Aggregation of Indistinguishability Operators

In many situations, there can be more than one indistinguishability operator or, more generally, a

T

-transitive relation defined on a universe. Let us suppose, for example that we have a set of instances defined by some features. We can generate an indistinguishability operator or a fuzzy preorder from each feature. Also, we can have some prototypes, and again we can define a relation from each of them in our universe. In these cases we may need to aggregate the relations obtained. This is usually done by calculating their minimum (or infimum). Although this has a very clear interpretation in fuzzy logic since the infimum is used to model the universal fuzzy quantifier ∀, it often leads to undesirable results in applications because the minimum has a drastic effect. If, for example, two objects of our universe are very similar or indistinguishable for all but one indistinguishability operator but are very different for this particular operator, then the application of the minimum will give this last measure all others will be forgotten. This can be reasonable and useful if we need a perfect matching with respect to all of our relations, but this is not the case in many situations. When we need to take all relations into account in a less dramatic way, we need other ways of aggregating them. Since if

R

and

S

are

T

-transitive fuzzy relations with respect to a t-norm

T

then

T

(

R

,

S

) is also a

T

-transitive fuzzy relation, it seems at first glance that this could be a good way to aggregate them. Nevertheless, if we aggregate in this way, we obtain relations with very low values. In the case of non-strict Archimedean t-norms, it is even worse, since in many cases almost all of the values of the obtained relation are equal to zero. Therefore, other ways need to be found.