# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 2 Imprecise Quantities--Considered Models
### Ordered Fuzzy Numbers--Some Basic Facts
For any \(z\in[b,c]\), the positively oriented OFN \(\stackrel{{\leftrightarrow}}{{\mathcal{L}}}(a,b,c,d,S_{L},E_{L})\) is a formal model of linguistic variable \"about or slightly above \(z\)\". The symbol \(\mathbb{K}^{+}\) denotes the space of all positively oriented OFN. If \(a>d\), then OFN \(\stackrel{{\leftrightarrow}}{{\mathcal{L}}}(a,b,c,d,S_{L},E_{L})\) has the negative orientation \(\stackrel{{\rightarrow}}{{a,d}}\). For any \(z\in[c,b]\), the negatively oriented TrOFN \(\stackrel{{\leftrightarrow}}{{\mathcal{L}}}(a,b,c,d,S_{L},E_{L})\) is a formal model of linguistic variable \"about or slightly below \(z\)\". The symbol \(\mathbb{K}^{-}\) denotes the space of all negatively oriented OFN. Understanding the phrases \"about or slightly above\(z\)\" and \"about or slightly below \(z\)\" depend on the applied pragmatics of the natural language. If \(a=d\), OFN \(\overleftrightarrow{\mathcal{L}}(a,a,a,a,S_{L},E_{L})=\llbracket a\rrbracket\) describes unoriented number \(a\in\mathbb{R}\). Summing up, we see that  
\[\mathbb{K}=\mathbb{K}^{+}\cup\mathbb{R}\cup\mathbb{K}^{-}. \tag{35}\]  
The minus operator \"\(-\)\" on \(\mathbb{R}\) is extended by [PERSON] [4] to the minus operator \(\boxplus\) on \(\mathbb{K}\) by means of the identity  
\[\boxplus\overleftarrow{\mathcal{L}}(a,b,c,d,S_{L},E_{L})=\overleftarrow{ \mathcal{L}}\Bigl{(}-a,-b,-c,-d,S_{L}^{(-)},\ E_{L}^{(-)}\Bigr{)}, \tag{36}\]  
where  
\[S_{L}^{(-)}(x)=S_{L}(-x) \tag{37}\]  
\[E_{L}^{(-)}(x)=E_{L}(-x) \tag{38}\]  
[PERSON] [1] defines the addition operator \(\boxplus_{K}\) on \(\mathbb{K}\) as the extension of operator \(\boxplus\) from \(\mathbb{F}\) to \(\mathbb{K}\). This extension is determined by extension the domain identities (18)-(22) from \(\mathbb{F}\) to \(\mathbb{K}\). In this way, [PERSON] defines addition of OFNs as an extension of results obtained by [PERSON] and [PERSON] [5] for addition of FNs. Moreover, [PERSON] [4] have shown that there exist such OFNs that their sum \(\boxplus_{K}\) does not exist. For this reason, [PERSON]'s operator \(\boxplus_{K}\) is replaced by addition operator \(\boxplus\) defined on \(\mathbb{K}\) by the identity [6]  
\[\overleftrightarrow{\mathcal{L}}(a_{K},b_{K},c_{K},d_{K},S_{K},E_{K}) \boxplus\overleftarrow{\mathcal{L}}(a_{M},b_{M},c_{M},d_{M},S_{M},E_{M})= \overleftrightarrow{\mathcal{J}}=\overleftrightarrow{\mathcal{L}}\Bigl{(}a_ {J},b_{J},c_{J},d_{J},S_{J},E_{J}\Bigr{)}, \tag{39}\]  
where we have  
\[\tilde{a}_{J}=a_{K}+a_{M}, \tag{40}\]  
\[b_{J}=b_{K}+b_{M}, \tag{41}\]  
\[c_{J}=c_{K}+c_{M}, \tag{42}\]  
\[\tilde{d}_{J}=d_{K}+d_{M}, \tag{43}\]  
\[a_{J}= \begin{cases}\min\bigl{\{}\tilde{a}_{J},b_{J}\bigr{\}},&(b_{J}<c_{J}) \vee(b_{J}=c_{J}\wedge\ \tilde{a}_{J}\leq\tilde{d}_{J}),\\ \max\bigl{\{}\tilde{a}_{J},b_{J}\bigr{\}},&(b_{J}>c_{J})\vee(b_{J}=c_{J}\wedge \ \tilde{a}_{J}>\tilde{d}_{J}),\end{cases}  \tag{44}\]  
\[d_{J}= \begin{cases}\max\bigl{\{}\tilde{d}_{J},c_{J}\bigr{\}},&(b_{J}<c_{J}) \vee(b_{J}=c_{J}\wedge\ \tilde{a}_{J}\leq\tilde{d}_{J}),\\ \min\bigl{\{}\tilde{d}_{J},c_{J}\bigr{\}},&(b_{J}>c_{J})\vee(b_{J}=c_{J}\wedge \ \tilde{d}_{J}>\tilde{d}_{J}),\end{cases}  \tag{45}\]  
\[\forall_{\alpha\in[0;1]}\ s_{J}(\alpha)= \begin{cases}S_{K}^{\star}(\alpha)+S_{ M}^{\star}(\alpha),&a_{J}\
eq b_{J},\\ b_{J},&a_{J}=b_{J}.\end{cases}  \tag{46}\]  
\[\forall_{\alpha\in[0;1]}\ e_{J}(\alpha)= \begin{cases}E_{K}^{\star}(\alpha)+E_ {M}^{\star}(\alpha),&c_{J}\
eq d_{J},\\ c_{J},&c_{J}=d_{J}.\end{cases}  \tag{47}\]  
\[\forall_{x\in[a_{J},b_{J}]}\ S_{J}(x)=s_{J}^{\star}(x), \tag{48}\]  
\[\forall_{x\in[c_{J},d_{J}]}\ E_{J}(x)=e_{J}^{\star}(x). \tag{49}\]  
In [6], the definition of addition operator \(\boxplus\) is justified in detail.