# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 5 Relations \"Greater Than\" and \"Equal to\" for Ordered Fuzzy Numbers  
In the last section, we explicitly define the preorder \"greater than or equal to\" \(\widetilde{GE}\) on the space \(\mathbb{K}\) of all OFNs. This relation may be applied as start point for determining other basic relations on \(\mathbb{K}\). Let us consider any pair \((\overset{\leftrightarrow}{\mathcal{K}},\overset{\leftrightarrow}{\mathcal{L }})\in\mathbb{K}^{2}\). On the space \(\mathbb{K}\) we introduce the relation \(\overset{\leftrightarrow}{\mathcal{K}}.\widetilde{GT}.\overset{\leftrightarrow}{\mathcal{L }}\), which reads:  
\[\text{``{\it OFN}}\overset{\leftrightarrow}{\mathcal{K}}\text{ is greater than OFN}\overset{ \leftrightarrow}{\mathcal{L}}.\text{''} \tag{98}\]This relation is a fuzzy strict order \(\widetilde{GT}\in\mathcal{F}(\mathbb{K}^{2})\) defined by its membership function \(v_{GT}\in\left[0,1\right]^{\mathbb{K}^{2}}\). From the point view of the multivalued logic, the value \(v_{GT}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\) is considered as a truth-value of the sentence (98) which is equivalent to the sentence:  
\[\text{``}\text{\emph{OFN}}\stackrel{{\leftrightarrow}}{{ \mathcal{L}}}\text{is not greater than or equal to OFN}\stackrel{{ \leftrightarrow}}{{\mathcal{K}}}\text{.''} \tag{99}\]  
It means that we have  
\[v_{GT}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})= \ell\boldsymbol{\sigma}(\text{``}\overleftrightarrow{\mathcal{K}}. \widetilde{GT}.\overleftrightarrow{\mathcal{L}}^{\prime\prime})=\ell \boldsymbol{\sigma}(\text{``}\overleftrightarrow{\mathcal{L}}. \widetilde{GE}.\overleftrightarrow{\mathcal{K}}^{\prime\prime})=1-\ell \boldsymbol{\sigma}(\text{``}\overleftrightarrow{\mathcal{L}}.\widetilde{GE}.\overleftrightarrow{\mathcal{K}}^{\prime\prime}). \tag{100}\]  
Therefore, the membership function \(v_{GT}\in\left[0,1\right]^{\mathbb{K}^{2}}\) is determined by the identity  
\[v_{GT}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})=1- v_{GE}(\overleftrightarrow{\mathcal{L}},\overleftrightarrow{\mathcal{K}}). \tag{101}\]  
Moreover, on the space \(\mathbb{K}\) we introduce the relation \(\overleftrightarrow{\mathcal{K}}.\widetilde{EQ}.\overleftrightarrow{\mathcal{L}}\), which reads:  
\[\text{``}\text{\emph{OFN}}\stackrel{{\leftrightarrow}}{{ \mathcal{K}}}\text{is equal to OFN}\stackrel{{\leftrightarrow}}{{ \mathcal{L}}}\text{.''} \tag{102}\]  
The relation \(\widetilde{EQ}\in\mathcal{F}(\mathbb{K}^{2})\) is fuzzy equivalence determined by membership function \(v_{EQ}\in\left[0,1\right]^{\mathbb{K}^{2}}\). From the point view of the multivalued logic, the value \(v_{EQ}(\overleftrightarrow{\mathcal{K}},\overleftrightarrow{\mathcal{L}})\) is considered as a truth-value of the sentence (102) which is equivalent to the sentence:  
\[\text{``}\text{\emph{OFN}}\stackrel{{\leftrightarrow}}{{ \mathcal{K}}}\text{is greater than or equal to OFN}\stackrel{{ \leftrightarrow}}{{\mathcal{L}}}\text{and OFN}\stackrel{{ \leftrightarrow}}{{\mathcal{L}}}\text{is greater than or equal to OFN}\stackrel{{ \leftrightarrow}}{{\mathcal{K}}}\text{''} \tag{103}\]  
It means that we have  
\[ \begin{split} v_{EQ}(\overleftrightarrow{\mathcal{K}}, \overleftrightarrow{\mathcal{L}})&=\ell\boldsymbol{\sigma}( \text{``}\overleftrightarrow{\mathcal{K}}.\widetilde{EQ}.\overleftrightarrow{ \mathcal{L}}^{\prime\prime})=\ell\boldsymbol{\sigma}(\text{``} \overleftrightarrow{\mathcal{K}}.\widetilde{EQ}.\overleftrightarrow{ \mathcal{L}}^{\prime\prime})\text{``}\overleftrightarrow{\mathcal{L}}. \widetilde{GE}.\overleftrightarrow{\mathcal{K}}^{\prime\prime})=\\ &=\min\{\ell\boldsymbol{\sigma}(\text{``}\overleftrightarrow{ \mathcal{K}}.\widetilde{EQ}.\overleftarrow{\mathcal{L}}^{\prime\prime}), \ell\boldsymbol{\sigma}(\text{``}\overleftrightarrow{\mathcal{L}}. \widetilde{GE}.\overleftrightarrow{\mathcal{K}}^{\prime\prime})\}.\end{split}