MADM Based on Generalized Interval Neutrosophic Schweizer-Sklar Prioritized Aggregation Operators

Abstract

The interval neutrosophic set (INS) can make it easier to articulate incomplete, indeterminate, and inconsistent information, and the Schweizer-Sklar (Sh-Sk) t-norm (tm) and t-conorm (tcm) can make the information aggregation process more flexible due to a variable parameter. To take full advantage of INS and Sh-Sk operations, in this article, we expanded the Sh-Sk and to IN numbers (INNs) in which the variable parameter takes values from $\left[\infty -,0\right)$, develop the Sh-Sk operational laws for INNs and discussed its desirable properties. After that, based on these newly developed operational laws, two types of generalized prioritized aggregation operators are established, the generalized IN Sh-Sk prioritized weighted averaging (INSh-SkPWA) operator and the generalized IN Sh-Sk prioritized weighted geometric (INSh-SkPWG) operator. Additionally, we swot a number of valuable characteristics of these intended aggregation operators (AGOs) and created two novel decision-making models to match with multiple-attribute decision-making (MADM) problems under IN information established on INSh-SkPWA and INSh-SkPRWG operators. Finally, an expressive example regarding evaluating the technological innovation capability for the high-tech enterprises is specified to confirm the efficacy of the intended models.

1. Introduction

The most important utility of multiple-attribute decision-making (MADM) problems is to go for the preeminent alternative from the set of finite alternatives as stated to the partiality values specified by decision makers (DMs) with admiration to the attributes. However, despite the complication of the decision situation, it is hard for DMs to convey the partiality values by a particular real number in realistic problems. To agree with such circumstances, an intuitionistic fuzzy set (IFS) initiated by Atanassov [1] is one of the most promising simplifications of the fuzzy set (FS) initiated by Zadeh [2] to articulate unsure and inaccurate information perfectly [3,4,5]. Yet, in several circumstances, only a positive-membership degree (TMD) and negative-membership degree (FMD) cannot depict the incompatible information precisely. To agree with the corresponding circumstances, Smarandache [6] created a neutrosophic set (NS) which depicts the vague, inaccurate, and incompatible information by TMD, neutral-membership degree (IMD), and FMD. The values of the said functions are taken independently and are standard or nonstandard subsets of $\left]0^{-},1^{+}\right[$. As the NS consists of the IMD, it can explain the vague information much better than FS and IFS, and in addition, it is more reliable when it comes to individual expected opinions and perceptions. However, NS is difficult to exploit in factual problems due to the included nonstandard subsets of $\left]0^{-},1^{+}\right[$. As a result, to employ NS effortlessly in factual problems, Wang et al. [7,8] initiated the conceptions of single-valued NS and interval neutrosophic set (INS), which are subclasses of NS.

In factual decision making, we require aggregation operators (AGOs) to incorporate the specified information. In a neutrosophic environment, a lot of researchers have anticipated a number of AGOs. For example, the operational laws of single-valued neutrosophic numbers (SVNNs) was initially anticipated by Ye [9] and established the SVN weighted averaging (SVNWA) operator and SVN weighted geometric (SVNWG) operator. Afterwards, Peng et al. [10] located various drawbacks in the operational laws presented by Ye [9] and established enhanced operational rules for SVNNs and anticipated various SVN ordered weighted averages and SVN ordered weighted geometric operators. Ye [11] further presented a number of SVN hybrid averaging (SVNHA) and SVN hybrid geometric (ACNHG) operators and used these AGOs to solve MADM problems. Zhang et al. [12] initiated operational laws for IN numbers and established some IN weighted averaging and IN weighted geometric AGOs and applied these AGOs to solve MADM problems. Ye [13] initiated some IN ordered weighted averaging operators and a possibility ranking method and initiated an approach established based on these AGOs and a possibility ranking method to solve a MADM problem under an IN environment. Sun et al. [14] studied some Choquet integral AGOs for INNs. Garg and Nancy [15] initiated a nonlinear programming model established on TOPSIS to solve MAM problems. Wei et al. [16] initiated several generalized IN Bonferroni mean operators and applied them in the evaluation of high-tech technology enterprises. Tan et al. [17] established various exponential AGOs and specified their application in typhoon disaster evaluation. Wang et al. [18] established a MADM method with IN probability established based on regret theory. Khan et al. [19] initiated the concept of IN power Bonferroni mean operators and applied these to solve MADM problems under IN information. Zhou et al. [20] established several Frank IN weighted and geometric averaging operators. Rani and Garg [21] discovered various drawbacks in division and subtraction operations of INS and established modified division and subtraction operations for INS. Liu et al. [22] presented a MAGDM established on IN power Hamy mean operators. Yang et al. [23] initiated various new similarities and entropies for INS. Meng et al. [24] presented the concept of IN preference and its application in the selection of virtual enterprise partners. Kakati et al. [25] presented various IN hesitant Choquet integral AGOs established on Einstein operational laws and applied them to MADM. Liu et al. [26] presented a number of generalized Hamacher AGOs for NS and applied them to MAGDM. Liu et al. [27] introduced the generalized IN power averaging (GINPA) operator by combining power AGOs with INS to gain the full advantage of power AGOs under an IN environment. Yang et al. [28] established various IN linguistic power AGOs based on Einstein’s operational laws.

