What are some simple examples of an order-preserving map $f \colon X \rightarrow Y$ between preordered sets where at least one of $X$ or $Y$ is neither partially ordered nor directed by its preorder?
I ask because the definition and terminology of order-preserving maps are typically introduced for partially ordered sets, but exactly the same definition and terminology apply to directed sets. Since partial orderings and directions are both instances of preorderings, it is desirable to introduce the notion just once, for preorderings. However, it is not especially meaningful in a pedagogical sense to do so if the only simple and "good" examples are, in fact, for partial orderings and directions!
As an order preserving map is defined between orders,
there are no order preserving maps between relations
that are not orders.
f:(X,R) -> (Y,S) is a relation preserving map when
R is relation for X, S is relation for Y, f:X -> Y
and for all x,y in X, (xRy implies f(x)S(y)).
For n, in the positive integers N, define
p(n) as the number of unique primes that divide n.
nPm when p(n) <= p(m) is a preorder.
nDm when n divides m is an order.
An example of a relation preserving map from an
order to a preorder is the identity map from (N,D) to (N,P).