For a long time, the only available methods to build up fuzzy transitive relations have been the transitive closure and related methods. As it has been pointed out repeteadly, these methods carry on a number of major problems, like the requirements of both storage and computer-time and, in spite of this, no one is satisfied with the results they yield, because there is no way to control the distorsion that its application produces on the data sample, so that the transitive closure methods do not fit the desiderata of having a method to specify a similarity measure which matches with the data.
To be more concrete, in order to apply the transitive closure method to construct a similarity relation and, in general, a fuzzy T-transitive relation, a reflexive and symmetric fuzzy relation has to be used as a starting point. In others words, an index of similarity relating each couple of elements in the sample set has to be given: each two elements should be matched, in some way, and then the method is applied to obtain either a similarity or dissimilarity measure. At this point, the first arising question is the following: Does it mean that, for instance, from a single criterion, or from the matching of all elements to one given, no similarity measure can be given? The obvious negative answer can be stated by assuming that as a result of the single criterion evaluation or the matching-to-one process, a function
is given, 
representing the degree to which x fits the given conditions; with such
assumption it is easy to check that
is a pseudo-distance on X (i.e. a dissimilarity measure) such that
for any 
;
so that such pseudo-distance may be considered as truly induced by the
data. It is also quite obvious, that
is a likeness relation on X for which 
,
for any 
,
i.e. 
itself
is the measure of similarity between the element y , and any perfect
prototype.
Such a construction can be extended in order to get T-transitive
fuzzy relations for any t-norm T . If 
stands for the quasi-inverse of the t-norm T , i.e.
then it is also easy to check that
is a T-fuzzy transitive relation, such that
for any 
.
Thus, for instance,
is a the similarity relation induced by h , i.e. 
is Min-transitive. And, consequently,
is an ultrametric. On its own part,
is a probabilistic relation, i.e. transitive with respect to the t-norm
and
is a generalized pseudo-metric with respect to the t-conorm 
.
Let it be noticed that, if the t-norm is strict, like the Min and Prod
t-norms and 
,
then the induced relation is nothing but the classical equivalence relation
associated with h, that is 
otherwise. In general, such relation can be found as the 1-level set of
the induced fuzzy relation.
Summing up, the above considerations show what to do in order to obtain a similarity (or disimilarity) measure which matches to the data from a single subjetive evaluation of the degrees of similarity in the sample set. Next, suppose that several criteria or prototypes are given in the form of a family of functions
in this case the most natural procedure seems, first, to get the similarity
measure -in the form of a fuzzy transitive relation for a fixed t-norm
T - associated with each 
,
, and then
to take as the degree of the similarity of two elements, 
,
the minimum of all the degrees 
,
i.e.
which, as it is easy to check, is also a T-transitive relation. Obviously, there are other ways to combine fuzzy transitive relations which also preserve the transitive character of the relation, but the one choose here is canonical, in the sense expressed by the following representation theorem:
In other words, any reflexive, symmetric and T -transitive fuzzy relation on a set X is generated by a family of fuzzy subsets of the given set through the procedure described in this section. In (Valverde and Ovchinnikov, 1986) it has been shown that the above representation also holds for left-continuous T -norms, this fact is specially interesting when the minimal T -norm Z is considered. As it is known, this T -norm is defined by
It is worth noting that reflexive, symmetric and Z-transitive fuzzy relations are simply those reflexive and symmetric relations for which the 1-level set is a classical equivalence relation. From this standpoint, the Z-transitive relation, S , obtained by applying the procedure implied by the representation theorem starting from a strict reflexive fuzzy relation, R , is simply the greatest symmetric relation contained in R , i.e.
On the other hand, when the representation theorem is applied to build
the T - indistinguishability relation 
generated by a reflexive and symmetric fuzzy relation R , i.e. when
the functions 
are the rows of R , then 
is either 
or the greatest number among those which satisfy both
for all z in X . The point is that from the representation
theorem both the existence of such a fuzzy relation and the method to compute
it follow. Moreover, the use of the representation theorem no longuer requires
a complete fuzzy binary relation; neither reflexivity nor symmetry are
required. As it has been shown the initial data may be just one function
from the set X into 
.
It is quite clear that, due to the duality between indistinguishabilities
and S-pseudometrics, Theorem 2.1 has an immediate counterpart for
S-pseudometrics, which have an appealing interpretation in terms
of the 
pseudometrics
as introduced in definition 1.3:
In other words, any S-pseudometric on a given set X comes
from a family of fuzzy subsets of the given set and a 
metric on the unit interval. So that, in the case of ordinary (bounded)
metrics, the corresponding S- metric is 
.
That is, once a "distance" on the unit interval is fixed, this
distance is carried to the given set X through the functions - fuzzy
subsets of X- h. Let it be noticed that such procedure is
implicitely used in (Bezdek and Harris, 1978) in order to associate a likenes
relation to a fuzzy partition. As it is known, at the very begining, a
fuzzy partition of a set X was defined as a finite family of fuzzy
subsets 
of
X such that
Given a fuzzy partition 
of a set X , then the above mentioned authors propose as the likeness
relation induced by the partition, the following one
which is equivalent to fix
as a base metric on the unit interval and, then, to sum the obtained
pseudometrics on X in order to get the desidered likeness relations.
After the representation theorem, it is quite clear that no structure on
the family 
is required in order to get a likeness relation. On the other hand, the
application of the representation theorem leads to finer measures of similarity
(or disimilarity), from the point of view of its matching with the initial
data.
Finally, the above results suggest the search for an alternative definition of fuzzy partition, which should be strongly related -as in the classical case- with the underlying indistinguishability relation. In the last section of this work, some results in this line are discussed.