  Quantifying the similarity between two mathematical structures or datasets
constitutes a particularly interesting and useful operation in several
theoretical and applied problems. Aimed at this specific objective, the Jaccard
index has been extensively used in the most diverse types of problems, also
motivating some respective generalizations. The present work addresses further
generalizations of this index, including its modification into a coincidence
index capable of accounting also for the level of relative interiority between
the two compared entities, as well as respective extensions for sets in
continuous vector spaces, the generalization to multiset addition, densities
and generic scalar fields, as well as a means to quantify the joint
interdependence between two random variables. The also interesting possibility
to take into account more than two sets has also been addressed, including the
description of an index capable of quantifying the level of chaining between
three structures. Several of the described and suggested eneralizations have
been illustrated with respect to numeric case examples. It is also posited that
these indices can play an important role while analyzing and integrating
datasets in modeling approaches and pattern recognition activities, including
as a measurement of clusters similarity or separation and as a resource for
representing and analyzing complex networks.
