2}\rangle\in\textbf{n-ScIFN}\):_  
\[\langle\mu_{1},v_{1}\rangle\leq_{n}\langle\mu_{2},v_{2}\rangle\text{ iff }\mu_{1}\leq\mu_{2},v_{1}\geq v_{2}. \tag{6}\]  
**Fact 3**. : _For any \(x,y\in[0,1]\),_  
1. \(\langle 0,1\rangle\leq_{n}\langle 0,x\rangle\leq_{n}\langle 0,0\rangle \leq_{n}\langle y,0\rangle\leq_{n}\langle 1,0\rangle\)_,_
2. \(\langle 0,1\rangle\leq_{n}\langle x,y\rangle\leq_{n}\langle 1,0\rangle\)_, when_ \(\langle x,y\rangle\in\textbf{n-ScIFN}\)_._
3. \(\langle 0,1\rangle=\min_{n}\textbf{n-ScIFN}\)_,_ \(\langle 1,0\rangle=\max_{n}\textbf{n-ScIFN}\)_._  
Analogous properties of the order relation \(\leq_{n}\) to the relation \(\leq\) on real numbers enable the generalization of triangular norms into the set _n-ScIFN_. **Definition 8**. :
1. _The operation_ \(\circ_{t}:\textbf{n-ScIFN}\times\textbf{n-ScIFN}\) \(\rightarrow\textbf{n-ScIFN}\) _is called the_ _t-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)_ \(\langle 0,1\rangle\circ_{t}\langle\mu_{1},v_{1}\rangle=\langle 0,1\rangle, \langle\mu_{1},v_{1}\rangle\circ_{t}\langle 1,0\rangle=\langle\mu_{1},v_{1}\rangle\)__--boundary conditions, _(b)_ \(\langle\mu_{1},v_{1}\rangle\circ_{t}\langle\mu_{2},v_{2}\rangle\leq_{n}\langle \mu_{3},v_{3}\rangle\circ_{t}\langle\mu_{2},v_{2}\rangle\)_, when_ \(\langle\mu_{1},v_{1}\rangle\leq_{n}\langle\mu_{3},v_{3}\rangle\)__--monotonicity, _(c)_ \(\langle\mu_{1},v_{1}\rangle\circ_{t}\langle\mu_{2},v_{2}\rangle=\langle\mu_{2}, v_{2}\rangle\circ_{t}\langle\mu_{1},v_{1}\rangle\)__--_commutativity, _(d)_ \(\langle\mu_{1},v_{1}\rangle\circ_{t}\langle(\langle\mu_{2},v_{2}\rangle)\circ_{t }\langle\mu_{3},v_{3}\rangle\rangle=(\langle\mu_{1},v_{1}\rangle\circ_{t} \langle\mu_{2},v_{2}\rangle)\circ_{t}\langle\mu_{3},v_{3}\rangle\)__--associativity. 2.