Studying fixed edges we start from a more general notion---p-pairs and p-points proving first that the set of all p-points of an antitone self-mapping of a complete lattice $L$ is a sublattice of $L$. In this way we obtain as a direct consequence J. Klimeš's Fixed edge Theorem and provide an easy proof of his Theorem 2. Besides, this approach sheds much more light on the treated problems. In the sequel (Theorem 2) we examine under which conditions a distinguished pair $(s,t)$ (see Notation) appearing in inconditionally complete posets is a fixed edge. In Theorem 3 the Problem in the text is solved in a special case.