How to Make $T$-Transitive a Proximity Relation

Three ways to approximate a proximity relation <i>R</i> (i.e., a reflexive and symmetric fuzzy relation) by a <i>T</i> -transitive one where <i>T</i> is a continuous Archimedean <i>t</i>-norm are given. The first one aggregates the transitive closure <i>R</i> <i>macr</i> of <i>R</i> with a (maximal) <i>T</i>-transitive relation <i>B</i> contained in <i>R</i> . The second one computes the closest homotecy of <i>R</i> <i>macr</i> or <i>B</i> to better fit their entries with the ones of <i>R</i>. The third method uses nonlinear programming techniques to obtain the best approximation with respect to the Euclidean distance for <i>T</i> the Lukasiewicz or the product <i>t</i>-norm. The previous methods do not apply for the minimum <i>t</i>-norm. An algorithm to approximate a given proximity relation by a min-transitive relation (a similarity) is given in the last section of the paper.