Let X be a non-empty set and 
a family of crisp subsets represented by its characteristic function 
i.e.:
If 
is
the discrete distance in the two point set 
,
then we can define a hard partition from a metrical point of view.
Remark. The relation R in 
whose characteristic function 
is defined by 
is, obviously, the equivalence relation associated to the hard partition
.
In order to extend the proposition 4.1 to the fuzzy framework, it is
important to notice the significant role played in it by the discrete distance
in the two
point set 
.
Consequently, it is reasonable to expect that, by taking different "metrics"
in the unit interval, different kinds of fuzzy clusters and therefore,
different types of associated relations, will be obtained. In this section,
it is shown that S-metrics are the suitable ones to do this job,
and one of their special types, the so-called 
-metrics,
yield as their associated relations, T-indistinguishability relations
which are the fuzzy counterpart to the classical equivalence relations.
Gathering all these ideas, enables us to give a "metrical" definition
of a "cluster coverage".
The next proposition gives us an insight into the structure of the m- clusters.
So, the equivalence relation R defined by 
,
if and only 
,
and its associated crisp partition 
are invariant with respect to the t-conorm S used to define
the s-metric m and every equivalence class 
has an associated m-cluster 
such that
therefore, a "m-cluster" 
has the graphical representation shown in fig. 1. Fig. 1
Let us turn our atention to the relation R associated to a m-cluster
coverage 
.
This relation is defined, as usual, by its characteristic function 
,
and is termed m-relation. In general, as it is shown in the next
proposition, m-relations are only reflexive and they will be symmetric
if and only if the section of the S- metric m, 
is injective.
Remark. If m is a 
-metric,
then 
is
a decreasing biyection in the unit interval therefore the relation associated
to a 
-cluster
coverage is always a T- indistinguishability relation. In this sense
-cluster coverages
are a proper subset of the set of m-cluster coverages whose associated
relation are T-transitive. As a matter of fact, it can be easily
proved that any m-cluster coverage such that 
is a decreasing biyection in the unit interval, is also a 
-cluster
coverage with 
.
Finally, even though, as it has been stated before, m-relations are not transitive in general, they lead, in a natural way, to T-indistinguishability relations. This result is a consequence of the next theorem
Given a m-partition 
,
last result allows us to build up the following schema:
where 
is the pseudometric on X defined by
This schema can be refined if we take into account the compressed version
of the representation theorem for S-metrics (Theorem 2.2) so that
can be rewritten
as
where
These ideas are summarized in the following schema
where we have a family of m-clusters given by 
and a second family of refined ones 
that leads to a S-metric in X and, therefore, to a T-indistinguishability
in it.
So, given a m-cluster coverage 
,
for any continuous and order-reversing bijection 
,
from the unit interval into itself then,
is a T-indistinguishability relation where 
.
If m is a 
-metric
then
and taking 
,
. That is,
T-transitive relations are a particular type of m-relations,
they are those associated to the above mentioned class of cluster coverages.