Generating Indistinguishability Operators

One of the most interesting issues related to indistinguishability operators is their generation, which depends on the way in which the data are given and the use we want to make of them. The four most common ways are:

By calculating the

T

-transitive closure of a reflexive and symmetric fuzzy relation (a proximity or tolerance relation).

By using the Representation Theorem.

By calculating a decomposable operator from a fuzzy subset.

By obtaining a transitive opening of a proximity relation.

In many situations, data come packed as a reflexive and symmetric fuzzy matrix or relation

R

, also known as a proximity or tolerance relation. When, for coherence, transitivity is also required, the relation

R

must be replaced by a new relation

R

′ that satisfies the transitivity property. The transitive closure of

R

is the smallest of such relations among those greater than or equal to

R

. It is the most popular approximation of

R

and there are several algorithms for calculating it. In Section 2.1, the sup− 

T

product is introduced to generate it. If a lower approximation is required, transitive openings are a possibility 2.4.