Model theory is the branch of mathematical logic that studies the
construction and classification of structures. Different methods are introduced for
building structures or families of structures, which have some feature
that interest us. Classifying a class of structures means grouping the structures into
subclasses in a useful way, and then proving that every structure in
the collection does belong in just one of the subclasses.

In this talk I present some basic notions and results of model theory for fuzzy predicate logics, focusing on the properties of equality-free languages. Strong influenced by its applications in geometry and algebra, equality was taken as a default in all classical predicate languages. The development of logics for computer science, lead logicians to study equality-free languages. For instance, the equality-free universal Horn fragment of first-order logic or, in description logics, where ACL and some of its extensions (without number restrictions) can be translated into equality-free formulae.

In a fuzzy setting the equality symbol can be taken as a binary predicate symbol not as a logical symbol, that is, the equality symbol is not necessarily present in all the languages and its interpretation is not fixed. In fuzzy predicate logics fuzzy equalities arise in a natural way, interpreted as similarities.

The most basic classification in classical model theory is given by the relation of isomorphism. We will introduce a notion equivalent to the isomorphism in equality-free languages: the relative relation, and show a characterization of this structure-preserving relation. We introduce a reduced semantics for fuzzy predicate logics. Reduced structures have the property that any two different elements of its domain can be distinguish by means of atomic formulae and parameters from the model.