## 4 Hasse Diagram for IVCI-Partial Order-FRs
\begin{array}{ll}\left((u,v),[0.07,0.12]e^{i2(0.15,0.25 )\pi},[0.56,0.66]e^{i2(0.63,0.74]\pi}\right),\\ \left((u,y),[0.07,0.12]e^{i2(0.24,0.29)\pi},[0.56,0.66]e^{i2(0.57,0.71]\pi} \right),\\ \left((u,z),[0.07,0.12]e^{i2(0.24,0.29)\pi},[0.56,0.66]e^{i2(0.57,0.71]\pi} \right),\\ \left((v,v),[0.63,0.71]e^{i2(0.03,0.15)\pi},[0.11,0.23]e^{i2(0.27,0.82)\pi} \right),\\ \left((v,z),[0.28,0.36]e^{i2(0.03,0.15)\pi},[0.48,0.61]e^{i2(0.27,0.82)\pi} \right),\\ \left((w,w),[0.37,0.46]e^{i2(0.15,0.25)\pi},[0.32,0.43]e^{i2(0.53,0.68)\pi} \right),\\ \left((w,y),[0.37,0.46]e^{i2(0.15,0.25)\pi},[0.32,0.43]e^{i2(0.53,0.68)\pi} \right),\\ \left((w,z),[0.28,0.36]e^{i2(0.15,0.25)\pi},[0.48,0.61]e^{i2(0.53,0.68)\pi} \right),\\ \left((x,u),[0.07,0.12]e^{i2(0.15,0.25)\pi},[0.56,0.66]e^{i2(0.63,0.74)\pi} \right),\\ \left((x,v),[0.26,0.34]e^{i2(0.03,0.15)\pi},[0.38,0.49]e^{i2(0.73,0.82)\pi} \right),\\ \left((x,w),[0.26,0.34]e^{i2(0.15,0.25)\pi},[0.38,0.49]e^{i2(0.63,0.74)\pi} \right),\\ \left((x,x),[0.26,0.34]e^{i2(0.15,0.25)\pi},[0.38,0.49]e^{i2(0.63,0.74)\pi} \right),\\ \left((x,y),[0.26,0.34]e^{i2(0.15,0.25)\pi},[0.38,0.49]e^{i2(0.63,0.74)\pi} \right),\\ \left((x,z),[0.26,0.34]e^{i2(0.15,0.25)\pi},[0.38,0.49]e^{i2(0.63,0.74)\pi} \right),\\ \left((y,y),[0.52,0.58]e^{i2(0.30,0.42)\pi},[0.18,0.37]e^{i2(0.06,0.13)\pi} \right),\\ \left((y,z),[0.26,0.34]e^{i2(0.43,0.53)\pi},[0.48,0.61]e^{i2(0.63,0.74)\pi} \right),\\ \left((z,z),[0.28,0.36]e^{i2(0.43,0.53)\pi},[0.48,0.61]e^{i2(0.32,0.42)\pi} \right)\end{array} \right.\]  
Figure 2 depicts the Hasse diagram of the IVCI-partial order-FR \(\bar{\mathbb{R}}\). **Definition 17**. : _Suppose that an IVCI-partial order-FR is represented by a Hasse diagram, then an element is known as:_  
1. _The maximal element if it appears at the top of the diagram._
2. _The minimal if it appears at the bottom of the diagram._
3. _The maximum or greatest element if every element related to it is smaller than it._
4. _The minimum or least element if every element related to it is greater than it._  
Figure 2: Hasse diagram for an IVCI-partial order-FR \(\bar{\mathbb{R}}\)  
**Definition 18**. : _Suppose \(\hat{\mathrm{F}}\) be the subset of an IVCI-partial order-FS \(\hat{\mathrm{F}}\), then an element \(x\in\hat{\mathrm{R}}\subseteq\hat{\mathrm{F}}\times\hat{\mathrm{F}}\) is known as the:_  
1. _Upper bound of_ \(\hat{\mathrm{F}}\) _if_ \((v,u)\in\hat{\mathrm{R}}\)_,_ \(\forall v\in\hat{\mathrm{F}}\)_._
2. _Lower bound of_ \(\hat{\mathrm{F}}\) _if_ \((u,v)\in\hat{\mathrm{R}}\)_,_ \(\forall v\in\hat{\mathrm{F}}\)_._
3. _Supremum of_ \(\hat{\mathrm{F}}\) _if it is the least upper bound of_ \(\hat{\mathrm{F}}\)_._
4. _Infimum of_ \(\hat{\mathrm{F}}\) _if it is the greatest upper bound of_ \(\hat{\mathrm{F}}\)_._  
**Example 15**. : _Assume that \(\hat{\mathrm{F}}=\{f,g,h,k,l,o,p,q,r,s,t,u\}\) is a complex intuitionistic partial order fuzzy set \(\hat{\mathrm{F}}\). For ease, we close your eyes to the degree of membership and degree of non-membership. Figure 3 is the illustration of the Hasse diagram of set \(\hat{\mathrm{F}}\)._  
In the above diagram,  
1. Maximal element is \(k\). 2. Minimal element is \(t\). 3. Maximum elements are \(k\) and \(p\). 4. Minimum elements are \(q\) and \(t\). Now considering a subset \(\hat{\mathrm{F}}=\{g,h,o,u\}\) of set \(\hat{\mathrm{F}}\) whose elements are enclosed in the green oval. 5. Upper bounds of \(\hat{\mathrm{F}}\) are \(h\) and \(k\). 6. Lower bounds of \(\hat{\mathrm{F}}\) are \(u\) and \(t\). 7. Supremum of \(\hat{\mathrm{F}}\) is \(h\). 8. Infimum of \(\hat{\mathrm{F}}\) is \(u\).