All the above-presented AGOs are recognized on the anticipation that the input arguments to be aggregated are independent. These managing AGOs have not measured the condition where the attributes have a priority relationship between them. To resolve this difficulty, Yager [29] initiated the concept of PA operator. Wei et al. [30] presented the concept of generalized PA operators. These AGOs were further enlarged by several researchers, such as Wu et al. [31], who enlarged PA operators to the SVN environment by anticipating the notions of SVN prioritized WA (SVNPWA) and SVN prioritized WG (SVNPWG) operators and used them on MADM problems with SVN information. Additionally, Liu et al. [32] anticipated a number of prioritized ordered WA/geometric operators to agree with IN information. Ji et al. [33] fused PA operators with BM operators and presented a number of SVN Frank prioritized BM AGOs by exploiting Frank operations. Recently, Wei et al. [34] put forward a number of PA operators established on Dombi TN and TCN and used them on MADM problems with SVN information. Sahin [35] anticipated some generalized PA operators for normal NS and applied these aggregation operators to MADM. Liu and You [36] studied some IN Muirhead mean operators and applied them to solve MADM problems with IN information. Sarkar et al. [37] developed an optimization technique for a national income determination model with stability analysis of differential equations in discrete and continuous processes under uncertain environment.

From the mentioned AGOs, the majority of these AGOs for NS or INS are established on algebraic, Hamacher, Frank, and Dombi operational rules, which are particular cases of Archimedean tn (Atn) and tcn (Atcn). Atn and Atcn are definitely the expansions of numerous TNs and TCNs, which have a number of particular cases which are preferrable for articulating the union and intersection of SVNS [38]. Sh–Sk operations [39] are the particular cases from Atn and Atcn, and they are with a changeable parameter, so they are additionally supple and better than the former operations. Still, the majority of research concerning Sh–Sk mostly determined the elementary theory and types of Sh-Sk TN (Sh-Sktn) and TCN (Sh-Sktcn) [40,41]. Recently, Liu et al. [42] and Zhang [43] merged Sh-Sk operations with interval-valued IFS (IVIFS) and IFS and anticipated power WA/geometric AGOs and weighted power WA/geometric AGOs for IVIFSs and IFSs, respectively. Wang and Liu [44] anticipated a Maclaurin symmetric mean operator for IFS established on Sh–Sk operational laws. Liu et al. [45] further presented Sh-Sk operational laws for SVNS and presented some Sh-Sk prioritized AGOs and applied these AGOs to solve MADM problems. Later on, Zhang et al. [46] anticipated some Muirhead mean operators for SVNS established on Sh-Sk operational laws and applied them to MADM problems. Nagarajan et al. [47] presented some Sh-Sk operational laws for INNs by taking the values of the variable parameter from $\left(0,+\infty \right]$. They also anticipated some weighted averaging and geometric AOs based on these Sh–Sk operational laws for INNs. In recent years, INSs have gained much attention from the researchers and a great number of achievement have been made, such as VIKOR [48,49,50], cross entropy [51], MABAC, EDAS [52], out ranking approach [53], distance and similarity measures [54], TOPSIS [55].

2. Literature Review

In this section, the literature discussing IN MADM aggregation operators is reviewed. It has been noticed that research of IN MADM aggregation operators has been rapidly published since 2013. Table 1 provides the recent literature of IN MADM based on different types of aggregation operators.

