  For a finite non-empty set $X$, let $\mathfrak{P}(X)$ denote the set of all
posets with carrier $X$, ordered by inclusion of their partial order relations.
We investigate properties of posets $P \in \mathfrak{P}(X)$ for which no lower
cover or no upper cover in $\mathfrak{P}(X)$ has the fixed point property. We
derive two conditions, one of them sufficient for that no lower cover of $P$
has the fixed point property, the other one sufficient for that no upper cover
of $P$ has the fixed point property, and we apply these results on several
types of posets.
