# Indistinguishability Operators Applied to Task Allocation Problems in Multi-Agent Systems
## 3 Preliminaries on Indistinguishability Operators
Let \(X\) be a nonempty set and let \(T\) be a \(t\)-norm, we will say that a fuzzy set \(E:X\times X\rightarrow[0,1]\) is a \(T\)-indistinguishability operator if it satisfies, for each \(x,y,z\in X\), the following:  
**(E1)**: \(E(x,x)=1\); (Reflexivity)
**(E2)**: \(E(x,y)=E(y,x)\); (Symmetry)
**(E3)**: \(E(x,z)\geq T(E(x,y),E(y,z))\). (Transitivity)  
A \(T\)-indistinguishability operator \(E\) is said to separate points provided that \(E(x,y)=1\Leftrightarrow x=y\) for all \(x,y\in X\). The notion of indistinguishability operators is essentially interpreted as a measure of similarity (in contrast to dissimilarity modeled by pseudo-metrics). Thus, \(E(x,y)\) matches up with the degree of indistinguishability between the objects \(x\) and \(y\). In fact, the greater \(E(x,y)\), the more similar are \(x\) and \(y\). In such a way that when \(x=y\), the measure of similarity is exactly \(E(x,x)=1\). Since [PERSON] introduced the notion of the indistinguishability operator, many authors have contributed to the development of a theory in which this concept plays an essential role. We focus our attention on a method to construct \(T\)-indistinguishability operators from distances and vice-versa (for a detailed treatment of the topic we refer the reader to [18,19]). The next proposition provides a technique that allows us to construct distances from indistinguishability operators. **Theorem 2**. : _Let \(X\) be a nonempty set and let \(T^{*}\) be a \(t\)-norm with additive generator \(f_{T^{*}}:[0,1]\to[0,\infty]\). Let \(d_{E}:X\times X\to[0,\infty]\) be the function defined by_  
\[d_{E}(x,y)=f_{T^{*}}(E(x,y))\]  
_for all \(x,y\in X\). If \(T\) is a \(t\)-norm, then the following assertions are equivalent._  
_(1)_ \(T^{*}\leq T\) _(i.e.,_ \(T^{*}(x,y)\leq T(x,y)\) _for all_ \(x,y\in X\)_)._  
_(2)_ _For any_ \(T\)_-indistinguishability operator_ \(E\) _on_ \(X\) _that separates points the function_ \(d_{E}\) _is a distance on_ \(X\)_._  
The following result develops a technique that allows the induction of indistinguishability operators from distances. **Theorem 3**. : _Let \(d\) be a distance on a nonempty set \(X\) and let \(T\) be a continuous Archimedean \(t\)-norm with additive generator \(f_{T}\). Then, the fuzzy set \(E_{T}:X\times X\to[0,1]\), given by \(E_{T}(x,y)=f_{T}^{(-1)}(d(x,y))\) for all \(x,y\in X\), is a \(T\)-indistinguishability operator that separates points on \(X\)._  
Recall that a distance on a nonempty set \(X\) is a fucntion \(d:X\times X\to[0,\infty]\) satisfying the following axioms for all \(x,y,z\in X\). **(d1)**: \(d(x,y)=0\Leftrightarrow x=y\);
**(d2)**: \(d(x,y)=d(y,x)\);
**(d3)**: \(d(x,z)\leq d(x,y)+d(y,z)\). **4. \(T\)-Indistinguishability Operators, Distances and Response Functions**  
In this section we will apply Theorems 2 and 3 for some particular \(t\)-norms in order to construct two indistinguishability operators that allow the reproduction of two celebrated response functions that appear in [11]. In this direction we first provide a few examples of \(T\)-indistinguishability operators that separate points with the aim of illustrating such a technique, due to the lack of this sort of examples in the literature.