An interesting problem within the theory of indistinguishability operators is how to approximate an arbitrary fuzzy subset by a similar extensional one. In this paper, we aim to solve this question and provide three methods to find extensional approximations of fuzzy subsets mu. These methods are exhaustively explained for different Archimedean t-norms, and an example is provided to illustrate them.
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 R macr of R with a (maximal) T-transitive relation B contained in R . The second one computes the closest homotecy of R macr 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 Lukasiewicz 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 min-transitive relation (a similarity) is given in the last section of the paper.