It can be seen that no research attempted to merge Sh-Sk operational laws and generalized prioritized aggregation operators to deal with IN information. Therefore, we suggest that:

(1)

INNs are superior in depicting tentative information by identifying the interval TMD, interval IMD, and interval FMD than FSs and IFSs in dealing with MADM problems.

(2)

The Sh-Sk operations are too flexible and better than the former operations by a variable parameter;

(3)

Conveniently, several MADM problems exist in which the attributes have a priority relationship, and a number of existing AGOs can moderate these circumstances only when the attributes take the form of real numbers. So far, there are no such AGOs to handle MADM problems under IN information established on Sh-Sktn and Sh-Sktcn. In response to this limitation, we merged the ordinary generalized PA operator with Sh-Sk operations to handle MADM problems with the IN information.

Therefore, from the above research inspirations, the objectives and offerings of this article are revealed as follows.

(1)

Anticipating a generalized IN Sh-Sk prioritized weighted averaging (GIN Sh-Sk PWA) operator and generalized IN Sh-Sk prioritized weighted geometric (GIN Sh-Sk PWG) operator.

(2)

Examining properties and precise cases of these anticipated AGOs.

(3)

Put forward two novel MADM approaches based on the anticipated AGOs.

(4)

Confirming the efficacy and realism of the anticipated approaches.

To achieve these objectives, this article is structured as follows. In Section 3, we initiate some basic ideas of INSs and score and accuracy functions of PA operators. In Section 4, we examine a number of Schweizer-Sklar operational laws for INNs where the variable parameter takes values from $\left[\infty -,0\right)$. In Section 5, we propose INSSPWA and INSSPWG operators and examine a number of properties and particular cases of the anticipated AGOs. In Section 6, we present two MADM approaches established on these AGOs. In Section 7, we resolve a numerical example to confirm the soundness and compensations of the anticipated approaches by contrasting with other existing approaches. Finally, a short conclusion is made in Section 8.

3. Preliminaries

In this part, some basic definitions about INSs, INN, the PA operator, Schweizer-Sklar TN and TCN, and their associated properties are argued. These definitions are given in Appendix A and Appendix B.

4. Sh-Sk Operations for INNs

The Sh-Sk (SS) operations contain Sh-Sk product and Sh-Sk sum, which are particular cases of Att.

Definition 1

[12]. Assume that ${\overline{\overline{VN}}}_1=\langle {\overline{\overline{TS}}}_1,{\overline{\overline{IS}}}_1,{\overline{\overline{FS}}}_1\rangle$ and ${\overline{\overline{VN}}}_2=\langle {\overline{\overline{TS}}}_2,{\overline{\overline{IS}}}_2,{\overline{\overline{FS}}}_2\rangle$ are any two arbitrary INSs, then the generalized union and intersection are identified as follows:

$${\overline{\overline{VN}}}_1{\displaystyle {\cup }_{\mathsf{\Theta },{\mathsf{\Theta }}^{*}}{\overline{\overline{VN}}}_2=\left\{\langle \wp ,\left[{\mathsf{\Theta }}^{*}\left({\overline{\overline{TS}}}_1^{LO}(\wp ),{\overline{\overline{TS}}}_2^{LO}(\wp )\right),{\mathsf{\Theta }}^{*}\left({\overline{\overline{TS}}}_1^{UP}(\wp ),{\overline{\overline{TS}}}_2^{UP}(\wp )\right)\right],\left[\mathsf{\Theta }\left({\overline{\overline{IS}}}_1^{LO}(\wp ),{\overline{\overline{IS}}}_2^{LO}(\wp )\right),\mathsf{\Theta }\left({\overline{\overline{IS}}}_1^{UP}(\wp ),{\overline{\overline{IS}}}_2^{UP}(\wp )\right)\right],\left[\mathsf{\Theta }\left({\overline{\overline{FS}}}_1^{LO}(\wp ),{\overline{\overline{FS}}}_2^{LO}(\wp )\right),\mathsf{\Theta }\left({\overline{\overline{FS}}}_1^{UP}(\wp ),{\overline{\overline{FS}}}_2^{UP}(\wp )\right)\right]\rangle ,\wp \in U\right\}}$$

