An Algorithm to Compute the Transitive Closure, a Transitive Approximation and a Transitive Opening of a Proximity*

Equivalence relations are important in many branches of knowledge and especially in Classification theories and Cluster Analysis since they generate a partition on the universe of discourse and permit to classify their elements and make clusters. In many cases the relation we start with is not an equivalence relation but only a reflexive and symmetric one. A very important family of fuzzy relations are T-indistinguishabilities (reflexive, symmetric and T-transitive fuzzy relations) since they generalize (fuzzify) the concepts of (crisp) equivalence relation and equality [Trillas and Valverde 1984] and are useful to represent the ideas of similarity and neighbourhood as well. Among T-indistinguishabilities, the ones which are transitive with respect to the Minimum t-norm are called similarities and are especially interesting and widely used in Taxonomy since they generate indexed hierarchical trees. How to obtain T-indistinguishabilities and especially similarities from a given proximity relation R, has become a very important task and there are many algorithms to do it. Many of them calculate the smallest T-indistinguishability