## 2 Introduction to Multidimensional Type-1 Fuzzy Arithmetic (MT1 EFA)
If T1 FNs are not trapezoidal, appropriate formulas for \(x_{A_{1}}(\mu,\alpha_{xA_{1}})\) and \(x_{A_{2}}(\mu,\alpha_{xA_{2}})\) should be used._  
If the support of \(A_{2}\) contains zero, then the operation of division is of multi-granular character. This case has been described in [13]. Note that single fuzzy numbers \(A_{1}(\mu,\alpha_{xA_{1}})\) and \(A_{2}(\mu,\alpha_{xA_{2}})\) are mathematical objects in 3D space, see Figure 4. However, their result \(Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})=A_{1}(\mu,\alpha_{xA_{1}})\odot A_{2} (\mu,\alpha_{xA_{2}})\) of basic arithmetic operation \(\odot\in\{+,-,\cdot,/\}\) is a mathematical object in 4D space and cannot be visualized. Examples of arithmetic operations for T1 FNs and a more detailed explanation of them can be found in articles [11; 12; 13; 14; 17]. Now, consider two T1 TFNs \(A_{1}=(1,2,3,4)\) and \(A_{2}=(3,4,5,7)\), Figure 3. Horizontal MFs of these FNs are given by (8). \[A_{1}(\mu,\alpha_{xA_{1}}):x_{A_{1}}(\mu,\alpha_{xA_{1}})\in A_{1}(\mu, \alpha_{xA_{1}}):x_{A_{1}}(\mu,\alpha_{xA_{1}})=(1+\mu)+\alpha_{xA_{1}}(3-2\mu)\]  
\[A_{2}(\mu,\alpha_{xA_{2}}):x_{A_{2}}(\mu,\alpha_{xA_{2}})\in A_{2}(\mu, \alpha_{xA_{2}});x_{A_{2}}(  
HMF in Figure 3 models in 2D space the entire surface inside the HMF. The 2D space \(M\times X_{A_{1}}\) is not the full space of this function, but only a subspace, because this function exists in the 3D space \(M\times X_{A_{1}}\times A_{xA_{1}}\), as shown in Figure 4.  
The realization of the multiplication \(A_{1}\cdot A_{2}\) is given by the Formula (9). \[ \begin{array}{l}Z=A_{1}\cdot A_{2},Z:z\in Z\\ Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})=A_{1}(\mu,\alpha_{xA_{1}})\cdot A_{2}( \mu,\alpha_{xA_{2}})\\ Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}}):z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}}) \in Z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})\\ z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})=x_{A_{1}}(\mu,\alpha_{xA_{1}})\cdot x_ {A_{2}}(\mu,\alpha_{xA_{2}})\\ z(\mu,\alpha_{xA_{1}},\alpha_{xA_{2}})=[(1+\mu)+\alpha_{xA_{1}}(3-2\mu)]\cdot[ (3+\mu)+\alpha_{xA_{2}}(4-3\mu)]\\ \mu,\alpha_{xA_{1}},\alpha_{xA_{2}}\in[0,1]\end{array}  \tag{9}\]  
The algebraic and universal result of the multiplication \(Z\) being the set of quadruples \((\mu,\alpha_{xA_{1}},\alpha_{xA_{2}},z)\) exists in 4D space and cannot be seen. Instead, you can define its span \(SP_{Z}\) in the direction of the \(z\) axis and see it in 2D space \(Z\times M\). The span \(SP_{Z}\) of the multiplication result \(Z\) is given by the Formula (10) and Figure 5.  
\[ \begin{array}{l}SP_{Z}(\mu)=[\min_{\alpha_{xA_{1}},\alpha_{xA_{2}}}z(\mu, \alpha_{xA_{1}},\alpha_{xA_{2}}),\max_{\alpha_{xA_{1}}\alpha_{xA_{2}}}z(\mu, \alpha_{xA_{1}},\alpha_{xA_{2}})]\\ \mu,\alpha_{xA_{1}},\alpha_{xA_{2}}\in[0,1]\end{array}  \tag{10}\]  
The examination of the function (9) shows that its minimum border is obtained for \(\alpha_{xA_{1}}=\alpha_{xA_{2}}=0\) and the maximum border for \(\alpha_{xA_{1}}=\alpha_{xA_{2}}=1\). The mathematical properties of MF arithmetic are described and proven in [42]. The following properties hold in type 1 multidimensional epistemic fuzzy arithmetic:  
- addition and multiplication of MFNs are commutative and associative,  
- there exist neutral elements of addition and multiplication,  
- there exist an additive and multiplicative inverse elements,  
- the distributive law holds,  
- the cancellation laws for addition and multiplication hold,  
- the result of arithmetic operations is multidimensional,  
- in MFA, the result of arithmetic operations is universal, it does not depend on the mathematical form of the problem. **Denotation principle of epistemic fuzzy numbers**Each FN occurring in the problem is usually marked with a letter symbol, e.g., \(X\) or \(A\). One letter symbol, e.g., \(A\) can usually be used only to denote one fuzzy number, because this number is a record of our knowledge about only one true value (although unknown exactly) of the variable \(a\), e.g., an uncertain amount of water in the tank.