(1)

$${\overline{\overline{VN}}}_1{\displaystyle {\cap }_{\mathsf{\Theta },{\mathsf{\Theta }}^{*}}{\overline{\overline{VN}}}_2=\left\{\langle \wp ,\left[\mathsf{\Theta }\left({\overline{\overline{TS}}}_1^{LO}(\wp ),{\overline{\overline{TS}}}_2^{LO}(\wp )\right),\mathsf{\Theta }\left({\overline{\overline{TS}}}_1^{UP}(\wp ),{\overline{\overline{TS}}}_2^{UP}(\wp )\right)\right],\left[{\mathsf{\Theta }}^{*}\left({\overline{\overline{IS}}}_1^{LO}(\wp ),{\overline{\overline{IS}}}_2^{LO}(\wp )\right),{\mathsf{\Theta }}^{*}\left({\overline{\overline{IS}}}_1^{UP}(\wp ),{\overline{\overline{IS}}}_2^{UP}(\wp )\right)\right],\left[{\mathsf{\Theta }}^{*}\left({\overline{\overline{FS}}}_1^{LO}(\wp ),{\overline{\overline{FS}}}_2^{LO}(\wp )\right),{\mathsf{\Theta }}^{*}\left({\overline{\overline{FS}}}_1^{UP}(\wp ),{\overline{\overline{FS}}}_2^{UP}(\wp )\right)\right]\rangle ,\wp \in U\right\}}$$

(2)

where $\mathsf{\Theta }$ and ${\mathsf{\Theta }}^{*}$ respectively represent TN and TCN.

The Sh-Sktn and TCN are explained as follows:

$${\mathsf{\Theta }}_{Sh-Sk,\mathfrak{A}}\left(\alpha ,\beta \right)={\left({\alpha }^{\mathfrak{A}}+{\beta }^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},$$

(3)

$${\mathsf{\Theta }}_{Sh-Sk,\mathfrak{A}}^{*}\left(\alpha ,\beta \right)=1-{\left({\left(1-\alpha \right)}^{\mathfrak{A}}+{\left(1-\beta \right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}$$

(4)

where $\mathfrak{A}<0,\alpha ,\beta \in \left[0,1\right].$

Moreover, when $\varsigma =0$, Sh-Sktn and Sh-Sktcn degenerate into algebraic TN and TCN.

Based on the tn and tcn of Sh-Sk operations, we can provide the following definition for Sh-Sk operations for INNs.

Definition 2

[12]. Let ${\overline{\overline{vn}}}_1=\langle \left[{\overline{\overline{ts}}}_1^{LO},{\overline{\overline{ts}}}_1^{UP}\right],\left[{\overline{\overline{is}}}_1^{LO},{\overline{\overline{is}}}_1^{UP}\right],\left[{\overline{\overline{fs}}}_1^{LO},{\overline{\overline{fs}}}_1^{UP}\right]\rangle$ and ${\overline{\overline{vn}}}_2=\langle \left[{\overline{\overline{ts}}}_2^{LO},{\overline{\overline{ts}}}_2^{UP}\right],\left[{\overline{\overline{is}}}_2^{LO},{\overline{\overline{is}}}_2^{UP}\right],\left[{\overline{\overline{fs}}}_2^{LO},{\overline{\overline{fs}}}_2^{UP}\right]\rangle$ be any two arbitrary INNs, then the following generalized union and generalized intersection of INN are established on the basis of Sh-Sk operations:

$${\overline{\overline{vn}}}_1{\oplus }_{{\mathsf{\Theta }}^{*},\mathsf{\Theta }}{\overline{\overline{vn}}}_2=\langle \left[{\mathsf{\Theta }}^{*}\left({\overline{\overline{ts}}}_1^{LO},{\overline{\overline{ts}}}_2^{LO}\right),{\mathsf{\Theta }}^{*}\left({\overline{\overline{ts}}}_1^{UP},{\overline{\overline{ts}}}_2^{UP}\right)\right],\left[\mathsf{\Theta }\left({\overline{\overline{is}}}_1^{LO},{\overline{\overline{is}}}_2^{LO}\right),\mathsf{\Theta }\left({\overline{\overline{is}}}_1^{UP},{\overline{\overline{is}}}_2^{UP}\right)\right],\left[\mathsf{\Theta }\left({\overline{\overline{fs}}}_1^{LO},{\overline{\overline{fs}}}_2^{LO}\right),\mathsf{\Theta }\left({\overline{\overline{fs}}}_1^{UP},{\overline{\overline{fs}}}_2^{UP}\right)\right]\rangle ;$$

