## 2 Preliminaries  
This Section provides novel operations for IFVs, which are established through the concepts of copula, co-copula, Archimedean copula and fuzzy negations. [PERSON] [28] presented the concept of copula, as a mathematical tool to combine probability distributions, framing the dependence structure within random variables with the usage of their CDFs. **Definition 1**. : [28] _A 2-dimensional copula is a function with domain \([0,1]\times[0,1]\) and range \([0,1]\), grounded and 2-increasing, i.e., satisfies the following boundary and monotonicity conditions:_  
1. \(C(x,0)=C(0,y)=0\)_._
2. \(C(x,1)=x,C(1,y)=y\)_._
3. \(C(x_{1},y_{1})+C(x_{2},y_{2})-C(x_{1},y_{2})-C(x_{2},y_{1})\geq 0\)_._  
_where, \(x,y,x_{1},x_{2},y_{1},y_{2}\in[0,1]\) and \(x_{1}\leq x_{2},y_{1}\leq y_{2}\)._  
Below, the definition of co-copula is given, according to [PERSON] et al. 's [29] work:  
**Definition 2**. : [29] _The co-copula of a copula is defined as:_  
\[C^{*}(x,y)=1-C(1-x,1-y) \tag{1}\]  
According to [PERSON] [24] the co-copula is not a copula, but may be constructed by the usage of a copula. [PERSON] [30] and [PERSON] [31] introduced the Frechet-Hoeffding [24] bounds of copulas for any \(x,y\in[0,1]\), \(\max\{x+y-1,0\}\leq C(x,y)\leq\min\{x,y\}\). At this point, the method follows the definition of Archimedean copulas, which may be constructed with significant ease and ensure some remarkable properties. **Definition 3**. : [24] _Let \(\varphi\) be a continuous, strictly decreasing function on \([0,1]\rightarrow[0,+\infty]\) with \(\varphi(1)=0\) and \(\varphi^{[-1]}\) the pseudo-inverse of \(\varphi\), which is defined as:_  
\[\varphi^{[-1]}(x)= \begin{cases}\varphi^{-1}(x),\ 0\leq x\leq\varphi(0),\\ 0,\qquad\varphi(0)\leq x\leq+\infty,\end{cases}  \tag{2}\]  
_Then, \(C:[0,1]^{2}\rightarrow[0,1]\), \(C(x,y)=\varphi^{[-1]}(\varphi(x)+\varphi(y))\) corresponds to the conditions of Definition 1, hence \(C\) is a copula and \(C\) is called Archimedean copula._  
_In the case, where \(C\) is strictly increasing, \(\varphi(0)=+\infty\) and \(\varphi^{[-1]}=\varphi^{-1}\), then \(C(x,y)=\varphi^{-1}(\varphi(x)+\varphi(y))\). In that case \(\varphi\) is called strict generator and \(C\) is called strict Archimedean copula._  
In Table 1, some Archimedean copulas are listed. The study of the applications of copulas in fuzzy sets is of great importance, given that specific copulas are t-norms and specific co-copulas are t-conorms and vice versa. Moreover, [PERSON] [23] mentioned in his work that the combination of probabilistic information or fuzzy information hardly matters. Below, the [PERSON]'s [1] definition for Intuitionistic Fuzzy Sets is provided:  
**Definition 4**. : [1] _Let \(X\) be a reference set, an Intuitionistic Fuzzy Set (IFS) \(A\) on \(X\) is defined as:_  
\[A=\{\langle x,\mu_{A}(x),\
u_{A}(x)\rangle\big{|}x\in X\}, \tag{3}\]  
_where, the functions \(\mu_{A}(x)\) and \(\
u_{A}(x)\) denote the degrees of membership and non-membership of the element \(x\in X\) to the set \(A\), respectively with:_  
\(0\leq\mu_{A}(x)\leq 1\)_, \(0\leq\
u_{A}(x)\leq 1\), \(0\leq\mu_{A}(x)+\
u_{A}(x)\leq 1\) and \(\pi_{A}(x)=1-\mu_{A}(x)-\
u_{A}(x)\) is called degree of indeterminacy of \(x\) to \(A\). Additionally, \(\alpha=\langle\mu_{\alpha},\
u_{\alpha}\rangle\) is called Intuitionistic Fuzzy Value (IFV) and the set of all IFVs is \(V\)._  
**Definition 5**. : [32] _A decreasing function \(n:[0,1]\rightarrow[0,1]\) is called a fuzzy negation if \(n(0)=1\) and \(n(1)=0\). A fuzzy negation \(n\) is called: Strict, if it is strictly decreasing and continuous and Strong, if it is an involution, i.e., \(n(n(x))=x\), \(\forall x\in[0,1]\)._  
Table 2, proposes two parametric families of fuzzy negations:  
Moreover, the operations of IFVs in [PERSON] et al. 's [9] work, are being extended through the adaptation of fuzzy negations.