least pre-fixed point

a pre-fixed point of a function $f:D\to D$, where $D$ is a poset, is some $d\in D$ such that $f(d)⊑d$.

the least pre-fixed point $\text{fix}(f)$ is the least such point. that is: $\forall d\in D$, if $d$ is a pre-fixed point then $\text{fix}(f)⊑d$.

notably, $\text{fix}(f)$ is also the least fixed point of $f$, since all fixed points are also pre-fixed points.