# Maps $f:X\to X$ with $x \geq f(x$)

In order theory, what do we call maps $$f:X\to X \mbox{ with } \forall x\in X:x \geq f(x)$$ (with or without the demand that it is order preserving)? I'm thinking of contraction or something of the like, but I don't find much for that term. Are there fixed point theorems for such maps?

It is clear that minimal points have to be fixed. But maybe something more can be said using infinite iterations of the map?

Background: The reason why I ask is because I am studying the homotopy theory of finite topological spaces. The category of finite topological spaces is (concretely) isomorphic to the category of finite posets, and a lot of the homotopy theory has clean order theoretic characterisations. One of these is the following:

To maps $$f,g: X\to Y$$ between finite posets/toplogical spaces are homotopic iff $$f = f_0 \geq f_1 \leq f_2 \geq ... fn = g$$

In particular, if we have an order preserving 'contraction' $$f$$ like above, then we have $$f \leq \text{id}_X$$, so $$f$$ is then homotopic to the identity. Reasoning a little further this means that $$X$$ is then homotopic to $$f(X)$$ and to further iterations $$f^n(X)$$.

So clearly these maps are of interest, and Stong uses 2 special cases in his article on finite topological spaces. I want to know if these maps are also studied in the order theoretic context, which they probalby are, since it is such a basic concept.