  We introduce a new class of real-valued monotones in preordered spaces,
injective monotones. We show that the class of preorders for which they exist
lies in between the class of preorders with strict monotones and preorders with
countable multi-utilities, improving upon the known classification of
preordered spaces through real-valued monotones. We extend several well-known
results for strict monotones (Richter-Peleg functions) to injective monotones,
we provide a construction of injective monotones from countable
multi-utilities, and relate injective monotones to classic results concerning
Debreu denseness and order separability. Along the way, we connect our results
to Shannon entropy and the uncertainty preorder, obtaining new insights into
how they are related. In particular, we show how injective montones can be used
to generalize some appealing properties of Jaynes' maximum entropy principle,
which is considered a basis for statistical inference and serves as a
justification for many regularization techniques that appear throughout machine
learning and decision theory.
