Take the 2-minute tour ×  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?  share|improve this question  1 Answer 1
up vote 1 down vote accepted  Withou