## 3 The P2 TLWBM Operator and P2 TLWGBM Operator
### P2 TLWBM Operator
\begin{array}{c}\mathop{\mathrm{P2 TLWBM}}\
olimits_{\omega}^{t,r}(p_{1},p_{2 },\cdots,p_{n})\\ =\Bigg{(}\Delta\Bigg{(}\Bigg{(}\mathop{\sum}\limits_{i,j=1}^{n}\omega_{i} \omega_{j}\big{(}\Delta^{-1}(S_{i},\rho_{i})\big{)}^{t}\big{(}\Delta^{-1}(S_{ j},\rho_{j})\big{)}^{r}\Bigg{)}^{1/(t+r)}\Bigg{)},\\ \Bigg{(}\Bigg{(}\sqrt{1-\mathop{\prod}\limits_{i,j=1}^{n}\Big{(}1-\mu_{i}^{2t} \mu_{j}^{2r}\Big{)}^{\omega_{i}\omega_{j}}\Big{)}^{1/(t+r)}},\\ \sqrt{1-\bigg{(}1-\mathop{\prod}\limits_{i,j=1}^{n}\Big{(}1-\big{(}1-v_{i}^{2} \big{)}^{t}\big{(}1-v_{j}^{2}\big{)}^{r}\Big{)}^{\omega_{i}\omega_{j}}\bigg{)} ^{1/(t+r)}}\Bigg{)}\Bigg{)}\end{array}  \tag{11}\]  
Hence, (5) is maintained. Thereafter:  
\[\omega_{i}\omega_{j}\Big{(}\Delta^{-1}(S_{i},\rho_{i})\Big{)}^{t}\Big{(} \Delta^{-1}\big{(}S_{j},\rho_{j}\big{)}\Big{)}^{r}\leq\omega_{i}\omega_{j} \Big{(}\Delta^{-1}(S_{\max},\rho_{\max})\Big{)}^{t+r}, \tag{12}\]  
\[ \begin{array}{c}\mathop{\sum}\limits_{i,j=1}^{n}\omega_{i}\omega_{j}\big{(} \Delta^{-1}(S_{i},\rho_{i})\big{)}^{t}\big{(}\Delta^{-1}\big{(}S_{j},\rho_{j} \big{)}\big{)}^{r}\leq\\ \mathop{\sum}\limits_{i,j=1}^{n}\omega_{i}\omega_{j}\big{(}\Delta^{-1}(S_{ \max},\rho_{\max})\big{)}^{t+r}=\big{(}\Delta^{-1}(S_{\max},\rho_{\max}) \big{)}^{t+r}\end{array} . \tag{13}\]  
\[\Delta\Bigg{(}\Bigg{(}\mathop{\sum}\limits_{i,j=1}^{n}\omega_{i}\omega_{j} \Big{(}\Delta^{-1}(S_{i},\rho_{i})\Big{)}^{t}\Big{(}\Delta^{-1}\big{(}S_{j},\rho_ {j}\big{)}\Big{)}^{r}\Bigg{)}^{1/(t+r)}\Bigg{)}\leq(S_{\max},\rho_{\max}). \tag{14}\]  
Similarly,  
\[(S_{\min},\rho_{\min})\leq\Delta\Bigg{(}\Bigg{(}\mathop{\sum}\limits_{i,j=1}^{ n}\omega_{i}\omega_{j}\Big{(}\Delta^{-1}(S_{i},\rho_{i})\Big{)}^{t}\Big{(}\Delta^{-1} \big{(}S_{j},\rho_{j}\big{)}\Big{)}^{r}\Bigg{)}^{1/(t+r)}\Bigg{)}. \tag{15}\]Thereafter,  
\[0\leq\left(\sqrt{1-\prod\limits_{i,j=1}^{n}\left(1-\mu_{i}^{2t}\mu_{j}^{2r} \right)^{\omega_{i}\omega_{j}}}\right)^{1/(t+r)}\leq 1, \tag{16}\]  
\[0\leq\sqrt{1-\left(1-\prod\limits_{i,j=1}^{n}\left(1-\left(1-v_{i}^{2}\right) ^{t}\left(1-v_{j}^{2}\right)^{r}\right)^{\omega_{i}\omega_{j}}}\right)^{1/(t+r )}}\leq 1. \tag{17}\]  
Because \(\mu_{i}^{2}+v_{i}^{2}\leq 1\),  
\[\left(\sqrt{1-\prod\limits_{i,j=1}^{n}\left(1-\mu_{i}^{2t}\mu_{j}^{2r}\right) ^{\omega_{i}\omega_{j}}}\right)^{1/(t+r)}\leq\sqrt{\left(1-\prod\limits_{i,j=1 }^{n}\left(1-\left(1-v_{i}^{2}\right)^{t}\left(1-v_{j}^{2}\right)^{r}\right)^ {\omega_{i}\omega_{j}}}\right)^{1/(t+r)}}. \tag{18}\]  
Therefore,  
\[\left(\left(\sqrt{1-\prod\limits_{i,j=1}^{n}\left(1-\mu_{i}^{2t} \mu_{j}^{2r}\right)^{\omega_{i}\omega_{j}}}\right)^{1/(t+r)}\right)^{2}+ \tag{19}\] \[\left(\sqrt{1-\left(1-\prod\limits_{i,j=1}^{n}\left(1-\left(1-v_ {i}^{2}\right)^{t}\left(1-v_{j}^{2}\right)^{r}\right)^{\omega_{i}\omega_{j}}} \right)^{1/(t+r)}\right)^{2}\] \[\leq\left(1-\prod\limits_{i,j=1}^{n}\left(1-\left(1-v_{i}^{2} \right)^{t}\left(1-v_{j}^{2}\right)^{r}\right)^{\omega_{i}\omega_{j}}\right)^{ 1/(t+r)}+1\] \[-\left(1-\prod\limits_{i,j=1}^{n}\left(1-\left(1-v_{i}^{2} \right)^{t}\left(1-v_{j}^{2}\right)^{r}\right)^{\omega_{i}\omega_{j}}\right)^ {1/(t+r)}=1\]  
\(\square\)  
**Property 2 (Idempotency). **_If \(p_{i}=p=((s,\rho),(\mu,v))\), then_  
\[\mathrm{P2 TLWBM}_{\omega}^{t,r}(p_{1},p_{2},\cdots,p_{n})=p. \tag{20}\]  
**Proof. **  
\[ \begin{array}{l}\mathrm{P2 TLWBM}_{\omega}^{t,r}(p_{1},p_{2},\cdots,p_{n})\\ =\left(\mathop{\oplus}\limits_{i,j=1}^{n}\omega_{i}\omega_{j}\right)^{1/(t+r) }\\ =p\left(\mathop{\oplus}\limits_{i,j=1}^{n}\omega_{i}\omega_{j}\right)^{1/(t+r )}\\ =p\end{array}  \tag{21}\]  
\(\square\)  
**Property 3 (Monotonicity). **_Let \(p_{i}=\left(\left(S_{p_{i}},\rho_{p_{i}}\right),\left(\mu_{p_{i}},v_{p_{i}} \right)\right)\). \((i=1,2,\cdots,n)\) and \(q_{i}=\left(\left(S_{q_{i}},\rho_{q_{i}}\right),\left(u_{q_{i}},v_{q_{i}} \right)\right)(i=1,2,\cdots,n)\) be two sets of P2 TLNs.