# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 3 Relation \"Greater than or Equal to\" for Fuzzy Numbers  
We consider the pair \((\mathcal{K},\mathcal{L})\in\mathbb{F}^{2}\) of FNs determined by membership functions \(\mu_{K},\mu_{L}\in[0,1]^{\mathbb{R}}\). On the space \(\mathbb{F}\), we can consider the relation \(\mathcal{K}.GE.\mathcal{L}\), which reads:  
\[\text{``FN $\mathcal{K}$ is greater than or equal to $FN $\mathcal{L}$.''} \tag{66}\]  
[PERSON] [61] shows that in agreement with the Zadeh's Extension Principle, this relation is a fuzzy preorder \([GE]\in\mathcal{F}\big{(}\mathbb{F}^{2}\big{)}\) described by membership function \(\
u_{[GE]}\in[0,1]^{\mathbb{F}^{2}}\) determined as follows  
\[\
u_{[GE]}(\mathcal{K},\mathcal{L})=\sup\{\min\{\mu_{K}(x),\mu_{L}(y)\}:x \geq y\}. \tag{67}\]  
From the multivalued logic point of view, the value \(\
u_{[GE]}(\mathcal{K},\mathcal{L})\) is considered as a truth-value of the sentence (66). It means that we have  
\[\
u_{[GE]}(\mathcal{K},\mathcal{L})=\epsilon\boldsymbol{\
u}(\text{``} \mathcal{K}.GE.\mathcal{L}^{\prime}). \tag{68}\]  
We prove that the fuzzy preorder \([GE]\in\mathcal{F}\big{(}\mathbb{F}^{2}\big{)}\) fulfils the following well-known properties. **Theorem 2**. : _For any pair \((\mathcal{K},\mathcal{L})\in\mathbb{F}^{2}\), we have:_  
\[\
u_{[GE]}(\mathcal{K},\mathcal{L})=\
u_{[GE]}(\ominus\mathcal{L},\ominus \mathcal{K}), \tag{69}\]  
\[\
u_{[GE]}(\mathcal{K},\mathcal{L})=\
u_{[GE]}(\mathcal{K}\ominus\mathcal{L}, \llbracket 0\rrbracket). \tag{70}\]  
Proof. : Take into account the quadruple \((\mathcal{K},\mathcal{L},\mathcal{M},\mathcal{N})\in\mathbb{F}^{4}\) of FNs represented respectively by their membership functions \(\mu_{K},\mu_{L},\ \mu_{M},\mu_{N}\in[0,1]^{\mathbb{R}}\). Let us assume that \(\mathcal{M}=\ominus\mathcal{K}\) and \(\mathcal{N}=\ominus\mathcal{L}\). Using the identities (11), (12), and (13) we obtain:  
\[\mu_{M}(y)=\mu_{K}(-y)\text{ and }\mu_{N}(x)=\mu_{L}(-x).\]  
Then the identity (67) implies  
\[\
u_{[GE]}(\ominus\mathcal{L},\ominus\mathcal{K})=\
u_{[GE]}( \mathcal{N},\mathcal{M})=\sup\{\min\{\mu_{N}(x),\mu_{M}(y)\}:x\geq y\}=\] \[=\sup\{\min\{\mu_{L}(-x),\mu_{K}(-y)\}:-x\leq-y\}=\sup\{\min\{\mu_{ L}(u),\mu_{K}(v)\}:u\leq v\}=\
u_{[GE]}(\mathcal{K},\ \mathcal{L}).\]Let us assume now that \(\mathcal{M}=\mathcal{K}\ominus\mathcal{L}\). Using the identity (7) we obtain:  
\[\mu_{M}(z)=\sup\{\min[\mu_{K}(x),\mu_{L}(y)]:z=x-y,(x,y)\in\mathbb{R}\}.\]  
Then the identity (67) implies  
\[ \begin{array}{l}\
u_{|[GE]}(\mathcal{K}\ominus\mathcal{L},\llbracket 0 \rrbracket)=\
u_{|[GE]}(\mathcal{M},\llbracket 0\rrbracket)=\sup[\mu_{M}(z):z\geq 0]=\\ =\sup\{\sup\{\min[\mu_{K}(x),\mu_{L}(y)]:z=x-y,(x,y)\in\mathbb{R}\}:z\geq 0 \}=\\ =\sup[\min\{\mu_{K}(x),\mu_{L}(y)\}:x-y\geq 0]=\
u_{|[GE]}(\mathcal{K},\mathcal{L}). \text{QED}\end{array} \]  
**Theorem 3**. _For any FNs \(\mathcal{L}(a,b,c,d,L_{K},R_{K})\), \(\mathcal{L}(e,f,g,h,L_{M},R_{M})\ \in\mathbb{F}\) we have_  
\[\
u_{|[GE]}(\mathcal{L}(a,b,c,d,L_{K},R_{K}),\ \mathcal{L}(e,f,g,h,L_{M},R_{M}))= \left\{ \begin{array}{rl}0&0<d-e,\\ R_{W}(0)&d-e\leq 0<c-f,\\ 1&0\leq c-f,\end{array} \right. \tag{71}\]  
_where the function \(R_{W}:[d-e,c-f]\to[0,1]\) is given by identity (28)._  
**Proof. ** For \(e>d\), using (67) and (8) we get  
\[ \begin{array}{l}\
u_{|[GE]}(\mathcal{L}(a,b,c,d,L_{K},R_{K}),\ \mathcal{L}(e,f,g,h,L_{M},R_{M}))=\sup\{\min\{\mu_{K}(x),\mu_{M}(y)\}:x\geq y \}=\\ =\max\{\sup\{\min[0,\mu_{M}(x)]:x\geq y\geq e\},\sup\{\min[\mu_{K}(x),0]:x\geq y \in|e,-\infty[]\}=0.\end{array}  \tag{72}\]  
For \(c\geq f\) we have  
\[ \begin{array}{l}1\geq\
u_{|[GE]}(\mathcal{L}(a,b,c,d,L_{K},R_{K}),\ \mathcal{L}(e,f,g,h,L_{M},R_{M}))=\sup\{\min\{\mu_{K}(x),\mu_{M}(y)\}:x\geq y \}\geq\\ \geq\sup\{\min\{\mu_{K}(x),\mu_{M}(y)\}:c\geq x\geq y\geq f\}=\sup\{\min\{1,1 \}]=1.\end{array}  \tag{73}\]  
For \(d\leq e\) and \(f<c\) we have \(d-e\leq 0<c-f\). Then from (24), (67) and (70) we obtain  
\[