Suppose $f:X\to Y$ preserves $x_1\geq x_2\implies y_1\geq y_2$ but does not necessarily preserve $x_1> x_2\implies y_1> y_2$.
In other words, an $f$ satisfying this relation might, upon iteration, reduce a set to equality, or it might reduce any pair and everything in-between to equality, but cannot ever exchange the orders of any elements.
Does this have a name?
A function $f$ is order embedding when
$\forall x,y: (x \leq y \iff f(x) \leq f(y))$.
$f$ is an order isomorphism when $f$ is an order embedding bijection.
$f$ is order preserving when
$\forall x,y: (x \leq y \implies f(x) \leq f(y))$.
Exercise. If the domain of $f$ is linear and
$\forall x,y: (x < y \implies f(x) < f(y))$,
prove $f$ is order embedding.
The glibness of your question, including use of y for f(.) and lack of clarity about whether the order is linear or partial, leaves us confused about what you are asking. The use of y for f(.) should be discarded.