(5)

$${\overline{\overline{vn}}}_1{\otimes }_{{\mathsf{\Theta }}^{*},\mathsf{\Theta }}{\overline{\overline{vn}}}_2=\langle \left[\mathsf{\Theta }\left({\overline{\overline{ts}}}_1^{LO},{\overline{\overline{ts}}}_2^{LO}\right),\mathsf{\Theta }\left({\overline{\overline{ts}}}_1^{UP},{\overline{\overline{ts}}}_2^{UP}\right)\right],\left[{\mathsf{\Theta }}^{*}\left({\overline{\overline{is}}}_1^{LO},{\overline{\overline{is}}}_2^{LO}\right),{\mathsf{\Theta }}^{*}\left({\overline{\overline{is}}}_1^{UP},{\overline{\overline{is}}}_2^{UP}\right)\right],\left[{\mathsf{\Theta }}^{*}\left({\overline{\overline{fs}}}_1^{LO},{\overline{\overline{fs}}}_2^{LO}\right),{\mathsf{\Theta }}^{*}\left({\overline{\overline{fs}}}_1^{UP},{\overline{\overline{fs}}}_2^{UP}\right)\right]\rangle$$

(6)

On the basis of Definitions 1 and 2, we can develop the SS operational laws of INNs shown as follows $\left(\mathfrak{A}<0\right)$:

$$\begin{array}{l}(1)\text{}\hspace{0.17em}\hspace{0.17em}{\overline{\overline{vn}}}_1{\oplus }_{Sh-Sk}{\overline{\overline{vn}}}_2=\langle \left[1-{\left({\left(1-{\overline{\overline{ts}}}_1^{LO}\right)}^{\mathfrak{A}}+{\left(1-{\overline{\overline{ts}}}_2^{LO}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},1-{\left({\left(1-{\overline{\overline{ts}}}_1^{UP}\right)}^{\mathfrak{A}}+{\left(1-{\overline{\overline{ts}}}_2^{UP}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}\right],\\ [{\left({\left({\overline{\overline{is}}}_1^{LO}\right)}^{\mathfrak{A}}+{\left({\overline{\overline{is}}}_2^{LO}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},{\left({\left({\overline{\overline{is}}}_1^{UP}\right)}^{\mathfrak{A}}+{\left({\overline{\overline{is}}}_2^{UP}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}],\left[{\left({\left({\overline{\overline{fs}}}_1^{LO}\right)}^{\mathfrak{A}}+{\left({\overline{\overline{fs}}}_2^{LO}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},{\left({\left({\overline{\overline{fs}}}_1^{UP}\right)}^{\mathfrak{A}}+{\left({\overline{\overline{fs}}}_2^{UP}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}\right]\rangle ;\end{array}$$

(7)

$$\begin{array}{l}(2)\hspace{0.17em}\hspace{0.17em}{\overline{\overline{vn}}}_1{\otimes }_{Sh-Sk}{\overline{\overline{vn}}}_2=\langle \left[{\left({\left({\overline{\overline{ts}}}_1^{LO}\right)}^{\mathfrak{A}}+{\left({\overline{\overline{ts}}}_2^{LO}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},{\left({\left({\overline{\overline{ts}}}_1^{UP}\right)}^{\mathfrak{A}}+{\left({\overline{\overline{ts}}}_2^{UP}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}\right],[1-{\left({\left(1-{\overline{\overline{is}}}_1^{LO}\right)}^{\mathfrak{A}}+{\left(1-{\overline{\overline{is}}}_2^{LO}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},\\ 1-{\left({\left(1-{\overline{\overline{is}}}_1^{UP}\right)}^{\mathfrak{A}}+{\left(1-{\overline{\overline{is}}}_2^{UP}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}],\left[1-{\left({\left(1-{\overline{\overline{fs}}}_1^{LO}\right)}^{\mathfrak{A}}+{\left(1-{\overline{\overline{fs}}}_2^{LO}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}},1-{\left({\left(1-{\overline{\overline{fs}}}_1^{UP}\right)}^{\mathfrak{A}}+{\left(1-{\overline{\overline{fs}}}_2^{UP}\right)}^{\mathfrak{A}}-1\right)}^{\frac{1}{\mathfrak{A}}}\right]\rangle ,\end{array}$$

