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Interval-Valued Picture Uncertain Linguistic Generalized Hamacher Aggregation Operators and Their Application in Multiple Attribute Decision-Making Process

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Abstract

This paper aims to present the concept of interval-valued picture uncertain linguistic set (IVPULS), which composes the grade of truth, abstinence, and falsity in the form of a subset of the unit interval. IVPULS is an extensive capable theory to manage awkward and unreliable information. To explore the theory, we firstly stated some basic operational laws of them and investigated their properties. Based on these stated laws, we defined several weighted and ordered weighted generalized Hamacher aggregation (GHA) operators. Several special cases are deduced from the proposed operators. Furthermore, a multi-attribute decision-making approach algorithm is stated by using the concept of proposed GHA operators under the IVPULSs environment. The presented algorithm has been explored with a numerical example and compares their results with the existing studies to examine and improve the quality and feasibility of the discovered theory.

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Authors and Affiliations

  1. School of Mathematics, Thapar Institute of Engineering and Technology (Deemed University), Patiala, Punjab, 147004, India

    Harish Garg

  2. Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, Pakistan

    Zeeshan Ali & Tahir Mahmood

Authors
  1. Harish Garg
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  2. Zeeshan Ali
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  3. Tahir Mahmood
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Correspondence to Harish Garg.

Appendix

Appendix

Proof of Theorem 2

We shall prove the result by using principle of mathematical induction on “n.”

For n = 1 with \({\Omega }_{W-1}=1\) and \({\Omega }_{W-1}^{{\prime}}=1\), then Eq. (36), such that

$$\begin{aligned} & IVPULGHHWA\left({\mathfrak{T}}_{IPU-1},{\mathfrak{T}}_{IPU-2},\dots ,{\mathfrak{T}}_{IPU-n}\right)\\ & \quad = {\mathfrak{T}}_{IPU}=\left(\begin{array}{c}\left[{\mathcal{L}}_{{\alpha }_{1}},{\mathcal{L}}_{{\beta }_{1}}\right],\\ \left(\left[{\mu }_{{\mathfrak{Q}}_{LG-1}}^{-},{\mu }_{{\mathfrak{Q}}_{UG-1}}^{+}\right],\left[{\psi }_{{\mathfrak{Q}}_{LG-1}}^{-},{\psi }_{{\mathfrak{Q}}_{UG-1}}^{+}\right],\left[{\eta }_{{\mathfrak{Q}}_{LG-1}}^{-},{\eta }_{{\mathfrak{Q}}_{UG-1}}^{+}\right]\right)\end{array}\right), \end{aligned}$$

then the right-hand side of Eq. (36), we have

$$=\left(\begin{array}{c}\left[{\mathcal{L}}_{{\alpha }_{1}},{\mathcal{L}}_{{\beta }_{1}}\right],\\ \left(\begin{array}{c}\left[\begin{array}{c}\left(\frac{\vartheta {\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}-\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}+\left({\vartheta }^{2}-1\right)\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)+\left(\vartheta -1\right)\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}-\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}\right),\\ \left(\frac{\vartheta {\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}-\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}+\left({\vartheta }^{2}-1\right)\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)+\left(\vartheta -1\right)\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}-\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}\right)\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}+\left({\vartheta }^{2}-1\right){\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}-{\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}+\left({\vartheta }^{2}-1\right){\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}-{\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}+\left({\vartheta }^{2}-1\right){\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}-{\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}+\left({\vartheta }^{2}-1\right){\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}-{\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}+\left({\vartheta }^{2}-1\right)\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}-\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}+\left({\vartheta }^{2}-1\right)\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}-\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}+\left({\vartheta }^{2}-1\right)\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}-\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}+\left({\vartheta }^{2}-1\right)\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}-\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right]\end{array}\right)\end{array}\right)$$
$$=\left(\begin{array}{c}\left[{\mathcal{L}}_{{\alpha }_{1}},{\mathcal{L}}_{{\beta }_{1}}\right],\\ \left(\left[{\mu }_{{\mathfrak{Q}}_{LG-1}}^{-},{\mu }_{{\mathfrak{Q}}_{UG-1}}^{+}\right],\left[{\psi }_{{\mathfrak{Q}}_{LG-1}}^{-},{\psi }_{{\mathfrak{Q}}_{UG-1}}^{+}\right],\left[{\eta }_{{\mathfrak{Q}}_{LG-1}}^{-},{\eta }_{{\mathfrak{Q}}_{UG-1}}^{+}\right]\right)\end{array}\right)$$

