## 2 The Norm Systems
### The Repeatability Norms
: Since \(f_{t}\) is a strictly descending function, there is only one value of \(f_{t}^{-1}[\lambda f_{t}(x)]\in[0,1]\), when \(\lambda f_{t}(x)\in[0,f_{t}(0)]\). Let us denote that \(\lambda\bullet_{p}x=_{df}f_{t}^{-1}[\lambda f_{t}(x)]\), for \(\lambda f_{t}(x)\in[0,f_{t}(0)]\), and \(\lambda\bullet_{p}x=_{df}0\), otherwise. We define the operation \(\bullet_{l}\) in the same way. The following notation agreement is assumed here:  
* _If_ \(x^{\lambda}=f_{t}^{-1}[\lambda f_{t}(x)]\)_, for_ \(\lambda f_{t}(x)\in[0,f_{t}(0)]\)_, then_ \(f_{t}(x^{\lambda})=\lambda f_{t}(x)\)_._ If_ \(\lambda x=f_{s}^{-1}[\lambda f_{s}(x)]\)_, for_ \(\lambda f_{s}(x)\in[0,f_{s}(1)]\)_, then_ \(f_{s}(\lambda x)=\lambda f_{s}(x)\)_._  
**Theorem 4**. : _The repetition operations \(\bullet_{p}\bullet_{l}\) for Theorem 3 satisfy the following conditions, for any \(x,y,\lambda,\lambda_{1},\lambda_{2}\in[0,1]\):_  
* \(\lambda(x\bullet_{s}y)=\lambda x\bullet_{s}\lambda y\)_,_
* \((x\bullet_{l}y)^{\lambda}=x^{\lambda}\bullet_{l}y^{\lambda}\)_,_
* \((\lambda_{1}+\lambda_{2})x=\lambda_{1}x\bullet_{s}\lambda_{2}x\)_,_
* \(x^{\lambda_{1}+\lambda_{2}}=x_{1}^{\lambda}\bullet_{l}x_{2}^{\lambda}\)_._  
_Thus, they are \(p\)-norm and \(l\)-norm._  
**Proof. **  
1. \(\lambda(x\bullet_{s}y)=f_{s}^{-1}[\lambda f_{s}(x\bullet_{s}y)]=f_{s}^{-1}[\lambda f _{s}(x)+\lambda f_{s}(y)]=f_{s}^{-1}[f_{s}(\lambda x)+f_{s}(\lambda y)]=\lambda x \bullet_{s}\lambda y\), for \(f_{s}(x\bullet_{s}y)\in[0,1]\), and because \(\lambda f_{s}(x\bullet_{s}y)\leq f_{s}(x\bullet_{s}y)\), so \(\lambda f_{s}(x\bullet_{s}y)\in[0,1]\). 2. \((x\bullet_{t}y)^{\lambda}=f_{t}^{-1}[\lambda f_{t}(x\bullet_{t}y)]=f_{t}^{-1}[ \lambda f_{t}(x)+\lambda f_{t}(y)]=f_{t}^{-1}[f_{t}(x^{\lambda})+f_{t}(y^{ \lambda})]=x^{\lambda}\bullet_{t}y^{\lambda}\), for \(f_{t}(x\bullet_{t}y)\in[0,1]\), and because \(\lambda f_{t}(x\bullet_{t}y)\leq f_{t}(x\bullet_{t}y)\), so \(\lambda f_{t}(x\bullet_{t}y)\in[0,1]\). 3. \((\lambda_{1}+\lambda_{2})x=f_{s}^{-1}[(\lambda_{1}+\lambda_{2})f_{s}(x)]=f_{s} ^{-1}[\lambda_{1}f_{s}(x)+\lambda_{2}f_{s}(x)]=f_{s}^{-1}[f_{s}(\lambda_{1}x)+ f_{s}(\lambda_{2}x)]\) \(=\lambda_{1}x\bullet_{s}\lambda_{2}x\), for \((\lambda_{1}+\lambda_{2})f_{s}(x)=f_{s}(\lambda_{1}x)+f_{s}(\lambda_{2}x)\in[0,1]\). 4. \(x^{\lambda_{1}+\lambda_{2}}=f_{t}^{-1}[(\lambda_{1}+\lambda_{2})f_{t}(x)]=f_{t }^{-1}[\lambda_{1}f_{t}(x)+\lambda_{2}f_{t}(x)]=f_{t}^{-1}[f_{t}(x^{\lambda_{1 }})+f_{t}(x^{\lambda_{2}})]=x^{\lambda_{1}}\bullet_{t}x^{\lambda_{2}}\), for \((\lambda_{1}+\lambda_{2})f_{t}(x)=f_{t}(x^{\lambda_{1}})+f_{t}(x^{\lambda_{2}}) \in[0,1]\). **3. The n-Scaling Intuitionistic Fuzzy Sets and Yager Aggregation Operators**  
In this section, a mathematical model of information granulation is presented. It is about the set \(X\) of the information instances obtained by measurements using a certain sequence of scaling functions \(\{Sc_{n}\}\). Definition 4: The sequence \(\{Sc_{n}\}\) of the function \(Sc_{n}:[0,1]\to[0,1]\) will be called the scale function sequence of the intuitionistic fuzzy sets when the following conditions are met:  
1. \(Sc_{1}\) is a function of identity, \(Sc_{n}\) is a continuous and strictly increasing function in the interval \([0,1]\), for \(n>1\),
2. \(Sc_{n}(0)=0,Sc_{n}(1)=1\), for any \(n\),
3. \(Sc_{n+1}(x)\leq Sc_{n}(x)\), for any \(n\) and \(x\in[0,1]\),
4. For any numbers \(u.v\in[0,1]\), there exists such function \(Sc_{k}\), that \(0\leq Sc_{k}(u)+Sc_{k}(v)\leq 1\). Example 3: The map scale. A sequence of map functions \(Sc_{n}(x)=x/n\), for \(n>0,x\in[0,1]\) is a sequence of scaling functions, since:  
1. \(Sc_{1}(x)=x\) and functions \(Sc_{n}\) are strictly growing,
2. \(Sc_{n}(0)=0,Sc_{n}(1)=1\), for any \(n\),
3. \(x/(n+1)\leq x/n\) iff \(Sc_{n+1}(x)\leq Sc_{n}(x)\), for any \(n\) and \(x\in[0,1]\),
4. For \(k=2,u,v\in[0,1]\): \(0\leq u/2+v/2\leq 1\),
5. \(Sc_{1}^{-1}(x)=x\) and for \(n>1\): \(Sc_{n}^{-1}(x)=nx\). Example 4: The logarithmic scale.