# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 2 Imprecise Quantities--Considered Models
### Ordered Fuzzy Numbers--Some Basic Facts  
The notion of OFN is intuitively introduced by [PERSON] [1; 2; 3; 4], as such model of imprecise number that subtraction of OFNs is the inverse operator to addition of OFNs. Therefore, OFNs can contribute to specific problems concerning the solution of fuzzy linear equations of the form or help with the interpretation of specific improper fuzzy arithmetic results. An important disadvantage of [PERSON]'s theory is that there exist such OFNs which are not linked to any membership function [4]. For this reason, the [PERSON]'s theory is revised in [6] where OFNs are defined as follows:  
**Definition 7**[6]. : _For any monotonic sequence \((a,b,c,d)\subset\mathbb{R}\), the ordered fuzzy number OFN \(\stackrel{{\leftrightarrow}}{{\mathcal{L}}}(a,b,c,d,S_{L},E_{L})= \stackrel{{\leftrightarrow}}{{\mathcal{L}}}\) is the pair of orientation \(\stackrel{{\rightarrow}}{{a,d}}=(a,d)\) and fuzzy set \({\mathcal{L}}\in{\mathcal{F}}(\mathbb{R})\) described by membership function \(\mu_{L}(\cdot|a,b,c,d,S_{L},E_{L})\in[0,1]^{\mathbb{R}}\) given by the identity_  
\[\mu_{L}(x|a,b,c,d,S_{L},E_{L})= \begin{cases}0,&x\
otin[a,d]\equiv[d,a],\\ S_{L}(x),&x\in[a,b]\equiv[b,a],\\ 1,&x\in[b,c]\equiv[c,b],\\ E_{L}(x),&x\in[c,d]\equiv[d,c].\end{cases}  \tag{33}\]  
_where the starting function \(S_{L}\in[0,1]^{[a,b]}\) and the ending function \(E_{L}\in[0,1]^{[c,d]}\) are upper semi-continuous monotonic ones meeting the conditions (6) and_  
\[S_{L}(b)=E_{L}(c)=1 \tag{34}\]  
The identity (33) additionally describes such modified notation of numerical intervals which is applied in this work. **Discussion about the terminology:** We see above that the notion of \"ordered fuzzy number\" is defined without applying any ordering relation between FNs. In original [PERSON]'s works \"ordered fuzzy number\" is also defined without use of any ordering relation between FN. In each of these cases, \"ordered fuzzy number\" is defined as FN completed by orientation. Therefore, in my opinion term \"ordered fuzzy number\" should be replaced by the term \"oriented fuzzy number\". The following premises support such a proposal for change:  
* Any discussion about the ordering of \"oriented fuzzy numbers\" is clearer than a discussion of ordering of \"ordered fuzzy numbers\". * Professor [PERSON]'s mother language is Polish. In Polish OFNs is called \"skierowana liczba rozmyta\". This term was proposed by Professor [PERSON]. Against, the quoted Polish term is translated into English as \"oriented fuzzy number\" or \"directed fuzzy number\". Moreover, the English term \"ordered fuzzy number\" is translated into Polish as \"uporzadkowana liczba rozmyta\". All this allows us to state that the meanings of the Polish term \"skierowana liczba rozmyta\" and the English term \"ordered fuzzy number\" are different. \"Ordered fuzzy numbers\" are the most important work of life for Professor [PERSON]. Therefore, the proposed change to the term OFN should be discussed with him. Because of Professor [PERSON] passed away, this is not possible. Therefore, I agree with other scientists [76, 77] that the OFN may be called the \"[PERSON]'s number\". Future scientific discussion will allow us to choose a \"oriented fuzzy number\" or \"Kosinski number\" or another term. Today we still use the term \"ordered fuzzy number\". No less in this work, the abbreviation OFN can be read \"ordered fuzzy number\" or \"oriented fuzzy numbers\". The use of the second term makes easier to read the section on the ordering relationship between OFNs. The symbol \(\mathbb{K}\) denotes the space of all OFNs. Any OFN describes an imprecise number with additional information about the location of the approximated number. This information is given as orientation of OFN. If \(a<d\) then OFN \(\stackrel{{\leftrightarrow}}{{\mathcal{L}}}(a,b,c,d,S_{L},E_{L})\) has the positive orientation \(\stackrel{{\rightarrow}}{{a,d}}\).