(8)

$$\begin{array}{l}(3)\hspace{0.17em}\hspace{0.17em}\partial \overline{\overline{vn}}=\langle \left[1-{\left(\partial {\left(1-{\overline{\overline{ts}}}^{LO}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}},1-{\left(\partial {\left(1-{\overline{\overline{ts}}}^{UP}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right],[{\left(\partial {\left({\overline{\overline{is}}}^{LO}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left(\partial {\left({\overline{\overline{is}}}^{UP}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}}],\left[{\left(\partial {\left({\overline{\overline{fs}}}^{LO}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}},{\left(\partial {\left({\overline{\overline{fs}}}^{UP}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right]\rangle ,\partial >0;\end{array}$$

(9)

$$\begin{array}{l}(4)\hspace{0.17em}\hspace{0.17em}{\overline{\overline{vn}}}^{\partial }=\langle \left[{\left(\partial {\left({\overline{\overline{ts}}}^{LO}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}},{\left(\partial {\left({\overline{\overline{ts}}}^{UP}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right],[1-{\left(\partial {\left(1-{\overline{\overline{is}}}^{LO}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}},\\ 1-{\left(\partial {\left(1-{\overline{\overline{is}}}^{UP}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}}]\left[1-{\left(\partial {\left(1-{\overline{\overline{fs}}}^{LO}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}},1-{\left(\partial {\left(1-{\overline{\overline{fs}}}^{UP}\right)}^{\mathfrak{A}}-\left(\partial -1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right]\rangle ,\partial >0.\end{array}$$

(10)

Theorem 1.

Let $\overline{\overline{vn}}=\langle \left[{\overline{\overline{ts}}}^{LO},{\overline{\overline{ts}}}^{UP}\right],\left[{\overline{\overline{is}}}^{LO},{\overline{\overline{is}}}^{UP}\right],\left[{\overline{\overline{fs}}}^{LO},{\overline{\overline{fs}}}^{UP}\right]\rangle ,{\overline{\overline{vn}}}_1=\langle \left[{\overline{\overline{ts}}}_1^{LO},{\overline{\overline{ts}}}_1^{UP}\right],\left[{\overline{\overline{is}}}_1^{LO},{\overline{\overline{is}}}_1^{UP}\right],\left[{\overline{\overline{fs}}}_1^{LO},{\overline{\overline{fs}}}_1^{UP}\right]\rangle$ and ${\overline{\overline{vn}}}_2=\langle \left[{\overline{\overline{ts}}}_2^{LO},{\overline{\overline{ts}}}_2^{UP}\right],\left[{\overline{\overline{is}}}_2^{LO},{\overline{\overline{is}}}_2^{UP}\right],\left[{\overline{\overline{fs}}}_2^{LO},{\overline{\overline{fs}}}_2^{UP}\right]\rangle$ be any three INNs. Then:

$${\overline{\overline{vn}}}_1{\oplus }_{SS}{\overline{\overline{vn}}}_2={\overline{\overline{vn}}}_2{\oplus }_{SS}{\overline{\overline{vn}}}_1$$

(11)

$${\overline{\overline{vn}}}_1{\otimes }_{SS}{\overline{\overline{vn}}}_2={\overline{\overline{vn}}}_2{\otimes }_{SS}{\overline{\overline{vn}}}_1$$

(12)

$$\partial \left({\overline{\overline{vn}}}_1{\oplus }_{SS}{\overline{\overline{vn}}}_2\right)=\partial {\overline{\overline{vn}}}_1{\oplus }_{SS}\partial {\overline{\overline{vn}}}_2,\partial \ge 0$$

(13)

$${\partial }_1\overline{\overline{vn}}{\oplus }_{Sh-Sk}{\partial }_2\overline{\overline{vn}}=\left({\partial }_1+{\partial }_2\right)\overline{\overline{vn}},{\partial }_1,{\partial }_2\ge 0$$

(14)

$${\overline{\overline{vn}}}^{{\partial }_1}{\otimes }_{Sh-Sk}{\overline{\overline{vn}}}^{{\partial }_2}={\left(\overline{\overline{vn}}\right)}^{{\partial }_1+{\partial }_2},{\partial }_1,{\partial }_2\ge 0$$

(15)

$${\overline{\overline{vn}}}_1^{\partial }{\otimes }_{Sh-Sk}{\overline{\overline{vn}}}_2^{\partial }={\left({\overline{\overline{vn}}}_1{\otimes }_{Sh-Sk}{\overline{\overline{vn}}}_2\right)}^{\partial },\partial >0.$$

(16)

5. Some Generalized Prioritized Aggregation Operators for INNs

In this part, we develop some generalized prioritized aggregation operators based on the developed operational laws for INNs.