For \({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}={\left(1+\left(\vartheta -1\right)\left(1-{\mu }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)\right)}^{{\Upsilon }_{SC}}+\left({\vartheta }^{2}-1\right){\mu }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{{{-}{\Upsilon }_{SC}}}\) and \({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}={\left(1+\left(\vartheta -1\right)\left(1-{\mu }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)\right)}^{{\Upsilon }_{SC}}+\left({\vartheta }^{2}-1\right){\mu }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{{{+}{\Upsilon }_{SC}}}\), \({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}={\left(1+\left(\vartheta -1\right)\left(1-{\psi }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)\right)}^{{\Upsilon }_{SC}}-{\psi }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{{{-}{\Upsilon }_{SC}}}\) and \({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}={\left(1+\left(\vartheta -1\right)\left(1-{\psi }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)\right)}^{{\Upsilon }_{SC}}-{\psi }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{{{+}{\Upsilon }_{SC}}}\), \(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}={\left(1+\left(\vartheta -1\right)\left(1-{\eta }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)\right)}^{{\Upsilon }_{SC}}-{\eta }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{{{-}{\Upsilon }_{SC}}}\) and \(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}={\left(1+\left(\vartheta -1\right)\left(1-{\eta }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)\right)}^{{\Upsilon }_{SC}}-{\eta }_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{{{+}{\Upsilon }_{SC}}}\). For \(n=1\), Eq. (36) is hold true.

Additionally, we can check for \(n=k\), we have

$$IVPULGHHWA\left({\mathfrak{T}}_{IPU-1},{\mathfrak{T}}_{IPU-2},\dots ,{\mathfrak{T}}_{IPU-n}\right)$$

\(=\left(\begin{array}{c}\left[{\mathcal{L}}_{\sum_{i=1}^{k}{\Omega }_{W-i}{\alpha }_{O\left(i\right)}},{\mathcal{L}}_{\sum_{i=1}^{k}{\Omega }_{W-i}{\beta }_{O\left(i\right)}}\right],\\ \left(\begin{array}{c}\left[\begin{array}{c}\left(\frac{\vartheta {\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)+\left(\vartheta -1\right)\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}\right),\\ \left(\frac{\vartheta {\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)+\left(\vartheta -1\right)\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}+\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)}\right)\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right]\end{array}\right)\end{array}\right)\);

