How to Make $T$-Transitive a Proximity Relation
Luis Garmendia, Jordi Recasens · IEEE Transactions on Fuzzy Systems · 2009
Three ways to approximate a proximity relation$R$(i.e., a reflexive and symmetric fuzzy relation) by a$T$-transitive one where$T$is a continuous Archimedean$t$-norm are given. The first one aggregates the transitive closure$\overline{R}$of$R$with a (maximal)$T$-transitive relation$B$contained in$R$. The second one computes the closest homotecy of$\overline{R}$or$B$to better fit their entries with the ones of$R$. The third method uses nonlinear programming techniques to obtain the best approximation with respect to the Euclidean distance for$T$the$\Lstrok$ukasiewicz or the product$t$-norm. The previous methods do not apply for the minimum$t$-norm. An algorithm to approximate a given proximity relation by a${\rm Min}$-transitive relation (a similarity) is given in the last section of the paper.