5.1. Generalized Interval Neutrosophic Schweizer-Sklar Prioritized Aggregation (GINSh-SkPA) Operator

In this subpart, we develop a generalized interval neutrosophic Schweizer-Sklar prioritized weighted averaging (GINSh-SkPWA) operator and discuss its enviable properties and some particular cases.

Definition 3.

For a group of INNs ${\overline{\overline{vn}}}_g\left(g=1,2,\dots ,s\right),$ the GINSh-SkPWA operator is a function ${\Im }^s\to \Im ,$

$$GINSh-SkPW{A}_{\mathsf{\Upsilon }}\left({\overline{\overline{vn}}}_1,{\overline{\overline{vn}}}_2,\dots ,{\overline{\overline{vn}}}_s\right)={\left(\underset{g=1}{\overset{s}{{\displaystyle {\oplus }_{{}^{Sh-Sk}}}}}\left(\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_g^{\mathsf{\Upsilon }}\right)\right)}^{\frac{1}{\mathsf{\Upsilon }}},$$

(17)

where $\mathsf{\Upsilon }\in \left(0,+\infty \right)$, $T_g=\underset{h=1}{\overset{g-1}{{\displaystyle \otimes }}}\hspace{0.17em}\overline{\overline{SE}}\left({\overline{\overline{vn}}}_h\right),(g=2,\dots ,s),T_1=1$, and$\overline{\overline{SE}}\left({\overline{\overline{vn}}}_h\right)$is the score of INN ${\overline{\overline{vn}}}_h$.

Theorem 2.

Let${\overline{\overline{vn}}}_g\left(g=1,2,\dots ,s\right)$be a set of INNs, then the aggregated value employing Definition (3) is still INN, and we have:

$$\begin{array}{l}GINSh-SkPW{A}_{\mathsf{\Upsilon }}\left({\overline{\overline{vn}}}_1,{\overline{\overline{vn}}}_2,\dots ,{\overline{\overline{vn}}}_s\right)=\hfill \\ \langle \left[{\left\{\frac{1}{\mathsf{\Upsilon }}{\left(1-{\left(\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{LO\hspace{0.17em}}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{1}{\mathsf{\Upsilon }}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\frac{1}{\mathsf{\Upsilon }}{\left(1-{\left(\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{1}{\mathsf{\Upsilon }}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\hfill \\ \hspace{0.17em}\left[{1-\{\frac{1}{\mathsf{\Upsilon }}{\left(1-{\left(\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{1}{\mathsf{\Upsilon }}-1\right)\}}^{\frac{1}{\mathfrak{A}}},{1-\{\frac{1}{\mathsf{\Upsilon }}{\left(1-{\left(\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{1}{\mathsf{\Upsilon }}-1\right)\}}^{\frac{1}{\mathfrak{A}}}\right]\hfill \\ \left[1-{\{\frac{1}{\mathsf{\Upsilon }}{\left(1-{\left(\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{1}{\mathsf{\Upsilon }}-1\right)\}}^{\frac{1}{\mathfrak{A}}},{1-\{\frac{1}{\mathsf{\Upsilon }}{\left(1-{\left(\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{UP}\right)}^{\varsigma }-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{1}{\mathsf{\Upsilon }}-1\right)\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle .\hfill \end{array}$$

(18)

Proof. 

In the following, we first prove:

$$\begin{array}{l}\frac{T_1}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\displaystyle {\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}}+\frac{T_2}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\displaystyle {\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}}+\dots +\frac{T_s}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\displaystyle {\overline{\overline{vn}}}_s^{\mathsf{\Upsilon }}}\hfill \\ =\langle \left[1-{\left\{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right],\hfill \\ \left[{\left\{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\varsigma }},{\left\{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\hfill \\ \left[{\left\{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{s}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle .\hfill \end{array}$$

(19)

by utilizing mathematical induction on $s$.

For $s=2$.

