Let $f:X \to Y$ be a function.
Suppose first that $\leq_X$ is a partial order on $X$.
Is it possible to define a partial order $\leq_Y$ on $Y$, induced by $\leq_X$ (and optimal in some way), so that $f:(X, \leq_X) \to (Y, \leq_Y)$ becomes order-prerserving (and thus can be viewed as a morphism of posets)?
Now suppose that $\leq_Y$ is a partial order on $Y$, and consider the converse problem:
Find a partial order for $X$, optimal in some sense, so that $f$ is order-preserving.
Are there standard names for these "induced" partial orders?
(I'm thinking of something in the spirit of weak/strong topologies.)
More generally, if $R$ is an arbitrary asymmetric binary relation, and $\bar{R}$ its transitive closure, then $I\cup \bar{R}$ is a partial order.
Is there a name for such order (vis-à-vis $R$)?
I've searched for likely candidates such as "the partial order generated by", etc., without much success?