# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 2 Imprecise Quantities--Considered Models
### Fuzzy NumbersSome Basic Notions  
The most general definition of fuzzy number is given as follows:  
**Definition 1**[69]. : _The fuzzy number (FN) is such a fuzzy subset \(\mathcal{L}\in\mathcal{F}(\mathbb{R})\) with bounded support closure \(\left[\mathcal{L}\right]_{0^{+}}\) that it is represented by its upper semi-continuous membership function \(\mu_{L}\in\left[0;1\right]^{\mathbb{R}}\) satisfying the conditions:_  
\[\exists_{x\in\mathbb{R}}\ \ \mu_{L}(x)=1 \tag{4}\]  
\[\forall_{(x,y,z)\in\mathbb{R}^{3}}\ \ \ x\leq y\leq z\ \Longrightarrow\ \mu_{L}(y)\geq\min\{\mu_{L}(x);\mu_{L}(z)\}. \tag{5}\]  
The set of all FN we denote by the symbol \(\mathbb{F}\). Let us consider any arithmetic operation \(\ast\) defined on \(\mathbb{R}\). The symbol \(\otimes\) denotes an extension of arithmetic operation \(\ast\) to \(\mathbb{F}\). In [70], arithmetic operations on FN are introduced in such way that they are coherent with the [PERSON]'s Extension Principle. In line with it, for any pair \((\mathcal{K},\mathcal{L})\in\mathbb{F}^{2}\) represented by their membership functions \(\mu_{K},\mu_{L}\in\left[0,1\right]^{\mathbb{R}}\), the FN  
\[\mathcal{M}=\mathcal{K}\otimes\mathcal{L} \tag{6}\]  
is described by its membership function \(\mu_{M}\in\left[0,1\right]^{\mathbb{R}}\) determined by means of the identity:  
\[\mu_{M}(z)=\sup\{\min\{\mu_{K}(x),\mu_{L}(y)\}:\ z=x\ast y,(x,y)\in\mathbb{R}\}. \tag{7}\]  
Thanks to the results obtained in [5], we have that any FN can be equivalently defined as follows:  
**Theorem 1**[71]. : _For any FN \(\mathcal{L}\) there exists such a non-decreasing sequence \((a,b,c,d)\ \subset\mathbb{R}\) that \(\mathcal{L}(a,b,c,d,L_{L},R_{L})=\mathcal{L}\in\mathcal{F}(\mathbb{R})\) is determined by its membership function \(\mu_{L}(\cdot|a,b,c,d,L_{L},R_{L})\in\left[0,1\right]^{\mathbb{R}}\) described by the identity_  
\[\mu_{L}(x|a,b,c,d,L_{L},R_{L})= \begin{cases}0,&x\
otin[a,d],\\ L_{L}(x),&x\in[a,b],\\ 1,&x\in[b,c],\\ R_{L}(x),&x\in[c,d],\end{cases}  \tag{8}\]  
_where the left reference function \(L_{L}\in\left[0,1\right]^{\left[a,b\right]}\) and the right reference function \(R_{L}\in\left[0,1\right]^{\left[c,d\right]}\) are upper semi-continuous monotonic ones meeting the conditions:_  
\[L_{L}(b)=R_{L}(c)=1, \tag{9}\]  
\[\left[\mathcal{L}\right]_{0^{+}}=[a,d]. \tag{10}\]The FN \(\mathcal{L}(a,a,a,a,L_{L},R_{L})=[\![a]\! ]\) represents the real number \(a\in\mathbb{R}\). Therefore, we can say \(\mathbb{R}\subset\mathbb{F}\). For any \(z\in[b,c]\), a FN \(\mathcal{L}(a,b,c,d,L_{L},R_{L})\) is a formal model of linguistic variable \"about \(z^{\prime}\)\". Understanding the phrase \"about \(z^{\prime}\)\" depends on the applied pragmatics of the natural language. Let us note that FN may be replaced by generalized FN [72] which does not meet the condition (4). In line with the identity (7), the unary minus operator \"\(-\)\" on \(\mathbb{R}\) is extended to the minus operator \(\odot\) on \(\mathbb{F}\) by the identity  
\[\odot\mathcal{L}(a,b,c,d,L_{L},R_{L})=\mathcal{L}\!\left(\!-d,-  
The difference \(\ominus\) between FNs is determined in determined in the following way  
\[\mathcal{L}(a,b,c,d,L_{K},R_{K})\ominus\mathcal{L}(e,f,g,h,L_{M},R_{M})=\mathcal{ L}(a,b,c,d,L_{K},R_{K})\oplus(\ominus\mathcal{L}(e,f,g,h,L_{M},R_{M})). \tag{23}\]  
Then identities (11)-(13) and (18)-(23) imply that  
\[\mathcal{L}(a-h,b-g,c-f,d-e,L_{W},R_{W})=\mathcal{L}(a,b,c,d,L_{K},R_{K}) \ominus\mathcal{L}(e,f,g,h,L_{M},R_{M}), \tag{24}\]  
where  
\[\forall_{\alpha\in[0,1]}\;l_{W}(\alpha)=L_{K}^{\star}(\alpha)-R_{M}^{\star}(\alpha) \tag{25}\]  
\[\forall_{\alpha\in[0,1]}\;r_{W}(\alpha)=R_{K}^{\star}(\alpha)-L_{M}^{\star}(\alpha) \tag{26}\]  
\[\forall_{x\in[a-h,b-g]}\;L_{W}(x)=l_{W}^{\sharp}(x), \tag{27}\]  
\[\forall_{x\in[c-f,d-e]}\;R_{W}(x)=r_{W}^{\sharp}(x). \tag{28}\]  
The above arithmetic operators may be generalized to the case of intuitionistic FNs [73]. On the other hand, the dependencies (18)-(28) are not met for discrete FNs [74].