# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 2 Imprecise Quantities--Considered Models
### Ordered Fuzzy Numbers vs. Fuzzy Numbers  
For the case \(a\geq d\) the membership function of OFN \(\overset{\leftrightarrow}{\mathcal{L}}(a,b,c,d,S_{L},E_{L})\) is equal to the membership function of FN \(\mathcal{L}(a,b,c,d,S_{L},E_{L})\). This fact implies the existence of isomorphism \(\Psi:(\mathbb{K}^{+}\cup\mathbb{R})\rightarrow\mathbb{F}\) given by the identity  
\[\mathcal{L}(a,b,c,d,S_{L},E_{L})=\Psi\bigg{(}\overset{\leftrightarrow}{ \mathcal{L}}(a,b,c,d,S_{L},E_{L})\bigg{)}. \tag{56}\]  
This isomorphism may be extended to the space \(\mathbb{K}\) by disorientation map \(\overset{\equiv}{\Psi}:\mathbb{K}\rightarrow\mathbb{F}\) given as follows  
\[\overset{\equiv}{\Psi}(\overset{\leftrightarrow}{\mathcal{L}})=\left\{ \begin{array}{ll}\Psi(\overset{\leftrightarrow}{\mathcal{L}})\overset{ \leftrightarrow}{\mathcal{L}}\in\mathbb{K}^{+}\cup\mathbb{R},\\ \underset{\leftrightarrow}{\Theta\Psi}(\overset{\leftrightarrow}{\Theta \mathcal{L}})\overset{\leftrightarrow}{\mathcal{L}}\in\mathbb{K}^{-}.\end{array} \right. \tag{57}\]Let us note, that the disorientation map \(\overset{\overline{\
abla}}{\mathbf{\Psi}}:\mathbb{K}\to\mathbb{F}\) may be equivalently defined by the identity  
\[\overset{\overline{\
abla}}{\mathbf{\Psi}}\!\left(\overset{\leftrightarrow}{ \mathcal{L}}\!\left(a,b,c,d,S_{L},E_{L}\right)\right)=\left\{ \begin{array}{ ll}\mathcal{L}\!\left(a,b,c,d,S_{L},E_{L}\right)\overset{\leftrightarrow}{ \mathcal{L}}\!\left(a,b,c,d,S_{L},E_{L}\right)\in\mathbb{K}^{+}\cup\mathbb{R},\\ \mathcal{L}\!\left(d,c,b,a,E_{L},S_{L}\right)\overset{\leftrightarrow}{ \mathcal{L}}\!\left(a,b,c,d,S_{L},E_{L}\right)\in\mathbb{K}^{-}.\end{array} \right. \tag{58}\]  
**Example 1**. : _Let us consider the OFN \(\overset{\leftrightarrow}{\mathcal{X}}=\overset{\leftrightarrow}{\mathcal{L} }\!\left(12,14,18,20,S_{X},E_{X}\right)\), where_  
\[\mu_{X}(x)= \begin{cases}0,&x\
otin[12,20],\\ S_{X}(x),&x\in[12,14],\\ 1,&x\in[14,18],\\ E_{X}(x),&x\in[18,20],\end{cases}\begin{cases}0,&x\
otin[12,20],\\ \frac{2\cdot x-24}{x-10},&x\in[12,14],\\ 1,&x\in[14,18],\\ \frac{7\cdot x-140}{2\cdot x-50},&x\in[18,20].\end{cases}  \tag{59}\]  
_and the \(\mathrm{OFN}\overset{\leftrightarrow}{\mathcal{Y}}=\overset{\leftrightarrow}{ \mathcal{L}}\!\left(13,11,6,5,\mathrm{S}_{Y},\mathrm{E}_{Y}\right)\), where_  
\[\mu_{Y}(x)=\left\{ \begin{array}{ll}0,&x\
otin[13,5],\\ S_{Y}(x),&x\in[13,11],\\ 1,&x\in[11,6],\\ E_{Y}(x),&x\in[6,5],\end{array} \right\}=\left\{ \begin{array}{ll}0,&x\
otin[1 3,5],\\ \frac{6\cdot x-30}{x},&x\in[13,11],\\ 1,&x\in[11,6],\\ \frac{2\cdot x-26}{x-15},&x\in[6,5].\end{array} \right. \tag{60}\]  
_Since \(\overset{\leftrightarrow}{\mathcal{X}}\in\mathbb{K}^{+}\), using (57) we get_  
\[\overset{\leftarrow}{\mathbf{\Psi}}\!\left(\overset{\leftrightarrow}{ \mathcal{X}}\right)=\overset{\leftarrow}{\mathbf{\Psi}}\!\left(\overset{ \leftrightarrow}{\mathcal{L}}\!\left(12,14,18,20,S_{X},E_{X}\right)\right)= \mathcal{L}\!\left(12,14,18,20,L_{U},R_{U}\right)=\mathcal{U}, \tag{61}\]  
_where FN \(\mathcal{U}\) is explicitly determined by the following membership function_  
\[\mu_{U}(x)=\left\{ \begin{array}{ll}0,&x\
otin[12,20],\\ L_{U}(x),&x\in[12,14],\\ 1,&x\in[14,18],\\ R_{U}(x),&x\in[18,20],\end{array} \right\}= \begin{cases}0,&x\
otin[12,20],\\ \frac{2\cdot x-24}{x-10},&x\in[12,14],\\ 1,&x\in[14,18],\\ \frac{7\cdot x-140}{2\cdot x-50},&x\in[18,20].\end{cases}  \tag{62}\]  
_Because \(\overset{\leftrightarrow}{\mathcal{Y}}\in\mathbb{K}^{-}\), using (57) we get_  
\[\overset{\leftarrow}{\mathbf{\Psi}}\!\left(\overset{\leftrightarrow}{ \mathcal{Y}}\right)=\overset{\leftarrow}{\mathbf{\Psi}}\!\left(\overset{ \leftrightarrow}{\mathcal{L}}\!\left(13,11,6,5,S_{Y},E_{Y}\right)\right)= \mathcal{L}\!\left(5,6,11,13,L_{Y},R_{V}\right)=\mathcal{V}, \tag{63}\]  
_where FN \(\mathcal{V}\) is described by the membership function_  
\[\mu_{V}(x)=\left\{ \begin{array}{ll}0,&x\
otin[5,13],\\ L_{V}(x),&x\in[5,6],\\ 1,&x\in[6,11],\\ R_{V}(x),&x\in[11,13],\end{array} \right\}=\left\{