# Indistinguishability Operators Applied to Task Allocation Problems in Multi-Agent Systems
## 3 Preliminaries on Indistinguishability Operators  
The concept of _triangular norm_, briefly \(t\)-norm, appeared in the literature as a tool to manage the triangle inequality in the construction of metric spaces that take, as values, a probability distribution instead of a positive real number. Since then, they have played an essential role in fuzzy logic and manyauthors have contributed to the development of this kind of binary operation. Our basic reference for \(t\)-norms and all related notions is [15]. Let us recall that a \(t\)-norm is a function \(T:[0,1]^{2}\rightarrow[0,1]\) such that for all \(x,y,z\in[0,1]\) the following four axioms are satisfied:  
1. \(T(x,y)=T(y,x)\); (Commutativity)
2. \(T(x,T(y,z))=T(T(x,y),z)\); (Associativity)
3. \(T(x,y)\geq T(x,z)\); where \(y\geq z\) (Monotonicity)
4. \(T(x,1)=x\). An interesting subclass of \(t\)-norms in our subsequent study are the so-called Archimedean, which are defined as follows. A \(t\)-norm is called Archimedean if for each \(x,y\in]0,1[\) there exists \(n\in\mathbb{N}\) such that \(x^{(n)}<y\), where \(x^{(n)}=T(x,\ldots,x)\)\(n\)-times. Archimedean \(t\)-norms are exactly those \(t\)-norms that satisfy \(T(x,x)<x\) for each \(x\in]0,1[\) whenever \(T\) is, in addition, continuous. Two well-known examples of continuous Archimedean \(t\)-norms are the usual product \(T_{P}\) and the Luckasievicz \(t\)-norm \(T_{L}\), where \(T_{P}(x,y)=x\cdot y\) and \(T_{L}(x,y)=\max\{x+y-1,0\}\) for all \(x,y\in[0,1]\). An example of continuous \(t\)-norm that is non-Archimedean is the minimum \(t\)-norm \(T_{M}\), i.e., \(T_{M}(x,y)=\min\{x,y\}\) for all \(x,y\in[0,1]\). These \(t\)-norms are the most commonly used in Fuzzy Logic. A concept related to a \(t\)-norm, which will play an important role in this paper, is the notion of the pseudo-inverse and an additive generator. The notion of the pseudo-inverse is given as follows:  
Let \(f:[0,1]\rightarrow[0,\infty]\) be a strictly decreasing continuous function provided that \(f(1)=0\). Then, the _pseudo-inverse_\(f^{(-1)}:[0,\infty]\rightarrow[0,1]\) of \(f\) is defined as follows. \[f^{(-1)}(y)=f^{-1}(\min\{f(0),y\})=\max\{0,f^{-1}(y)\}. \tag{4}\]  
Moreover, given a \(t\)-norm \(T\), a strictly decreasing continuous function \(f_{T}:[0,1]\rightarrow[0,\infty]\) is said to be an additive generator of \(T\) provided that \(f_{T}(1)=0\) and  
\[T(x,y)=f_{T}^{(-1)}(f_{T}(x)+f_{T}(y)). \tag{5}\]  
Note that in this case, the \(t\)-norm is continuous. It is known that each \(t\)-norm with an additive generator is Archimedean. However, the converse of the former assertion is not true in general. The next result states that continuous \(t\)-norms always admits an additive generator. Theorem 1: _A mapping \(T:[0,1]\times[0,1]\rightarrow[0,1]\) is a continuous Archimedean \(t\)-norm if and only if there exists a continuous additive generator \(f_{T}\) of \(T\)._  
Let us note that if \(T\) is a \(t\)-norm and \(f_{T}\) is an additive generator of \(T\), then this additive generator multiplied by a positive constant is again an additive generator of \(T\), i.e., if \(f_{T}\) is an additive generator of \(T\), then the function \(f_{T,\theta}\) is again an additive generator of \(T\), where \(f_{T,\theta}(x)=\theta\cdot f_{T}(x)\) for all \(x\in[0,1]\) and each \(\theta\in]0,\infty[\). Furthermore, if \(f_{T}^{(-1)}\) is the pseudo-inverse of an additive generator \(f_{T}\), then it is easy to verify that the pseudo-inverse of the additive generator \(f_{T,\theta}\) is given by \(f_{T,\theta}^{(-1)}(y)=f_{T}^{(-1)}(\frac{y}{y})\) for all \(y\in[0,\infty]\). We have introduced some details about \(t\)-norms that will be necessary later on. Now, we are able to recall the concept of the _\(T\)-indistinguishability operator_. This concept was introduced in 1982 by [PERSON] as a way to measure the degree of equivalence, in fuzzy logic, between the elements of a set \(X\) (see [16; 17]).