We can prove for \(n=k+1\), such that

$$IVPULGHHWA\left({\mathfrak{T}}_{IPU-1},{\mathfrak{T}}_{IPU-2},\dots ,{\mathfrak{T}}_{IPU-k+1}\right)$$
$$=IVPULGHHWA\left({\mathfrak{T}}_{IPU-1},{\mathfrak{T}}_{IPU-2},\dots ,{\mathfrak{T}}_{IPU-k}\right){\oplus }_{IPU}{\Omega }_{W-k+1}{\mathfrak{T}}_{IPU-k+1}$$
$$=\left(\begin{array}{c}\left[{\mathcal{L}}_{\sum_{i=1}^{k}{\Omega }_{W-i}{\alpha }_{O\left(i\right)}},{\mathcal{L}}_{\sum_{i=1}^{k}{\Omega }_{W-i}{\beta }_{O\left(i\right)}}\right],\\ \left(\begin{array}{c}\left[\begin{array}{c}\left(\frac{\vartheta {\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)+\left(\vartheta -1\right)\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}\right),\\ \left(\frac{\vartheta {\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)+\left(\vartheta -1\right)\left(\prod_{i=1}^{k}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}+\prod_{i=1}^{k}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)}\right)\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{k}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{k}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right]\end{array}\right)\end{array}\right){\oplus }_{IPU}$$
$$\left(\begin{array}{c}\left[{\mathcal{L}}_{{\alpha }_{k+1}},{\mathcal{L}}_{{\beta }_{k+1}}\right],\\ \left(\begin{array}{c}\left[\begin{array}{c}\frac{{\left(1+\left(\vartheta -1\right){\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}-{\left(1-{\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}}{{\left(1+\left(\vartheta -1\right){\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}+\left(\vartheta -1\right){\left(1-{\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}},\\ \frac{{\left(1+\left(\vartheta -1\right){\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}-{\left(1-{\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}}{{\left(1+\left(\vartheta -1\right){\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}+\left(\vartheta -1\right){\left(1-{\mu }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}}\end{array}\right],\\ \left[\begin{array}{c}\frac{\vartheta {\left({\psi }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}}{{\left(1+\left(\vartheta -1\right)\left(1-{\psi }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)\right)}^{{\Omega }_{W-k+1}}+\left(\vartheta -1\right){\left({\psi }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}},\\ \frac{\vartheta {\left({\psi }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}}{{\left(1+\left(\vartheta -1\right)\left(1-{\psi }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)\right)}^{{\Omega }_{W-k+1}}+\left(\vartheta -1\right){\left({\psi }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}}\end{array}\right],\left[\begin{array}{c}\frac{\vartheta {\left({\eta }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}}{{\left(1+\left(\vartheta -1\right)\left(1-{\eta }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)\right)}^{{\Omega }_{W-k+1}}+\left(\vartheta -1\right){\left({\eta }_{{\mathfrak{Q}}_{LG-k+1}}^{-}\right)}^{{\Omega }_{W-k+1}}},\\ \frac{\vartheta {\left({\eta }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}}{{\left(1+\left(\vartheta -1\right)\left(1-{\eta }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)\right)}^{{\Omega }_{W-k+1}}+\left(\vartheta -1\right){\left({\eta }_{{\mathfrak{Q}}_{LG-k+1}}^{+}\right)}^{{\Omega }_{W-k+1}}}\end{array}\right]\\ \end{array}\right)\end{array}\right).$$
$$IVPULGHHWA\left({\mathfrak{T}}_{IPU-1},{\mathfrak{T}}_{IPU-2},\dots ,{\mathfrak{T}}_{IPU-n}\right)=$$
$$=\left(\begin{array}{c}\left[{\mathcal{L}}_{\sum_{i=1}^{n}{\Omega }_{W-i}{\alpha }_{O\left(i\right)}},{\mathcal{L}}_{\sum_{i=1}^{n}{\Omega }_{W-i}{\beta }_{O\left(i\right)}}\right],\\ \left(\begin{array}{c}\left[\begin{array}{c}\left(\frac{\vartheta {\left(\prod_{i=1}^{n}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left(\prod_{i=1}^{n}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)+\left(\vartheta -1\right)\left(\prod_{i=1}^{n}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\prod_{i=1}^{n}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}\right),\\ \left(\frac{\vartheta {\left(\prod_{i=1}^{n}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{\left(\prod_{i=1}^{n}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)+\left(\vartheta -1\right)\left(\prod_{i=1}^{n}{\left({\widehat{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}+\prod_{i=1}^{n}{\left(\bar{\bar{\mu }}_{{\mathfrak{Q}}_{LG-i}}^{+}\right)}^{{\Omega }_{W-i}}\right)}\right)\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{n}{\left({\widehat{\psi }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left({\bar{\bar{\psi }}}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right],\\ \left[\begin{array}{c}\frac{{\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{-}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}},\\ \frac{{\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}-{\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}{{\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}+\left({\vartheta }^{2}-1\right)\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}+\left(\vartheta -1\right){\left(\prod_{i=1}^{n}{\left({\widehat{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}-\prod_{i=1}^{n}{\left(\bar{\bar{\eta }}_{{\mathfrak{Q}}_{LG-O\left(i\right)}}^{+}\right)}^{{\Omega }_{W-i}}\right)}^{\frac{1}{{\Upsilon }_{SC}}}}\end{array}\right]\end{array}\right)\end{array}\right)$$

Therefore, Eq. (36) is also holds for \(n=k+1\). Hence, from the above two conditions, we get Eq. (36) is hold true for \(n\).

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Garg, H., Ali, Z. & Mahmood, T. Interval-Valued Picture Uncertain Linguistic Generalized Hamacher Aggregation Operators and Their Application in Multiple Attribute Decision-Making Process. Arab J Sci Eng 46, 10153–10170 (2021). https://doi.org/10.1007/s13369-020-05313-9

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