  In the early 80's, Alain Quilliot presented an approach of ordered sets and
graphs in terms of metric spaces, where instead of positive real numbers, the
values of the distance are elements of an ordered monoid equipped with an
involution. This point of view was further developed in a series of papers by
Jawhari, Misane, Pouzet, Rosenberg and Kabil. Some results are currently
published by Rosenberg, Kabil and Pouzet, Bandelt, Pouzet and Sa\"idane, Khamsi
and Pouzet. Special aspects were developed by the authors of the present paper.
A survey on generalized metric spaces is in print. In this paper, we review
briefly the salient aspects of the theory of generalized metric spaces, then we
illustrate the properties of the preservation, by operations, of sets of
relations, notably binary relations, and particularly equivalence relations.
