Following the recent introduction of new classes of monotones, like injective
monotones or strict monotone multi-utilities, we present the classification of
preordered spaces in terms of both the existence and cardinality of real-valued
monotones and the cardinality of the quotient space. In particular, we take
advantage of a characterization of real-valued monotones in terms of separating
families of increasing sets in order to obtain a more complete classification
consisting of classes that are strictly different from each other