Let $X$ and $Y$ be sets and let $f$ be a function from $X$ to $Y$. Then the kernel of $f$ is the equivalence relation $\sim$ defined by $x \sim y$ if and only if $f(x) = f(y)$. I would like to know whether there is an accepted terminology in the ordered case, although I would be happy to use kernel also in this case.
More precisely, let $(X, \leqslant)$ and $(Y, \leqslant)$ be (pre)ordered sets and let $f:X \to Y$ be a monotone map. Is there a standard name for the preorder $\preccurlyeq$ on $X$ defined by $x \preccurlyeq y$ if and only if $f(x) \leqslant f(y)$?