From the operational rules defined for INNs in Definition (2), we have:

$$\begin{array}{ll}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}\hfill & =\langle \left[{\left\{\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\left[1-{\left\{\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ & \left[1-{\left\{\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle ,\end{array}$$

and:

$$\begin{array}{ll}\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}& =\langle \left[1-{\left\{\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ & \left[{\{\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}},{\{\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}}\right],\\ & \left[{\{\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}},{\{\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle .\end{array}$$

Similarly:

$$\begin{array}{ll}\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}& =\langle \left[1-{\left\{\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ & \left[{\{\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}},{\{\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}}\right],\\ & \left[{\{\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}},{\{\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle .\end{array}$$

Then:

$$\begin{gathered} \begin{array}{l}\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}+\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}\\ =\langle [1-{\left\{{\left(1-\left(1-{\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)\right)}^{\mathfrak{A}}+{\left(1-\left(1-{\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}},\\ 1-{\left\{{\left(1-\left(1-{\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)\right)}^{\mathfrak{A}}+{\left(1-\left(1-{\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}}],\\ [{\left\{{\left({\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}+{\left({\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}},\\ {\left\{{\left({\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}+{\left({\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}}],\\ [{\left\{{\left({\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}+{\left({\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}},\\ {\left\{{\left({\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}+{\left({\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_2^{Up}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}}]\rangle ,\\ =\langle [1-{\left\{{\left({\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}+{\left({\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}},\\ 1-{\left\{{\left({\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}+{\left({\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-1\right\}}^{\frac{1}{\mathfrak{A}}}],\end{array} \\ \begin{array}{l}[{\left\{\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)+\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)-1\right\}}^{\frac{1}{\mathfrak{A}}},\\ {\left\{\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)+\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)-1\right\}}^{\frac{1}{\mathfrak{A}}}]\hspace{0.17em},\\ [{\left\{\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)+\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_2^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)-1\right\}}^{\frac{1}{\mathfrak{A}}},\\ {\left\{\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_1^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)+\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_2^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right)-1\right\}}^{\frac{1}{\mathfrak{A}}}]\hspace{0.17em}\rangle ;\end{array} \\ \begin{array}{l}\frac{T_1}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}+\frac{T_2}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}\\ =\langle \left[1-{\left\{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathfrak{M}-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\\ \hspace{0.17em}\left[{\left\{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{2}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle ;\end{array} \end{gathered}$$

If Equation (23) holds for $s=m$:

$$\begin{array}{l}\frac{T_1}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}+\frac{T_2}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}+\dots +\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_m^{\mathsf{\Upsilon }}\\ =\langle \left[1-{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^U\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\\ \left[{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle .\end{array}$$

Then, when $s=m+1,$ by the operational rules given in Definition (5), we have:

$$\begin{array}{l}\frac{T_1}{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}+\frac{T_2}{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}+\dots +\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_{m+1}^{\mathsf{\Upsilon }}=\frac{T_1}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}+\frac{T_2}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}+\dots +\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_m^{\mathsf{\Upsilon }}+\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_{m+1}^{\mathsf{\Upsilon }}\\ =\langle \left[1-{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}\frac{T_m}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle +\\ \langle \left[1-{\left\{\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_{m+1}^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_{m+1}^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_{m+1}^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_{m+1}^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_{m+1}^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_{m+1}^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\left(\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}-1\right)\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle ,\end{array}$$

$$\begin{array}{l}\frac{T_1}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_1^{\mathsf{\Upsilon }}+\frac{T_2}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_2^{\mathsf{\Upsilon }}+\dots +\frac{T_{m+1}}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\overline{\overline{vn}}}_{m+1}^{\mathsf{\Upsilon }}\\ =\langle \left[1-{\left\{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},1-{\left\{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left({\overline{\overline{ts}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right],\\ \left[{\left\{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{is}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},\right]\\ \hspace{0.17em}\left[{\left\{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{LO}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}},{\left\{\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}{\left(1-{\left(\mathsf{\Upsilon }{\left(1-{\overline{\overline{fs}}}_g^{UP}\right)}^{\mathfrak{A}}-\left(\mathsf{\Upsilon }-1\right)\right)}^{\frac{1}{\mathfrak{A}}}\right)}^{\mathfrak{A}}-\underset{g=1}{\overset{m+1}{{\displaystyle \oplus }}}\frac{T_g}{\underset{g=1}{\overset{m}{{\displaystyle \oplus }}}{T}_g\hspace{0.17em}}+1\right\}}^{\frac{1}{\mathfrak{A}}}\right]\rangle .\end{array}$$

That is, Equation (19) is true for $g=m+1.$ Thus, Equation (19), is true for all $g.$ Then: