The initial work in this direction is found in the paper [95] by Rival, which has since been extended in various directions. We will present the best folklore version of Rival's original result in Theorem 3.2 and a modification in Theorem 3.4. Notice that chain-completeness is not needed in Theorem 3.4 and Scholium 3.5.

**Proof:**
The direction `` " follows from part 1 of
Theorem 3.1. To prove `` " let
be a comparative retraction and let be
order-preserving. Then
has a fixed point *p* and since
*r* is a comparative retraction
*f*(*p*) is related to *p*. Thus by the
Abian-Brown-Pelczar Theorem we have that *f* has a fixed point. \

Bernd.S.W.Schroder