  The study of complexity and optimization in decision theory involves both
partial and complete characterizations of preferences over decision spaces in
terms of real-valued monotones. With this motivation, and 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. As a result, we gain new
insight into both complexity and optimization, and clarify their interplay in
preordered spaces.
