## 2 The Norm Systems
### The Repeatability Norms  
For any generators of triangular norms \(f_{t},f_{s}\), and any number \(x\in[0,1]\), assuming notation \(x=x_{1}=x_{2}=...=x_{k}\):  
\(y=x_{1}\bullet_{t}x_{2}\bullet_{t}...\bullet_{t}x_{k}=f_{t}^{-1}[kft_{t}(x)]\) for \(kft_{t}(x)\in[0,f_{t}(0)]\),  
\(z=x_{1}\bullet_{s}x_{2}\bullet_{s}...\bullet_{s}x_{k}=f_{s}^{-1}[kft_{s}(x)]\) for \(kft_{s}(x)\in[0,f_{s}(1)]\). Thus, the \(k\)-times execution of the triangular norm for the number \(x\) is a number which generator value is \(k\)-times the generator value for the number \(x:f_{t}(x)=(1/k)f_{t}(y)\) and \(f_{s}(x)=(1/k)f_{s}(z)\). Hence \(if_{t}(x)=(i/k)f_{t}(y)\) and \(if_{s}(x)=(i/k)f_{s}(z)\). Moreover, for \(i<k\):  
\(x_{1}\bullet_{t}x_{2}\bullet_{t}...\bullet_{t}x_{i}=f_{t}^{-1}[(i/k)f_{t}(y)]\),  
\(x_{1}\bullet_{s}x_{2}\bullet_{s}...\bullet_{s}x_{i}=f_{s}^{-1}[(i/k)f_{s}(y)]\). Since for any \(y\in[0,1]\) there exists a number \(x\in[0,1]\) satisfying the above formulas, we can assume the notation:  
\((i/k)\bullet_{p}y=_{df}f_{t}^{-1}[(i/k)f_{t}(y)]\),  
\((i/k)\bullet_{t}z=_{df}f_{s}^{-1}[(i/k)f_{s}(z)]\). If we assume that the procedures of repeating norms (calculating the values of the generators of these norms) for any \(x\) can be used only with a certain degree \(\lambda\in[0,1]\), it will be natural to adopt following Definitions of repeatability norms by applying these procedures. **Definition 2**. : _The operation \(\bullet_{p}:[0,1]\times[0,1]\to[0,1]\) will be called the power repeatability standardoft the \(t\)-norm \(\bullet_{t}:[0,1]\times[0,1]\to[0,1]\) (briefly: \(p\)-norm), if the following conditions are met, for any \(\lambda_{1},\lambda_{2},x,y\in[0,1]\):_  
* \(1\bullet_{p}x=x,0\bullet_{p}x=1\)_,_
* _if_ \(x\
eq 0\) _and_ \(\lambda_{1}\bullet_{p}x=\lambda_{2}\bullet_{p}x\)_, then_ \(\lambda_{1}=\lambda_{2}\)_,_
* _if_ \(x\
eq 0,y\
eq\lambda,\lambda_{2}\
eq 0\)_, and_ \(\lambda_{1}\bullet_{p}x=\lambda_{2}\bullet_{p}y\)_, then_ \((\lambda_{1}/\lambda_{2})\bullet_{p}x=y\)_,_
* \((\lambda_{1}*\lambda_{2})\bullet_{p}x=\lambda_{1}\bullet_{p}(\lambda_{2} \bullet_{p}x)\)_,_
* \((\lambda_{1}+\lambda_{2})\bullet_{p}x=(\lambda_{1}\bullet_{p}x)\bullet_{t}( \lambda_{2}\bullet_{p}x)\)_,_
* \(\lambda_{1}\bullet_{p}(x\bullet_{t}y)=(\lambda_{1}\bullet_{p}x)\bullet_{t}( \lambda_{1}\bullet_{p}y)\)_._  
_The operation \(\bullet_{l}:[0,1]\times[0,1]\to[0,1]\) will be called the linear repeatability standardoft the \(s\)-norm \(\bullet_{s}:[0,1]\times[0,1]\to[0,1]\) (briefly: \(l\)-norm), if the following conditions are met, for any \(\lambda_{1},\lambda_{2},x,y\in[0,1]\):_  
* \(1\bullet_{l}x=x,0\bullet_{l}x=0\)_,_
* _if_ \(x\
eq 0\) _and_ \(\lambda_{1}\bullet_{l}x=\lambda_{2}\bullet_{l}x\)_, then_ \(\lambda_{1}=\lambda_{2}\)_,_
* _if_ \(x\
eq 0,y\
eq 0,\lambda_{2}\
eq 0\)_, and_ \(\lambda_{1}\bullet_{l}x=\lambda_{2}\bullet_{l}y\)_, then_ \((\lambda_{1}/\lambda_{2})\bullet_{l}x=y\)_,_
* \((\lambda_{1}*\lambda_{2})\bullet_{l}x=\lambda_{1}\bullet_{l}(\lambda_{2} \bullet_{l}x)\)_,_
* \((\lambda_{1}+\lambda_{2})\bullet_{l}x=\lambda_{1}\bullet_{l}x\)_,_
* \(\lambda_{1}\bullet_{l}(x\bullet_{s}y)=(\lambda_{1}\bullet_{l}x)\bullet_{s}( \lambda_{1}\bullet_{l}y)\)_._  
**Definition 3**. : _The system \(\mathbf{S_{Vager}}=\langle[0,1],\bullet_{t},\bullet_{s},\bullet_{p}\bullet_{l},0,1\rangle\) is called the Yager system of norms (triangular and repeatability)._  
**Theorem 3**. : _Let \(f_{t}\) be a generator of the triangular norm \(\bullet_{t}\). Then there are operations \(\bullet_{p}:[0,1]\times[0,1]\to[0,1]\) defined by formula:_  
\[\lambda\bullet_{p}x= \begin{cases}f_{t}^{-1}[\lambda f_{t}(x)]\text{ for }\lambda f_{t}(x)\in[0,f_{t}(0)],\\ 0\text{, otherwise.}\end{cases}  \tag{4}\]  
_Let \(f_{s}\) be a generator of the triangular norm \(\bullet_{s}\). Then there are operations \(\bullet_{l}:[0,1]\times[0,1]\to[0,1]\) defined by formula:_  
\[\lambda\bullet_{l}x= \begin{cases}f_{s}^{-1}[\lambda f_{s}(x)]\text{ for }\lambda f_{s}(x)\in[0,f_{s}(1)],\\ 1\text{, otherwise.}\end{cases}  \tag{5}\]  
Proof.